% 2008-07-29
% $Id: thesis.tex,v 3.0 2008/08/20 10:10:29 jn226 Exp $

\documentclass[a4paper,oneside,12pt]{report}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{amssymb}
\usepackage{mathrsfs}
\usepackage[latin1]{inputenc}
\usepackage{diagrams}
%\usepackage[notref,notcite]{showkeys}
\usepackage{a4wide}
\usepackage{setspace}


\setlength\oddsidemargin{3.6mm}
%\setlength\evensidemargin{-6.4mm}


\def\mytitle{Deformations and gluing of asymptotically cylindrical manifolds
with exceptional holonomy}
\def\myname{Lars Peter Johannes Nordström}


\title{\mytitle}
\author{\myname}

\newcommand{\ignore}[1]{}

\newcommand{\half}{{\textstyle\frac{1}{2}}}
\newcommand{\third}{{\textstyle\frac{1}{3}}}
\newcommand{\quart}{{\textstyle\frac{1}{4}}}
\newcommand{\sixth}{{\textstyle\frac{1}{6}}}
\newcommand{\fourthird}{{\textstyle\frac{4}{3}}}
\newcommand{\threequart}{{\textstyle\frac{3}{4}}}
\newcommand{\eightth}{{\textstyle\frac{1}{8}}}
\newcommand{\seveneightth}{{\textstyle\frac{7}{8}}}

\newcommand{\bbz}{\mathbb{Z}}
\newcommand{\bbr}{\mathbb{R}}
\newcommand{\bbc}{\mathbb{C}}
\newcommand{\bbo}{\mathbb{O}}
\newcommand{\bbp}{\mathbb{P}}
\newcommand{\bbrp}{\mathbb{R}^{+}}

\newcommand{\into}{\hookrightarrow}
\newcommand{\isomap}{\stackrel{\sim}{\longrightarrow}}
\newcommand{\galg}{\mathfrak{g}}
\newcommand{\gtalg}{\mathfrak{g}_{2}}
\newcommand{\halg}{\mathfrak{h}}
\newcommand{\glalg}{\mathfrak{gl}}
\newcommand{\soalg}{\mathfrak{so}}
\newcommand{\gstr}{$G$\nobreakdash-\hspace{0pt}structure}
\newcommand{\gtstr}{$G_{2}$\nobreakdash-\hspace{0pt}structure}
\newcommand{\slstr}{$SL(\bbc^{3})$-\hspace{0pt}structure}
\newcommand{\gmfd}{$G$\nobreakdash-\hspace{0pt}manifold}
\newcommand{\gtmfd}{$G_{2}$\nobreakdash-\hspace{0pt}manifold}
\newcommand{\gmetric}{$G$\nobreakdash-\hspace{0pt}metric}
\newcommand{\gtmetric}{$G_{2}$\nobreakdash-\hspace{0pt}metric}
\newcommand{\spmfd}{$Spin(7)$-manifold}
\newcommand{\spstr}{$Spin(7)$-structure}
\newcommand{\spmetric}{$Spin(7)$-metric}
\newcommand{\ltwoorth}{$L^{2}$\nobreakdash-\hspace{0pt}orthogonal}
\newcommand{\cytre}{Calabi-Yau $3$\nobreakdash-\hspace{0pt}structure}
\newcommand{\cystr}{Calabi-Yau structure}
\newcommand{\frch}{Fr\'echet}
\newcommand{\sob}[1]{L^{2}_{#1}}
\newcommand{\sobs}[2]{L^{2}_{#2}[#1]}
\newcommand{\sobd}[1]{L^{2}_{#1,\delta}}
\newcommand{\sobsd}[2]{L^{2}_{#2,\delta}[#1]}
\newcommand{\sobx}[2]{L^{2}_{#1,#2}}
\newcommand{\sobf}[1]{L^{14}_{#1}}
\newcommand{\sobfd}[1]{L^{14}_{#1,\delta}}
\newcommand{\sobp}[1]{L^{p}_{#1}}
\newcommand{\holda}[1]{C^{#1, \alpha}}
\newcommand{\holdad}[1]{C^{#1, \alpha}_{\delta}}
\newcommand{\holdax}[2]{C^{#1, \alpha}_{#2}}
%\newcommand{\holda}[1]{C^{#1,\alpha}}
%\newcommand{\holdad}[1]{C^{#1,\alpha}_{\delta}}
%\newcommand{\holdax}[2]{C^{#1,\alpha}_{#2}}
\newcommand{\wrad}[1]{W^{#1,\alpha}_{\delta}}
\newcommand{\harm}{\mathcal{H}}
\newcommand{\norm}[1]{\Vert #1 \Vert}
\newcommand{\lnorm}[1]{\Vert #1 \Vert_{L^{2}}}
\newcommand{\lxnorm}[1]{\Vert #1 \Vert_{L^{2}_{X}}}
%\newcommand{\cal}[1]{\mathcal{#1}}

\newcommand{\kcal}[1]{\mathcal{#1}_{k}}
\newcommand{\mcal}[2]{\mathcal{#1}^{#2}}
\newcommand{\kpcal}[1]{\mathcal{#1}_{k+1}}
\newcommand{\pcal}[1]{\mathcal{#1}_{\scriptscriptstyle+}}
\newcommand{\xcal}[1]{\mathcal{#1}_{X}}
\newcommand{\dcal}[1]{\mathcal{#1}_{\delta}}
\newcommand{\kdcal}[1]{\mathcal{#1}_{k,\delta}}
\newcommand{\mdcal}[2]{\mathcal{#1}^{#2}_{\delta}}
\newcommand{\mkcal}[2]{\mathcal{#1}^{#2}_{k}}
\newcommand{\mkdcal}[2]{\mathcal{#1}^{#2}_{k,\delta}}
\newcommand{\kpdcal}[1]{\mathcal{#1}_{k+1,\delta}}
\newcommand{\qcal}[1]{\mathcal{#1}_{\mathcal{Q}}}
\newcommand{\tcal}[1]{\tilde{\mathcal{#1}}}

\newcommand{\orbt}{T}
\newcommand{\slt}{K}
\newcommand{\harmgta}{\harm^{4}_{G_{2},A}}
\newcommand{\calxa}{\mathcal{X}'_{\delta}}
\newcommand{\ztwo}{\mathcal{Z}^{2}}
\newcommand{\ztre}{\mathcal{Z}^{3}}
\newcommand{\ztreq}{\mathcal{Z}^{3}_{\mathcal{Q}}}
\newcommand{\zfive}{\mathcal{Z}^{5}}
\newcommand{\cald}{\mathcal{D}}
\newcommand{\caldk}{\mathcal{D}_{k+1}}
\newcommand{\dkaq}{\mathcal{D}^{k+1}_{\mathcal{Q}}}
\newcommand{\caldkpz}{\mathcal{D}^{k+1}_{Z}}

\newcommand{\calr}{\mathcal{R}}
\newcommand{\calq}{\mathcal{Q}}
\newcommand{\calc}{\mathcal{C}}
\newcommand{\calx}{\mathcal{X}}
\newcommand{\cals}{\mathcal{S}}
\newcommand{\calu}{\mathcal{U}}
\newcommand{\calm}{\mathcal{M}}
\newcommand{\caln}{\mathcal{N}}
\newcommand{\calw}{\mathcal{W}}
\newcommand{\cali}{\mathcal{I}}
\newcommand{\calg}{\mathcal{G}}
\newcommand{\cala}{\mathcal{A}}
\newcommand{\cale}{\mathcal{E}}
\newcommand{\calb}{\mathcal{B}}
\newcommand{\call}{\mathcal{L}}
\newcommand{\calf}{\mathcal{F}}

\newcommand{\txcali}{\tilde\cali_{X}}
\newcommand{\calqa}{\mathcal{Q}_{A}}
\newcommand{\calcf}{\mathcal{C}^{4}}
\newcommand{\calcfk}{\mathcal{C}^{4}_{k}}
\newcommand{\calcfq}{\mathcal{C}^{4}_{\mathcal{Q}}}


\newcommand{\defstr}{\calm}
\newcommand{\defmet}{\calw}
\newcommand{\defstrp}{\defstr_{+}}
\newcommand{\defmetp}{\defmet_{+}}
%\newcommand{\defstrg}{\defstr_{G}}
\newcommand{\defstrg}{\defstr}
\newcommand{\defmetg}{\defmet_{G}}
%\newcommand{\defstrsp}{\defstr_{Spin(7)}}
\newcommand{\defstrsp}{\defstr}
\newcommand{\defmetsp}{\defmet_{Spin(7)}}
%\newcommand{\defstrgt}{\defstr_{G_{2}}}
\newcommand{\defstrgt}{\defstr}
\newcommand{\defmetgt}{\defmet_{G_{2}}}
%\newcommand{\defstrsu}{\defstr_{SU(3)}}
\newcommand{\defstrsu}{\defstr}
\newcommand{\defmetsu}{\defmet_{SU(3)}}
%\newcommand{\qcalg}{\mathcal{Q}_{G}}
\newcommand{\qcalg}{\calr}
\newcommand{\defmetz}{\defmet}
%\newcommand{\defmetz}{\defmet_{0}}
\newcommand{\qcalz}{\calq}
%\newcommand{\qcalz}{\mathcal{Q}_{0}}
%\newcommand{\qcalsp}{\mathcal{Q}_{Spin(7)}}
\newcommand{\qcalsp}{\calq}
\newcommand{\defstrbar}{\overline\defstr}
\newcommand{\calgbar}{\overline\calg}
\newcommand{\defstrgi}{\defstr_{\gi}}
\newcommand{\scrmkz}{\mathscr{M}^{k}_{Z}}

\newcommand{\gen}[1]{\langle#1\rangle}
\newcommand{\inner}[1]{<\!\!#1\!\!>}
\newcommand{\linner}[1]{<\!\!#1\!\!>_{L^{2}}}
\newcommand{\lxinner}[1]{<\!\!#1\!\!>_{L^{2}_{X}}}
\newcommand{\lgnorm}[1]{\Vert #1 \Vert_{L^{2}(g)}}
\newcommand{\lgpnorm}[1]{\Vert #1 \Vert_{L^{2}(g')}}
\newcommand{\lginner}[1]{<\!\!#1\!\!>_{L^{2}(g)}}
\newcommand{\lgpinner}[1]{<\!\!#1\!\!>_{L^{2}(g')}}
\newcommand{\ind}{\mathrm{ind}}
\newcommand{\tr}{\mathrm{tr}}
\newcommand{\re}{\mathrm{re}}
\newcommand{\ad}{\mathrm{ad}}
\newcommand{\vol}{\mathrm{vol}}
\newcommand{\divergence}{\mathrm{div}}
\newcommand{\im}{\mathrm{im}\,}
\newcommand{\diam}{\mathrm{diam}\;}
\newcommand{\contrax}[1]{\textstyle\frac{\partial}{\partial x^{#1}}}
\newcommand{\contrath}[1]{\textstyle\frac{\partial}{\partial \theta^{#1}}}
\newcommand{\contrat}{\textstyle\frac{\partial}{\partial t}}
\newcommand{\contral}{\textstyle\frac{\partial}{\partial L}}
\newcommand{\ddt}{\textstyle\frac{d}{dt}}
\newcommand{\dddt}{\frac{d}{dt}}
\newcommand{\dsubx}{d^{}_{\! \scriptscriptstyle X}}
\newcommand{\dcsubx}{d^{*}_{\! \scriptscriptstyle X}}
\newcommand{\sinfty}{{\textstyle \scriptscriptstyle \infty}}
\newcommand{\tildm}{\tilde M}
\newcommand{\unim}{\overline M}
\newcommand{\tv}{\tilde \varphi}
\newcommand{\tp}{\tilde \psi}
\newcommand{\inc}{e}
\newcommand{\tbtmatrix}[4]{\left( {\setlength\arraycolsep{2pt}
\begin{array}{cc} #2 & #4 \\ #1 & #3 \end{array}} \right)}
\newcommand{\tbomatrix}[2]{\left( \!{%\setlength\arraycolsep{2pt}
\begin{array}{c} #2 \\ #1 \end{array}} \!\right)}
\newcommand{\tbomatrixsmall}[2]{\displaystyle \binom{#2}{#1}}
\newcommand{\tbtmatrixl}[4]{\left( {\setlength\arraycolsep{2pt}
\begin{array}{c|c} #1 & #2 \\ \hline #3 & #4 \end{array}} \right)}
\newcommand{\hcup}{\cup}
\newcommand{\diag}{d}
\newcommand{\gi}{\pm}

\newcommand{\diagcal}[1]{\mathcal{#1}_{\diag}}
\newcommand{\ycal}[1]{\mathcal{#1}_{y}}
\newcommand{\gical}[1]{\mathcal{#1}_{\gi}}
\newcommand{\onecal}[1]{\mathcal{#1}_{+}}
\newcommand{\twocal}[1]{\mathcal{#1}_{-}}

\newcommand{\contratgi}{\textstyle\frac{\partial}{\partial t_{\gi}}}
\newcommand{\dirac}{\eth}
\newcommand{\diracp}{\dirac_{+}}
\newcommand{\diracm}{\dirac_{-}}
\newcommand{\diracx}{\dirac_{X}}
\newcommand{\diracpm}{\dirac_{\pm}}
\newcommand{\diracmp}{\dirac_{\mp}}
\newcommand{\ahat}{\hat A}
\newcommand{\cliff}[1]{r_{#1}}
\newcommand{\spinr}[1]{\sigma_{#1}}
\newcommand{\spinrp}[1]{\sigma_{#1}^{+}}
\newcommand{\spinrm}[1]{\sigma_{#1}^{-}}
\newcommand{\spinrpm}[1]{\sigma_{#1}^{\pm}}
\newcommand{\spinrmp}[1]{\sigma_{#1}^{\mp}}
\newcommand{\spinrcp}[1]{\sigma_{#1}^{\bbc+}}
\newcommand{\spinfp}{\sigma_{x}}
\newcommand{\natrep}{\tau}
\newcommand{\spinor}{S}
\newcommand{\spinorp}{\spinor^{+}}
\newcommand{\spinorm}{\spinor^{-}}
\newcommand{\spinorpm}{\spinor^{\pm}}
\newcommand{\spinormp}{\spinor^{\mp}}
\newcommand{\spinorx}{\spinor_{X}}
\newcommand{\spinorxp}{\spinorp_{X}}
\newcommand{\spinorxm}{\spinorm_{X}}
\newcommand{\spinorcp}{\spinor^{\bbc+}}
\newcommand{\harms}{\harm^{\spinor}}
\newcommand{\harmsp}{\harm^{\spinorp}}
\newcommand{\harmsm}{\harm^{\spinorm}}
\newcommand{\harmspm}{\harm^{\spinorpm}}
\newcommand{\lmg}{\Lambda_{G}}
\newcommand{\lmgt}{\Lambda_{G_{2}}}
\newcommand{\lmsp}{\Lambda_{Spin(7)}}
\newcommand{\lmsu}{\Lambda_{SU(3)}}
\newcommand{\gspace}{\Lambda_{G}T^{*}M}
\newcommand{\gtspace}{\Lambda_{G_{2}}T^{*}M}
\newcommand{\spspace}{\Lambda_{Spin(7)}T^{*}M}
\newcommand{\suspace}{\Lambda_{SU(3)}T^{*}M}
\newcommand{\spl}{u}

 
\newtheorem{thm}{Theorem}[section]
\newtheorem{prop}[thm]{Proposition}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{conj}[thm]{Conjecture}
\newtheorem{cor}[thm]{Corollary}
\theoremstyle{definition}
\newtheorem{defn}[thm]{Definition}
\theoremstyle{remark}
\newtheorem{rmk}[thm]{Remark}
\newtheorem{ex}[thm]{Example}

\renewcommand{\theenumi}{\textup{(\roman{enumi})}}
\renewcommand{\labelenumi}{\theenumi}

\frenchspacing
\widowpenalty=1000
\clubpenalty=1000

\begin{document}


\pagestyle{empty}
% --------------------------------------------------------------
%\maketitle

\bigskip % so vfill works

%\vfill
\vspace*{0cm}
 
\begin{center}
{\Huge \mytitle}

\bigskip
\vspace*{3cm}
\bigskip
\bigskip

{\Large\bf \myname}\\
{\large King's College}

\bigskip

\vfill

{This dissertation is submitted for the degree of\\
Doctor of Philosophy at the University of Cambridge\\
April 2008}

\bigskip

\end{center}

\cleardoublepage

% --------------------------------------------------------------

\onehalfspacing

\begin{center}
{\large \mytitle}\\
{Johannes Nordström}
\end{center}

\bigskip

\centerline{\bf Abstract}

In Berger's classification of Riemannian holonomy groups
there are several infinite families and two exceptional cases:
the groups $Spin(7)$ and $G_{2}$.
This thesis is mainly concerned with $7$-dimensional manifolds
with holonomy $G_{2}$.
A metric with holonomy contained in $G_{2}$ can be defined in terms of
a torsion-free \gtstr, and a \gtmfd{} is a $7$-dimensional manifold
equipped with such a structure.

There are two known constructions of compact manifolds with holonomy
exactly $G_{2}$. Joyce found examples by resolving singularities of
quotients of flat tori.
Later Kovalev found different examples by gluing pairs of exponentially
asymptotically cylindrical (EAC) \gtmfd s (not necessarily with holonomy
exactly $G_{2}$) whose cylinders match. The result of this gluing construction
can be regarded as a generalised connected sum of the EAC components, and has
a long approximately cylindrical neck region.

We consider the deformation theory of EAC \gtmfd s and show, generalising from
the compact case, that there is a smooth moduli space of torsion-free EAC
\gtstr s.
As an application we study the deformations of the gluing construction for
compact \gtmfd s, and find that the glued torsion-free \gtstr s form an open
subset of the moduli space on the compact connected sum.  For a fixed pair of
matching EAC \gtmfd s the gluing construction provides a path of torsion-free
\gtstr s on the connected sum with increasing neck length.
Intuitively this defines a boundary point for the moduli space on the connected
sum, representing a way to `pull apart' the compact \gtmfd{} into a pair of EAC
components. We use the deformation theory to make this more precise.

We then consider the problem whether compact \gtmfd s constructed by Joyce's
method can be deformed to the result of a gluing construction.
By proving a result for resolving singularities of EAC \gtmfd s we show
that some of Joyce's examples can be pulled apart in the above sense.
Some of the EAC \gtmfd s that arise this way satisfy a necessary and
sufficient topological condition for having holonomy exactly~$G_{2}$.
%that is proved.

We prove also deformation results for EAC \spmfd s, i.e. dimension $8$
manifolds with holonomy contained in $Spin(7)$. On such manifolds there is
a smooth moduli space of torsion-free EAC \spstr s.
Generalising a result of Wang for compact manifolds we show that for
EAC \gtmfd s and \spmfd s the special holonomy metrics form an open subset of
the set of Ricci-flat metrics.



% --------------------------------------------------------------

\cleardoublepage

\chapter*{Preface}
\thispagestyle{empty}  % necessary because of chapter command

$\mathbf{Originality.}$ \hspace{0pt}
This dissertation is my own work and includes nothing which is the outcome of
work done in collaboration except where specifically indicated in the text.


\bigskip

\noindent $\mathbf{Acknowledgements.}$ \hspace{0pt}
First and foremost I wish to thank Alexei Kovalev for all his help and
encouragement. I am grateful to Jonny Evans for constructive criticism of
a late draft, and to the Gates Cambridge Trust and the EPSRC for financial
support. I thank also my family and friends for being there.

\cleardoublepage


\setcounter{page}{1}
\pagenumbering{roman}
\pagestyle{plain}

\onehalfspacing

\tableofcontents


\cleardoublepage
\pagenumbering{arabic}


% --------------------------------------------------------------

\raggedbottom
\chapter{Introduction}

The holonomy group of a Riemannian manifold is the group of isometries of a
tangent space generated by parallel transport around closed paths based at a
point using the Levi-Civita connection.
In the classification of Riemannian holonomy groups due to Berger
\cite{berger55} there are two exceptional cases: $G_{2}$ and $Spin(7)$.
This thesis is concerned with manifolds of exceptional holonomy.
They can be described as Riemannian manifolds with different versions of
parallel `octonionic structures' on the tangent spaces.
We will study in particular the deformation theory of manifolds
with exceptional holonomy which are asymptotically cylindrical.

The main focus of the thesis is on the $G_{2}$ case.
We apply the deformation results to study the properties of a gluing
construction that uses asymptotically cylindrical \gtmfd s.
This leads to the idea that one can attempt to reverse the gluing construction,
and `pull apart' a compact manifold with holonomy $G_{2}$ into a pair of
asymptotically cylindrical connected summands.
We apply this idea to some of the compact \gtmfd s that Joyce constructed
by a Kummer-type method. In particular we find some examples of
asymptotically cylindrical manifolds with holonomy exactly $G_{2}$.

$G_{2}$ is a compact simple simply-connected Lie group of dimension $14$.
It is one of the exceptional simple Lie groups. It can be defined
as the automorphism group of the normed algebra of octonions~$\bbo$,
or equivalently as the group
of linear transformations of $\bbr^{7}$ that preserve the Euclidean metric
and a vector cross product. A \emph{\gtstr{}} on a manifold $M^{7}$ is
a smoothly varying identification of the tangent space at each point of $M$
with $\bbr^{7}$
equipped with a metric and vector cross product. A \gtstr{} can be defined by
a differential $3$-form $\varphi$, and is said to be \emph{torsion-free} if
$\nabla \varphi = 0$.
A metric on $M^{7}$ has holonomy contained in $G_{2}$ if and only if it can
be defined by a torsion-free \gtstr.
\gtmfd s, i.e. dimension 7 manifolds equipped with a
torsion-free \gtstr, are Ricci-flat.

Similarly, $Spin(7)$ is a compact simple simply-connected Lie group of
dimension $21$. It is the double cover of $SO(7)$. Its spinor representation
is faithful and real of rank $8$, so it can be considered as a subgroup of
$SO(8)$.
This action of $Spin(7)$ on $\bbr^{8}$ preserves a certain triple cross
product, which can be defined in terms of the algebraic structure of the
octonions.
A \emph{\spstr{}} on a manifold $M^{8}$ is an identification of the tangent
space of $M$ at each point with $\bbr^{8}$ equipped with a metric and
triple cross product.
A \spstr{} can be defined by a differential $4$-form $\psi$,
and is said to be \emph{torsion-free} if $\nabla \psi = 0$.
A metric on $M^{8}$ has holonomy contained in $Spin(7)$ if and only if it can
be defined by a torsion-free \spstr.
\spmfd s too are Ricci-flat.

$G_{2}$ is the stabiliser in $Spin(7)$ of a vector in $\bbr^{8}$.
Similarly the stabiliser in $G_{2}$ of a vector in $\bbr^{7}$ is isomorphic to
$SU(3)$. We will therefore
also be interested in $6$-dimensional manifolds with holonomy $SU(3)$.
These are Calabi-Yau $3$-folds. They have an integrable complex structure
and a Ricci-flat Kähler metric. They also have a holomorphic volume form
(i.e. a nowhere vanishing $(3,0)$-form)
so the first Chern class vanishes.

Constructing manifolds with holonomy group exactly $G_{2}$ or $Spin(7)$ is
complicated. The first local examples were found by Bryant \cite{bryant87},
and the first complete examples by Bryant and Salamon \cite{bryant89}.
The first compact examples were constructed by Joyce in \cite{joyce96-I},
by desingularising quotients of flat tori.

By contrast it is easy to find compact Calabi-Yau manifolds.
A large supply of compact Kähler manifolds with vanishing first
Chern class can be found using complex algebraic geometry,
and such manifolds have Ricci-flat Kähler metrics by Yau's
solution of the Calabi conjecture \cite{yau78}.
Taking the product of a Calabi-Yau $3$-fold with a circle gives a \gtmfd,
but a reducible one. Generally any \gtmfd{} whose holonomy is a proper
subgroup of $G_{2}$ is locally reducible, in the sense that any point has 
a neighbourhood that is a product of two Riemannian manifolds of non-zero
dimension, while a $7$-dimensional manifold with holonomy exactly $G_{2}$
is irreducible.
%The analogous statement for $Spin(7)$ holds too. False!

Another way to obtain irreducible compact \gtmfd s is by gluing a pair of
non-compact \gtmfd s which are asymptotically cylindrical.
A manifold is said to have cylindrical ends if it is homeomorphic to a cylinder
outside a compact piece. An asymptotically cylindrical manifold is a Riemannian
manifold with cylindrical ends for which the metric is asymptotic to
a product metric on the cylindrical ends.
Asymptotically cylindrical manifolds are easier to work
with than arbitrary non-compact manifolds, e.g. many analysis results for
elliptic operators on compact manifolds can be generalised to statements
about asymptotically translation-invariant elliptic operators
acting on suitable spaces of sections on an asymptotically cylindrical
manifold.
Such results are proved in e.g. \cite{lockhart85} and~\cite{mazya78}.
In some arguments it is helpful to impose a stronger condition, requiring
the manifold to be \emph{exponentially asymptotically cylindrical} (EAC).

Given a pair of EAC \gtmfd s whose cylinders match one can form a generalised
connected sum by truncating the cylinders after some large but finite length
and gluing them together. If the `neck length' is sufficiently large then the
EAC \gtstr s can be glued to form a torsion-free \gtstr{} on the connected
sum. This is a \emph{gluing construction} for compact \gtmfd s.
Kovalev \cite{kovalev03} proves an EAC
version of the Calabi conjecture to produce EAC Calabi-Yau $3$-folds.
By multiplying with circles reducible EAC \gtmfd s are obtained, which can
be glued to form irreducible compact \gtmfd s of different topological types
from those constructed by Joyce.

In chapter \ref{bgchap} we review the basic properties of the
exceptional holonomy groups, and of EAC manifolds.
Chapter \ref{cptchap} discusses the local deformation problem for compact
manifolds with exceptional holonomy, which was solved previously by
Joyce \cite{joyce96-I}.
The group of diffeomorphisms
isotopic to the identity on a compact \gtmfd{} $M^{7}$ acts on the space
of torsion-free \gtstr s by pull-backs, and the quotient is the moduli space
of torsion-free \gtstr s. We prove that the moduli space is a smooth manifold.
Since the torsion-free \gtstr s are induced by closed $3$-forms there is a
natural projection map to the de Rham cohomology $H^{3}(M)$,
and this is a local diffeomorphism. 
The argument used here is a modification of that used by
Hitchin \cite{hitchin00},
and we adapt it to deal also with deformations of compact \spmfd s and
Calabi-Yau $3$-folds (in these cases the projection to de Rham cohomology is 
an immersion rather than a local diffeomorphism).
The deformation problem for Calabi-Yau manifolds (in any dimension $\geq 3$)
was solved independently by Tian \cite{tian87} and Todorov \cite{todorov89},
using the Kodaira-Spencer theory for deformations of complex manifolds.
Goto \cite{goto04} has also studied the deformations of compact \gmfd s
when $G$ is any of the Ricci-flat holonomy groups $SU(n)$, $Sp(n)$, $Spin(7)$
or $G_{2}$.

In chapter \ref{eacgchap} we discuss the properties of EAC \gmfd s, where
$G$ is one of the exceptional holonomy groups $G_{2}$ and $Spin(7)$.
The main result here is that the moduli spaces of EAC torsion-free \gstr s
are smooth manifolds. The proofs are generalisations from 
the compact case. 
The extra difficulties are largely to do with the analysis
and understanding the boundary conditions.

For an EAC \gtmfd{} $M^{7}$ with cross-section $X^{6}$
let $\defstrp$ be the moduli space of torsion-free EAC \gtstr s on $M$.
This is the quotient of the space of EAC torsion-free \gtstr s (with any
exponential rate of decay) by the group of EAC diffeomorphisms isotopic to
the identity. The cross-section $X$ of $M$ is a compact Calabi-Yau $3$-fold.
By the results in chapter \ref{cptchap} the moduli space $\caln$ of
Calabi-Yau structures on $X$ is a smooth manifold.
There is a natural boundary map
\begin{equation}
\label{bdmapeq}
B : \defstrp \to \caln .
\end{equation}
In order to obtain an immersion of $\defstrp$ it is not enough to project
the EAC torsion-free \gtstr s to the de Rham cohomology $H^{3}(M)$, but we
must also use the image in $H^{2}(X)$ of the Kähler class of the induced
Calabi-Yau structure on $X$. Theorem \ref{maincylthm} states that
$\defstrp$ is a smooth manifold, and that the map
\[ %\pi_{\mathcal{M}} :
\pcal{M} \to H^{3}(M) \times H^{2}(X) \]
is an immersion.
The dimension of the moduli space is given by the formula
\[ \dim \pcal{M} = b^{4}(M) + \half b^{3}(X) - b^{1}(M) - 1 . \]
Moreover the boundary map $B$ is a submersion onto its image, which is a
submanifold of $\caln$ defined locally by a topological condition.

Theorem \ref{mainspcylthm} is the corresponding result for \spmfd s.
The moduli space $\defstrp$ of EAC torsion-free \spstr s on an EAC
\spmfd{} $M^{8}$ is a smooth manifold, and the projection to de Rham cohomology
\[ %\pi_{\defstr} :
\defstrp \to H^{4}(M) \]
is an immersion. In this case the cross-section $X$ is a compact \gtmfd{},
and we rely on the result that the moduli space $\caln$ of
torsion-free \gtstr s on $X$ is a smooth manifold.
Again there is a natural boundary map $B : \defstrp \to \caln$, and it is a
submersion onto a submanifold. The dimension is given by
\[ \dim \pcal{M} = b^{4}(M) - b^{4}_{+}(M) + b^{1}(X) - b^{1}(M) - 1 , \]
where $b^{4}_{+}(M)$ denotes the dimension of the positive part of the
compactly supported subspace of $H^{4}(M)$.

We also prove theorems \ref{topcondthm} and \ref{sptopcondthm}, which
give necessary and sufficient topological conditions for an asymptotically
cylindrical \gtmfd{} or \spmfd{} $M$ to have a holonomy exactly
$G_{2}$ or $Spin(7)$, respectively, and not a proper subgroup.
In the $G_{2}$ case the condition for full holonomy is that the fundamental
group $\pi_{1}(M)$ is finite and that neither $M$ nor any double cover of $M$
is homeomorphic to a cylinder.
This is a generalisation of a known result for compact \gtmfd s.
In the $Spin(7)$ case one must additionally assume that the cross-section
$\tilde X$ of the universal cover has $b^{1}(\tilde X) = 0$ to ensure that the
holonomy is exactly $Spin(7)$.

It is usually most effective to study manifolds with exceptional holonomy
$G = Spin(7)$ or $G_{2}$ in terms of the \gstr s, but in
chapter \ref{gricchap} we consider the deformations of the metrics themselves.
A result due to Wang \cite{wang91} states that small Ricci-flat
deformations of \gmetric s still have holonomy contained in $G$. We give a
careful account of this result,
and explain how it can be extended to the EAC case by using the
deformation theory of EAC \gmfd s from chapter~\ref{eacgchap}.
We also show that the moduli space $\defmetg$ of \gmetric s is smooth,
and that the moduli space of \gstr s is a locally trivial fibre bundle
over~$\defmetg$ (for manifolds with holonomy exactly $G$ the two moduli spaces
are diffeomorphic).

In chapter \ref{gluechap} we discuss how the gluing of EAC \gtmfd s behaves
under deformations.
If $M^{7}$ is a compact \gtmfd{} produced by the gluing construction
then the glued \gtstr s form an open subset of
the moduli space $\defstr$ of torsion-free \gtstr s on~$M$.
To set up the proof of this claim we first note that the data required for the
gluing construction is a pair of torsion-free EAC \gtstr s with matching
asymptotic limits together with a `neck length' parameter. The parameter
specifies how far along the infinite ends the EAC \gtmfd s are truncated before
gluing, so it controls the diameter of the glued \gtstr. 
We use the results about the boundary map (\ref{bdmapeq})
to show that there is a smooth moduli space $\calg$ of data %arguments
for the gluing construction.
%This is a fibre bundle over a moduli space $\calb$
%of matching pairs of EAC \gtstr s. The typical fibre is $\bbrp$, corresponding
%to the neck length parameter in the construction.
Then the construction gives a smooth gluing map
\begin{equation}
\label{introglueeq}
Y : \calg \to \defstr .
\end{equation}
Theorem \ref{maingluethm} states that this is a local diffeomorphism,
so in particular its image is open.

Intuitively, a path given by gluing a matching pair of EAC \gtstr s
with an increasing neck length defines a boundary point of $\defstr$,
corresponding to `pulling apart' $M$ into a pair of EAC connected-summands.
We can use the gluing map (\ref{introglueeq}) to make this precise.
Theorem \ref{gluecptfythm} states that $\defstr$
can be partially compactified by inclusion in a manifold
$\defstrbar$ with boundary, so that paths of increasing neck
length converge to boundary points of $\defstrbar$.

In chapter \ref{pullchap} we study how some of the examples of compact \gtmfd s
constructed by Joyce's method can be pulled apart in the
above sense.
We explain how Joyce's method of resolving singularities of
$G_{2}$-orbifolds can be extended to the EAC setting.
We can then pull apart some of the simpler of Joyce's examples,
by first performing a resolution of singularities to construct their
EAC components.
Some of the EAC \gtmfd s that arise this way are simply-connected,
and have holonomy exactly $G_{2}$ by the topological condition stated above.
%In particular we find some examples of irreducible EAC \gtmfd s.

Joyce's examples come with a path of torsion-free \gtstr s degenerating
to the orbifold metric the construction started from. When it can be pulled
apart we therefore find that a connected component of the moduli space
of torsion-free \gtstr s has boundary points of both orbifold and connected-sum
type.



% --------------------------------------------------------------

\cleardoublepage
%\pagestyle{headings}
\pagestyle{plain}

\chapter{Background material}
\label{bgchap}

In this chapter we review some background material.
We define the holonomy group of a Riemannian manifold, describe
the exceptional holonomy groups $G_{2}$ and $Spin(7)$, and explain how
a metric with holonomy $G = Spin(7)$, $G_{2}$ or $SU(3)$ can be defined
in terms of a torsion-free \gstr.
We explain what exponentially asymptotically cylindrical
(EAC) manifolds are, and give some results about elliptic differential
operators and Hodge theory on such manifolds.


\section{Preliminaries}

To begin with we introduce the notion of the holonomy of a Riemannian manifold,
and explain how a holonomy reduction gives rise to decompositions of spaces
of harmonic forms similar to the Hodge decomposition on a Kähler manifold.
We also summarise the elements of spin geometry.



\subsection{Riemannian holonomy}
\label{holsub}

%We define the holonomy group of a Riemannian manifold.
For a fuller discussion of holonomy see e.g. Joyce \cite[\S $2$]{joyce00}
or Besse \cite[\S 10]{besse87}.

\begin{defn}
Let $M^{n}$ be a manifold with a Riemannian metric $g$.
If $x \in M$ and $\gamma$ is a closed piecewise $C^{1}$ loop in $M$ based at
$x$ then the parallel transport around~$\gamma$
(with respect to the Levi-Civita connection of the metric)
defines an orthogonal linear map
$P_{\gamma} : T_{x}M \to T_{x}M$. The \emph{holonomy group}
$Hol(g, x) \subseteq O(T_{x}M)$ at $x$ is the group 
\mbox{$\{P_{\gamma} : \gamma \textrm{ is a closed loop based at } x\}$}.
The \emph{restricted holonomy group} $Hol^{0}(g, x)$ is the subgroup
of $Hol(g, x)$ consisting of parallel transport maps
around null-homotopic loops.
\end{defn}

If $x, y \in M^{n}$ and $\tau$ is a path from $x$ to $y$ then we can define a
group isomorphism $Hol(g, x) \to Hol(g, y)$ by 
$P_{\gamma} \mapsto P_{\tau} \circ P_{\gamma} \circ P^{-1}_{\tau}$.
Provided that $M$ is connected we can therefore identify $Hol(g, x)$ with
a subgroup of $O(n)$, independently of $x$ up to conjugacy,
and talk simply of the holonomy group of~$g$.
Another way to express this is that the holonomy is well-defined up
to isomorphism as an orthogonal group representation.
The restricted holonomy group is the identity component of the holonomy group,
and it is a compact Lie subgroup of $SO(n)$ (see \cite[\S 10.E]{besse87}).

There is a correspondence between tensors fixed by the holonomy group and
parallel tensor fields on the manifold.

\begin{prop}[{\cite[Proposition $2.5.2$]{joyce00}}]
\label{simpleholprop}
Let $M^{n}$ be a Riemannian manifold, $x \in M$
and $E$ a vector bundle on $M$ associated to $TM$.
If $s$ is a parallel section of $E$ then $s(x)$ is fixed by
$Hol(g, x)$.
Conversely if $s_{0} \in E_{x}$ is fixed by $Hol(g, x)$ then there is a
parallel section $s$ of $E$ such that $s(x) = s_{0}$.
\end{prop}

If $M$ is a simply-connected Riemannian manifold then the holonomy group must
be connected, so $Hol(M) \subseteq SO(n)$. One can ask which subgroups
of $SO(n)$ can occur as the holonomy group of a simply-connected Riemannian
manifold. The problem simplifies in two
cases. If $M$ is a symmetric space it is a homogeneous space $G/H$, and the
holonomy is the adjoint action of $H$.
If $M$ is a Riemannian product then the holonomy group
is the product of the holonomy groups of the factors. The list of possible
holonomy groups of non-symmetric irreducible Riemannian manifolds is
known as \emph{Berger's list}.

\begin{thm}
\label{bergerthm}
Let $(M,g)$ be a simply-connected non-symmetric irreducible Riemannian manifold.
Then one of the following 7 cases holds:
\begin{itemize}
\item $Hol(M) = SO(n)$ and $\dim M = n$,
\item $Hol(M) = U(n)$ and $\dim M = 2n$,
\item $Hol(M) = SU(n)$ and $\dim M = 2n$,
\item $Hol(M) = Sp(n)$ and $\dim M = 4n$,
\item $Hol(M) = Sp(n)Sp(1)$ and $\dim M = 4n$,
\item $Hol(M) = G_{2}$ and $\dim M = 7$,
\item $Hol(M) = Spin(7)$ and $\dim M = 8$.
\end{itemize}
\end{thm}

In particular we may note that the Riemannian holonomy groups occur in
enumerated infinite families, except for $G_{2}$ and $Spin(7)$.
%
For a non-symmetric irreducible manifold $M$ that is not simply-connected
we can consider its universal cover and deduce that the restricted holonomy
$Hol^{0}(M)$ must be one of the groups on the list.

If a simply-connected neighbourhood $U$ of $x \in M$ is isometric to a product
manifold then $Hol(U,x) \subseteq Hol^{0}(M,x)$ acts reducibly on $T_{x}M$.
A converse also holds.

\begin{prop}[{cf. \cite[Theorem 10.38]{besse87}}]
Let $M$ be a Riemannian manifold. If $T_{x}M$ splits as a sum of irreducible
representations $V_{1}, \ldots, V_{k}$ under the action of $Hol^{0}(M,x)$ then
there are submanifolds $N_{i} \subseteq M$ with $T_{x}N_{i} = V_{i}$ such that a
neighbourhood of $x$ in $M$ is isometric to $N_{1} \times \cdots \times N_{k}$.
\end{prop}

\noindent
If $M$ is complete and simply-connected and $Hol(M)$ acts reducibly then
there is in fact a corresponding global isometry
$M \cong N_{1} \times \cdots \times N_{k}$.
%Furthermore $Hol^{0}(M)$ has the following nice property in general.
%
%\begin{thm}[{\cite[Theorem 3.2.8]{joyce00}}]
%\label{restcptthm}
%Let $M$ be a Riemannian manifold. Then $Hol^{0}(M)$ is a compact Lie subgroup
%of $SO(n)$.
%\end{thm}
%
%\noindent

The holonomy of $M$ imposes algebraic constraints on the Riemannian curvature.
In particular

%flag: reference?

\begin{thm}
If $M$ is a Riemannian manifold and $Hol(M)$ is contained in
$SU(n)$, $Sp(n)$, $G_{2}$ or $Spin(7)$, then $M$ is Ricci-flat.
\end{thm}


\subsection{Laplacians}
\label{harmholsub}

For a Riemannian manifold with holonomy $H$ one can define a
\emph{Lichnerowicz Laplacian} on vector bundles associated to
the tangent bundle.
On differential forms this agrees with the usual Hodge Laplacian, as is
explained in \cite[\S 1I]{besse87}. It was originally noted by Chern
\cite{chern57} that this may be used to study how the Hodge Laplacian interacts
with the decomposition of $\Lambda^{m}T^{*}M$ into $H$-invariant subbundles.

Suppose that $M$ is a Riemannian manifold with holonomy group
$Hol(M) \subseteq H$. Then the frame bundle of $M$ can be reduced to
a principal $H$-bundle, i.e. $M$ has an $H$-structure.
Let $\rho : H \to GL(E)$ be a representation
of $H$, and $E_{\rho}$ the corresponding associated vector bundle.
Let $\halg_{\ad}$ be the vector bundle induced by the adjoint representation
of $H$.
$\halg_{\ad}$~can be identified with a subbundle of~$\Lambda^{2}T^{*}M$,
and because $Hol(M) \subseteq H$ the Riemannian curvature tensor $R$ is a
symmetric section of $\halg_{\ad} \otimes \halg_{\ad}$.
We use the Lie algebra representation $D\rho : \halg \to End(E)$
to define
\[ (D\rho)^{2} : \halg \otimes \halg \to End(E), \;\:
a \otimes b \mapsto D\rho(a) \circ D\rho(b) . \]
This induces a bundle map $\halg_{\ad} \otimes \halg_{\ad} \to End(E_{\rho})$.
Symmetry of $R$ implies that $(D\rho)^{2}(R)$ is self-adjoint.

\begin{defn}
\label{lichdef}
Let $M$ be a Riemannian manifold with $Hol(M) \subseteq H$ and $\rho$
a representation of $H$.
The \emph{Lichnerowicz Laplacian} on $E_{\rho}$ is the formally
self-adjoint operator
\[ \triangle_{\rho} = \nabla^{*}\nabla - 2(D\rho)^{2}(R) \; : \;
\Gamma(E_{\rho}) \to \Gamma(E_{\rho}) , \]
where $\nabla$ is the connection on $E_{\rho}$ induced by the
Levi-Civita connection on $M$.
\end{defn}

\begin{lem}
\label{lichhodgelem}
Let $M^{n}$ be a Riemannian manifold.
The Lichnerowicz Laplacian corresponding to the standard representation
of $O(n)$ on $\Lambda^{m}(\bbr^{n})^{*}$ is the usual Hodge Laplacian.
\end{lem}

\begin{proof}
See \cite[\S 1I]{besse87}.
\end{proof}

The expression of the Hodge Laplacian in terms of the Lichnerowicz
Laplacian of the standard representation of $O(n)$ is often called the
\emph{Weitzenböck formula}. The formula is particularly simple on 1-forms.

\begin{prop}
\label{weitzenprop}
If $M$ is a Riemannian manifold and $\phi$ is a 1-form then
\[ \triangle \phi = \nabla^{*}\nabla \phi + Ric(\phi) , \]
where the metric is used to interpret the Ricci curvature as a section of
$End(TM)$.
\end{prop}

\begin{proof}
See \cite[$(1.155)$]{besse87}.
\end{proof}

\begin{cor}
\label{bochnercor}
Let $M$ a Ricci-flat manifold and $\phi$ a $1$-form. If $\phi$ is parallel
then $\phi$ is harmonic. If $M$ is compact then the converse also holds.
\end{cor}

\begin{lem}[{cf. \cite[Theorem $3.5.3$]{joyce00}}]
\label{lichlem}
Let $M$ be a Riemannian manifold and suppose that $Hol(M) \subseteq H$.
Let $\phi : E \to F$ be an equivariant map of $H$-representations
$(E, \rho)$, $(F, \sigma)$. Then $\phi$ induces a bundle map
$E_{\rho} \to F_{\sigma}$, and the diagram below commutes.

\begin{diagram}[PostScript=Rikicki]
\Gamma(E_{\rho}) &\rTo^{\phi}
& \Gamma(F_{\sigma}) \\
\dTo^{\triangle_{\rho}} & & \dTo_{\triangle_{\sigma}} \\
\Gamma(E_{\rho}) &\rTo^{\phi} & \Gamma(F_{\sigma}) 
\end{diagram}
In particular, if $\rho_{1}$ and $\rho_{2}$ are $H$-representations then
$\triangle_{\rho_{1} \oplus \rho_{2}} = \triangle_{\rho_{1}}
\oplus \triangle_{\rho_{2}}$.
\end{lem}

\begin{proof}
This is obvious from the fact that the Lichnerowicz Laplacian is defined
naturally by the representations.
\end{proof}

When $\Lambda^{m}\bbr^{n}$ decomposes as a direct sum of
subrepresentations
$\Lambda^{m}\bbr^{n} = \bigoplus \Lambda^{m}_{d}\bbr^{n}$ under the action
of $H \subseteq O(n)$ there is a
corresponding $H$-invariant decomposition of the exterior product bundle
\begin{equation}
\label{extdecompeq}
\Lambda^{m}T^{*}M = \bigoplus \Lambda^{m}_{d}T^{*}M .
\end{equation}
We write $\Omega^{m}_{d}(M)$ for the space of sections of
$\Lambda^{m}_{d}T^{*}M$, the `forms of type $d$'.
We can define bundle projections
$\pi_{d} : \Lambda^{m}T^{*}M \to \Lambda^{m}_{d}T^{*}M$. These extend to
maps of sections $\pi_{d} : \Omega^{m}(M) \to \Omega^{m}_{d}(M)$, and
allow us to decompose forms into type components.

We can combine lemmas \ref{lichhodgelem} and \ref{lichlem} to see that the
Hodge Laplacian respects the type decompositions.

\begin{cor}
\label{hodgelapcor}
Let $M^{n}$ be a Riemannian manifold with $Hol(M) \subseteq H$.
If $\phi : \Lambda^{m}_{d}\bbr^{n} \to \Lambda^{k}_{e}\bbr^{n}$
is an $H$-equivariant map of $H$-subrepresentations of
$\Lambda^{m}\bbr^{n}, \Lambda^{k}\bbr^{n}$ then
\begin{diagram}[PostScript=Rikicki]
\Omega^{m}_{d}(M) &\rTo^{\phi} & \Omega^{k}_{e}(M) \\
\dTo^{\triangle} & & \dTo_{\triangle} \\
\Omega^{m}_{d}(M) &\rTo^{\phi} & \Omega^{k}_{e}(M)
\end{diagram}
commutes.
In particular, if $\Lambda^{m}_{d}T^{*}M$ is an $H$-invariant subbundle of
$\Lambda^{m}T^{*}M$ then $\triangle$ commutes with $\pi_{d}$ on
$\Omega^{m}(M)$, and maps $\Omega^{m}_{d}(M)$ to itself.
\end{cor}

\noindent
It follows that given an $H$-invariant decomposition (\ref{extdecompeq})
of $\Lambda^{m}T^{*}M$ into subbundles there is a corresponding
decomposition of the harmonic forms
\[ \harm^{m} = \bigoplus \harm^{m}_{d} . \]
If $M$ is compact then by Hodge theory the natural map $\harm^{m} \to H^{m}(M)$
is an isomorphism. If we let $H^{m}_{d}(M)$ be the image of $\harm^{m}_{d}$
then we obtain a decomposition of the de Rham cohomology
\begin{equation}
\label{hmdecompeq}
H^{m}(M) = \bigoplus H^{m}_{d}(M) .
\end{equation}


\subsection{Spinors}
\label{spinorsub}

We collect here some facts about spin representations, spinors and the
Dirac operator.
For background see e.g. Lawson and Michelsohn \cite{lawson89},
or Roe \cite{roe88} for the analysis side.

One reason why this is relevant for the study of the Ricci-flat holonomy groups
is that, as we will see, harmonic spinors on a compact scalar-flat manifold
are parallel, while the existence of parallel spinors implies a holonomy
reduction by proposition \ref{simpleholprop}. Indeed Wang \cite{wang89} gives
an explicit characterisation of torsion-free Ricci-flat holonomy structures in
terms of parallel spinors on a Riemannian manifold. Determining the dimension
of the space of harmonic spinors on a manifold in terms of topological
invariants is therefore of interest.

Another reason for us to review spinors is that $Spin(7)$ acting on its spin
representation is one of the exceptional holonomy representations.
We will also use the ellipticity of the Dirac operator in some of the technical
slice arguments when constructing moduli spaces.

For $n \geq 3$ the fundamental group of $SO(n)$ is $\bbz_{2}$. Its
universal cover is a Lie group, called the spin group $Spin(n)$.
We call the action of $Spin(n)$ on $\bbr^{n}$ that factors through $SO(n)$ the
\emph{vector representation}.
$Spin(n)$ also has a natural complex representation $\spinr{n}$ called the
\emph{spin representation}. Without going into the details of how these are
constructed we summarise some of their properties.
\begin{itemize}
\item If $n = 2k+1$ then $\spinr{n}$ is irreducible of rank $2^{k}$.
\item If $n = 2k$ then $\spinr{n}$ is reducible of rank $2^{k}$.
It has two distinct irreducible components~$\spinrpm{n}$ of rank
$2^{k-1}$ each.
\begin{equation}
\label{spinrspliteq}
\spinr{n} = \spinrp{n} \oplus \spinrm{n}
\end{equation}
\end{itemize}
$\spinr{n}$ has a natural Hermitian metric, so that it is a unitary
representation of $Spin(n)$. There is a $Spin(n)$-equivariant
map
\begin{equation*}
\bbr^{n} \otimes \spinr{n} \to \spinr{n} .
\end{equation*}
called \emph{Clifford multiplication}. For $v \in \bbr^{n}$ the multiplication
map $\cliff{v} \in End(\spinr{n})$ is skew-Hermitian. If $n$ is even then
$\cliff{v}$ maps $\spinrpm{n} \to \spinrmp{n}$ (so $\spinr{n}$ is irreducible
considered as an $\bbr^{n}$-module). Moreover the \emph{Clifford relations}
hold.
\begin{itemize}
\item $\cliff{v}\cliff{w} + \cliff{w}\cliff{v} = -2\inner{v,w}id_{\spinr{n}}$
for any $v,w \in \bbr^{n}$.
\end{itemize}

There is an obvious inclusion $Spin(n) \into Spin(n+1)$ by splitting
$\bbr^{n+1} = \bbr^{n} \oplus \bbr$, and $\spinr{n+1}$ can be restricted
to a representation of $Spin(n)$.
\begin{itemize}
\item If $n$ is even then $\spinr{n+1} \cong \spinr{n}$ as
$Spin(n)$-representations.
\item If $n$ is odd then $\spinrpm{n+1} \cong \spinr{n}$ as
$Spin(n)$-representations.
\end{itemize}
If $n \equiv m \mod 8$ then the spin representations of $Spin(n)$ and $Spin(m)$
have similar properties, a phenomenon sometimes referred to as
Bott periodicity. For example
\begin{itemize}
\item If $n \equiv 0 \mod 8$ then $\spinr{n+1}$, $\spinrpm{n}$ and
$\spinr{n-1}$ are real representations.
\end{itemize}

Let $M^{n}$ be an oriented Riemannian manifold. $M$ is said to be spin
if it has a \emph{spin structure}, i.e. a lift of the principal $SO(n)$-bundle
defined by the metric to a principal $Spin(n)$-bundle.
This condition is independent of the metric;
it is equivalent to the vanishing of the second Stiefel-Whitney
class $w_{2}(M) \in H^{2}(M, \bbz_{2})$. If $M$ is spin then the spin
structures are in bijection with $H^{1}(M, \bbz_{2})$.

Given a spin structure on $M^{n}$ one may for
each representation of $Spin(n)$ define an associated vector bundle on $M$.
In particular one can define the \emph{spinor bundle} $\spinor$ associated to
the spin representation $\spinr{n}$.
The Levi-Civita connection of $M$ induces a connection on $\spinor$
\begin{equation*}
\nabla : \Gamma(\spinor) \to \Gamma(T^{*}M \otimes \spinor) .
\end{equation*}
Composing this with the bundle map $T^{*}M \otimes \spinor \to \spinor$ given
by fibre-wise Clifford multiplication defines the \emph{Dirac operator}
\begin{equation*}
\dirac : \Gamma(\spinor) \to \Gamma(\spinor) .
\end{equation*}
This is a first-order partial differential operator, which is elliptic and
formally self-adjoint.
When $n$ is even the spinor bundle splits as
$\spinor = \spinorp \oplus \spinorm$ modelled on (\ref{spinrspliteq}),
and $\dirac$ maps $\Gamma(\spinorpm) \to \Gamma(\spinormp)$.
In other words it can be written as a sum of two formally adjoint operators
$\diracpm : \Gamma(\spinorpm) \to \Gamma(\spinormp)$.
If $n \equiv 2 \mod 4$ then $\ind \: \diracpm = 0$ because
$\spinorp \cong (\spinorm)^{*}$.
When $n \equiv 0 \mod 4$ the Atiyah-Singer index theorem (see \cite{atiyah68})
gives that $\ind \: \diracp = \ahat(M)$, where $\ahat(M)$ is a characteristic
class called the \emph{A-hat genus} of $M$. In particular it is a topological
invariant of $M$.

The second-order elliptic formally self-adjoint operator
$\dirac^{2} : \Gamma(\spinor) \to \Gamma(\spinor)$ is called the
\emph{Dirac Laplacian}. Note that if $M$ is closed then
$\dirac^{2}\psi = 0 \Leftrightarrow \dirac\psi = 0$.
There is an expression for the Dirac Laplacian similar to those discussed
in \S \ref{harmholsub}, called the \emph{Lichnerowicz formula}.
\begin{equation}
\label{licheq}
\dirac^{2} = \nabla^{*}\nabla + \quart s ,
\end{equation}
where $s$ is the scalar curvature of $M$.
If $M$ is closed and scalar-flat then an integration by parts argument like
in proposition \ref{weitzenprop} implies that
any solution of the Dirac equation $\dirac \psi = 0$ is parallel.

\begin{rmk}
\label{lichrmk}
In the notation of \S \ref{harmholsub}
\[ s\, id_{S} = 16(D\spinr{n})^{2}(R)  \]
(cf. \cite[1.142]{besse87}).
Therefore when $M$ is scalar-flat the Dirac Laplacian equals
$\nabla^{*}\nabla$, just like the Lichnerowicz Laplacian
(proposition \ref{weitzenprop}).
\end{rmk}


\section{The exceptional holonomy groups}
\label{groupsec}

We discuss the groups $G_{2}$ and $Spin(7)$ which occur as exceptional cases
in the classification Riemannian holonomy groups, and how manifolds with such
holonomy can be defined in terms of certain closed differential forms.
We also consider manifolds with holonomy $SU(3)$, which are
Calabi-Yau $3$-folds. $SU(3)$ is not an exceptional holonomy group since it
belongs to the infinite family $SU(n)$, but the fact that $SU(3)$ is the
stabiliser in $G_{2}$ of a vector means that it still has a place in the
exceptional holonomy story.

For more detail see e.g. Harvey \cite{harvey90}, Joyce \cite{joyce00} or
Salamon \cite{salamon89}.

\subsection{The group $G_{2}$}
\label{gtsub}

In this section we recall some elementary properties of the
Lie group $G_{2}$, and explain some related linear algebra
(taken from Hitchin \cite[\S 7.1]{hitchin00}).

One way to define $G_{2}$ is as the automorphism group of $\bbo$,
the normed algebra of octonions.
The automorphisms preserve the splitting $\bbo = \bbr \oplus \im \bbo$
and act trivially on $\bbr$, so can therefore be identified with a subgroup
of $GL(\bbr^{7})$.
Since the inner product on $\im \bbo$ is defined in terms of the normed
algebra structure
it is preserved by the automorphisms. It is also the case that the
automorphisms preserve the orientation, so $G_{2} \subseteq SO(7)$.

$G_{2}$ is a compact Lie subgroup of $SO(7)$. It is 14-dimensional, simple,
connected and simply-connected.

If we identify $\im \bbo$ with $\bbr^{7}$ we can define a vector product
on $\bbr^{7}$ by
\[ a \times b = \im ab . \]
The algebra structure on $\bbr \oplus \im \bbo$ can be recovered from the
vector product $\times$ and the standard inner product $g_{0}$ by
\[ (x,a)(y,b) = (xy \: - \inner{a,b}, \: xb + ya + a \times b) . \]
An equivalent definition of $G_{2}$ is therefore that it is the subgroup
of $GL(\bbr^{7})$ that preserves both $g_{0}$ and $\times$.
From $g_{0}$ and $\times$ we can define the trilinear form
\begin{equation}
\label{multtableeq}
\varphi_{0}(a,b,c) = \:\inner{a \times b, c} .
\end{equation}
In fact this is alternating, so $\varphi_{0} \in \Lambda^{3}(\bbr^{7})^{*}$.
Explicitly in standard coordinates
\begin{equation}
\label{gtformeq}
\varphi_{0} = dx^{123} + dx^{145} + dx^{167} + dx^{246}
 - dx^{257} - dx^{347} - dx^{356} .
\end{equation}
If $V$ is a dimension $7$ real vector space
let $\Lambda^{3}_{+}V^{*} \subseteq \Lambda^{3}V^{*}$ denote the
set of $3$-forms $\varphi$ that are 
equivalent to $\varphi_{0}$ under
some isomorphism $V \cong \bbr^{7}$. We will see below that the diffeomorphism
orbit $\Lambda^{3}_{+}V^{*}$ is open in $\Lambda^{3}V^{*}$, and we call such
forms \emph{stable}.

For $\varphi \in \Lambda^{3}_{+}V^{*}$ we can define an inner product
$g_{\varphi}$ and a volume form $vol_{\varphi}$ on $V$ in the following way:
For $v,w \in V$ let
\[ B_{\varphi}(v,w) = \frac{1}{6} (v \lrcorner \varphi) \wedge
(w \lrcorner \varphi) \wedge \varphi . \]
$B_{\varphi}$ is a symmetric bilinear form on $V$ with values in
$\Lambda^{7}V^{*}$.
$B_{\varphi}$ induces a linear map
$K_{\varphi} : V \to V^{*} \otimes \Lambda^{7}V^{*}$, which has a determinant
$\det K_{\varphi} \in (\Lambda^{7}V^{*})^{9}$. Since $\varphi$ is stable
$B_{\varphi}$ is non-degenerate, and we can define $vol_{\varphi}$ and
$g_{\varphi}$ by
\begin{gather*}
(vol_{\varphi})^{9} = \det K_{\varphi} , \\
g_{\varphi} \otimes vol_{\varphi} = B_{\varphi} .
\end{gather*}

For $\varphi_{0}$ we can compute that $g_{\varphi_{0}} = g_{0}$, so the metric
can be recovered from~$\varphi_{0}$, and hence so can the vector
product $\times$.
Thus the stabiliser of $\varphi_{0}$ in $GL(\bbr^{7})$ preserves $g_{0}$
and~$\times$, and must equal~$G_{2}$.
This gives yet another possible definition of $G_{2}$.
Since it is in terms of an alternating 3-form it is --
as noted by Bryant \cite{bryant87} --
a useful one for the purposes of differential geometry.

Since the action of $GL(V)$ (dim 49) on $\Lambda^{3}V^{*}$
(dim 35) has stabiliser $G_{2}$ (dim~14) at a stable form $\varphi$ it follows
by dimension-counting that $\Lambda^{3}_{+}V^{*}$ is open in
$\Lambda^{3}V^{*}$.
%
$\Lambda^{3}_{+}V^{*}$ splits into two orbits under the action of $GL_{+}(V)$.
When $V$ is oriented we can label as positive those stable $3$-forms whose
associated volume form agrees with the orientation.

\begin{defn}
If $V$ is oriented we let $\lmgt V^{*} \subseteq \Lambda^{3}V^{*}$
be the $3$-forms equivalent to $\varphi_{0}$ under some oriented linear
isomorphism $V \cong \bbr^{7}$, and call its elements
\emph{positive $3$-forms}.
\end{defn}

\begin{rmk}
This definition of `positive' agrees with that in \cite{joyce00}, while in
\cite{hitchin00} `positive' refers to the elements of $\Lambda^{3}_{+}V^{*}$.
\end{rmk}

The natural representation of $G_{2}$ on $\bbr^{7}$ is irreducible.
The splitting of the action of $G_{2}$ on $\Lambda^{m}\bbr^{7}$ into
irreducible representations is as follows:
\begin{equation}
\label{gtdecompeq}
\begin{aligned}
\Lambda^{2}\bbr^{7} &= \Lambda^{2}_{7}\bbr^{7}
\oplus \Lambda^{2}_{14}\bbr^{7} , \\
\Lambda^{3}\bbr^{7} &= \Lambda^{3}_{1}\bbr^{7} \oplus \Lambda^{3}_{7}\bbr^{7}
\oplus \Lambda^{3}_{27}\bbr^{7} , \\
\Lambda^{4}\bbr^{7} &= \Lambda^{4}_{1}\bbr^{7} \oplus \Lambda^{4}_{7}\bbr^{7}
\oplus \Lambda^{4}_{27}\bbr^{7} , \\
\Lambda^{5}\bbr^{7} &= \Lambda^{5}_{7}\bbr^{7} \oplus
\Lambda^{5}_{14}\bbr^{7} .
\end{aligned}
\end{equation}

$\Lambda^{m}_{d}\bbr^{7}$ is an irreducible subrepresentation of
$\Lambda^{m}\bbr^{7}$ of rank~$d$.
Like in \S \ref{harmholsub} we denote the projection
$\Lambda^{m}\bbr^{7} \to \Lambda^{m}_{d}\bbr^{7}$ by $\pi_{d}$.
%
Note that
$\Lambda^{2}_{7}(\bbr^{7})^{*} = \bbr^{7} \lrcorner \varphi_{0}$ and
$\Lambda^{3}_{7}(\bbr^{7})^{*} = \bbr^{7} \lrcorner (*\varphi_{0})$,
where $\lrcorner$ denotes the contraction of a vector with an alternating form.
$\Lambda^{2}_{14}(\bbr^{7})^{*}$ is isomorphic to the adjoint
representation $\gtalg$ of $G_{2}$.

Because $G_{2}$ is a simply-connected subgroup of $SO(7)$ it can also be
considered as a subgroup of $Spin(7)$, so the spin representation $\spinr{7}$
can be restricted to a representation of~$G_{2}$. As $G_{2}$-representations
\begin{equation}
\label{gtspineq}
\spinr{7} \cong \bbr \oplus \bbr^{7} .
\end{equation}

It would be possible to define a \gtstr{} in terms of the dual $4$-form
$*_{\varphi}\varphi$. Because the Hodge star depends on $\varphi$ the
relation between $\varphi$ and $*_{\varphi}\varphi$ is non-linear.
At times we will want to use a notation that does less to disguise the
non-linearity.

\begin{defn}
For $\varphi \in \lmgt V^{*} $ let $\Theta(\varphi) = *_{\varphi}\varphi$.
\end{defn}

\begin{prop}
\label{derthetaprop}
The derivative of $\Theta : \lmgt V^{*} \to \Lambda^{4}V^{*}$
at $\varphi \in \lmgt V^{*}$ is
\begin{equation}
\label{derthetaeq}
%D\Theta_{\varphi} : \Lambda^{3}(\bbr^{7})^{*} \to \Lambda^{4}(\bbr^{7})^{*}, 
D\Theta_{\varphi} : \Lambda^{3}V^{*} \to \Lambda^{4}V^{*}, 
\;\: \chi \mapsto *\fourthird\pi_{1}\chi + *\pi_{7}\chi
- *\pi_{27}\chi ,
\end{equation}
where the Hodge star and type decomposition are defined by $\varphi$.
\end{prop}

\begin{proof}
See \cite[Lemma $3.1.1$]{joyce96-I} or \cite[Lemma $20$]{hitchin00}.
\end{proof}


\subsection{$G_{2}$-manifolds}
\label{g2reviewsec}

An effective approach to \gtstr s, due to Bryant, is to define them in terms
of positive 3-forms.

\begin{defn}
A \emph{\gtstr{}} on an oriented manifold $M^{7}$ is a section
$\varphi$ of $\gtspace$.
\end{defn}

If $\varphi \in \Gamma(\gtspace)$ we can define a subbundle $Q$ of the frame
bundle of $M$ consisting of all frames $T_{p}M \to \bbr^{7}$ ($p \in M$)
which identify $\varphi_{p}$ with $\varphi_{0}$.
$Q$~is a principal $G_{2}$-bundle, so any positive $3$-form
defines a \gtstr{} on $M$ in the usual sense.
Conversely for any principal $G_{2}$-bundle $Q$ which
induces the correct orientation on $M$ we can find
a corresponding positive 3-form.

A \gtstr{} $\varphi$ induces a Riemannian metric $g_{\varphi}$ on $M$,
and hence also a Levi-Civita connection $\nabla_{\varphi}$, a Hodge star
$*_{\varphi}$ and a codifferential $d^{*}_{\varphi}$. We may drop the
subscripts if the \gtstr{} is clear from the context.

\begin{defn}
\label{torsionfreedef}
A \gtstr{} defined by a positive 3-form $\varphi$ is \emph{torsion-free}
if $\nabla_{\varphi} \varphi = 0$. 
\end{defn}

\begin{rmk}
There is a notion of the \emph{intrinsic torsion} of a $G$-structure on $M$ for
a general structure group $G$ (see e.g. \cite[\S $2.6$]{joyce00}).
A \gtstr{} has zero intrinsic torsion in this sense if and only if it
is torsion-free according to definition \ref{torsionfreedef}.
\end{rmk}

As an immediate application of proposition \ref{simpleholprop}
we have that metrics with holonomy contained in $G_{2}$ are
equivalent to torsion-free \gtstr s.

\begin{cor}
\label{g2holcor}
Let $(M^{7},g)$ be a Riemannian manifold. Then $Hol(g)$ is a subgroup
of $G_{2}$ if and only if there is a torsion-free \gtstr{}
$\varphi$ on $M$ such that $g = g_{\varphi}$.
\end{cor}

\begin{defn}
A \emph{\gtmfd} is a manifold $M^{7}$ equipped with a torsion-free
\gtstr{} $\varphi$ and the associated Riemannian metric $g_{\varphi}$.
\end{defn}

A \gtstr{} on $M$ induces decompositions of $\Lambda^{m}T^{*}M$
into subbundles modelled on~(\ref{gtdecompeq}).
Considering how $d\varphi$ and
$d^{*}_{\varphi}\varphi$ are obtained algebraically from
$\nabla_{\varphi}\varphi$ shows that

\begin{thm}[{\cite[Lemma $11.5$]{salamon89}}]
\label{graythm}
For $\varphi \in \gtspace$
\begin{enumerate}
\item \label{torfreeitem}
$\nabla_{\varphi}\varphi = 0$ if and only if $d\varphi = 0$ and
$d^{*}_{\varphi}\varphi = 0$,
\item \label{pi7item}
$\pi_{7}d\varphi = 0$ if and only if $\pi_{7}d^{*}_{\varphi}\varphi = 0$.
\end{enumerate}
\end{thm}

Because $G_{2} \subseteq Spin(7)$ a \gtstr{} on a manifold $M^{7}$
induces a natural spin structure.
The point-wise isomorphism (\ref{gtspineq}) implies that
the spinor bundle splits as
\begin{equation}
\label{gtspinoreq}
\spinor \cong \bbr \oplus T^{*}M ,
\end{equation}
where $\bbr$ is the trivial line bundle.
Conversely, Gray \cite{gray69} observed that any $7$-dimensional spin manifold
admits a \gtstr. For if $M^{7}$ is an oriented
Riemannian manifold with a spin structure then,
because the rank of the spinor bundle $\spinor$ on $M$ is greater than
the dimension of $M$, there is a global unit norm section of $\spinor$.
Since the stabiliser in $Spin(7)$ of any non-zero element
of the spin representation $\spinr{7}$ is conjugate to $G_{2}$
(see (\ref{sppointmodeleq}) below), this defines a \gtstr{} on~$M$. 

If $M^{7}$ has a torsion-free \gtstr{} then (\ref{gtspinoreq})
identifies parallel spinors with parallel elements of
$\Omega^{0}(M) \oplus \Omega^{1}(M)$. In particular any \gtmfd{} has a
non-zero parallel spinor.
Conversely, proposition \ref{simpleholprop} implies
%that any Riemannian
%$7$-manifold with a spin structure and a parallel spinor is a \gtmfd{}.

\begin{thm}
An oriented Riemannian manifold $M^{7}$ has a spin structure with a non-zero
parallel spinor if and only if $Hol(M) \subseteq G_{2}$.
\end{thm}

\noindent
Wang \cite{wang89} gives an explicit way to construct
a parallel positive $3$-form from a parallel spinor.

For compact \gtmfd s there is a known necessary and sufficient
condition for the holonomy group to be exactly $G_{2}$.

\begin{thm}[{\cite[Proposition $10.2.2$]{joyce00}}]
\label{cpttopcondthm}
Let $M^{7}$ be a compact \gtmfd. Then the holonomy
$Hol(M) = G_{2}$ if and only if the fundamental group $\pi_{1}(M)$ is finite.
\end{thm}


\subsection{The group $Spin(7)$}
\label{sp7sub}

$Spin(7)$ is the double cover of $SO(7)$. It is a compact simple
simply-connected Lie group of dimension $21$.
Its spin representation $\spinr{7}$ is a real irreducible representation of
rank $8$.
It is faithful, so $Spin(7)$ can be regarded as a subgroup of $SO(8)$.
$Spin(7)$ is conjugate to
the stabiliser in $GL(\bbr^{8})$ of
\begin{multline}
\label{sp7formeq}
\psi_{0} = dx^{1234} + dx^{1256} + dx^{1278} + dx^{1357} - dx^{1368}
-dx^{1458} - dx^{1467} \\ - dx^{2358} - dx^{2367} - dx^{2457} + dx^{2468}
+ dx^{3456} + dx^{3478} + dx^{5678} \in \Lambda^{4}(\bbr^{8})^{*} .
\end{multline}
To see that $Spin(7)$ leaves $\psi_{0}$ invariant consider first $Spin(8)$
and its three real rank $8$ representations.
As in \S \ref{spinorsub} we use $\bbr^{8}$ to denote the
vector representation, i.e. the representation that factors through the natural
action of $SO(8)$, and let $\spinrpm{8}$ be the two irreducible spin
representations. $Spin(8)$ has an exceptional triality property -- its
outer automorphism group is isomorphic to $S_{3}$, permuting $\bbr^{8}$,
$\spinrp{8}$ and $\spinrm{8}$ (cf. \cite[\S 14]{harvey90}).
Another manifestation of the triality is that the Clifford multiplication
\begin{equation*}
\bbr^{8} \times \spinrpm{8} \to \spinrmp{8}
\end{equation*}
is non-degenerate. This implies that each of $\bbr^{8}$, $\spinrp{8}$ and
$\spinrm{8}$ can be identified (but not equi\-variantly) with a division
algebra -- necessarily the octonions $\bbo$ -- in such a way that the Clifford
multiplication $\bbr^{8} \times \spinrp{8} \to \spinrm{8}$ is identified with
the multiplication map $\bbo \times \bbo \to \bbo$, $\;(x,y) \mapsto xy$
(in order to satisfy the Clifford relations
$\bbr^{8} \times \spinrm{8} \to \spinrp{8}$ must then be identified with
$\bbo \times \bbo \to \bbo, \; (x,y) \mapsto -\bar{x}y$).

Since $Spin(7)$ is a connected simply-connected subgroup of $SO(8)$ it can
be regarded as a subgroup of $Spin(8)$.
$\bbr^{8} \cong \spinr{7}$ as representations of $Spin(7)$ by our choice of
inclusion $Spin(7) \into Spin(8)$. We choose the orientation on $\bbr^{8}$
so that as $Spin(7)$-representations
\begin{subequations}
\label{spspineq}
\begin{align}
\label{spspinpeq}
\spinrp{8} & \cong \bbr \oplus \bbr^{7} , \\
\label{spspinmeq}
\spinrm{8} & \cong \bbr^{8} .
\end{align}
\end{subequations}
The double Clifford multiplication
$\bbr^{8} \times \bbr^{8} \times \spinrm{8} \to \spinrm{8}$ then gives a
$Spin(7)$-equivariant trilinear map $(\bbr^{8})^{\otimes 3} \to \bbr^{8}$.
We call its alternation the \emph{triple cross product}. Identifying
$\bbr^{8}$ with the octonions as above, the triple cross product is
given by (see Harvey \cite[p. 145]{harvey82})
\begin{equation*}
\bbo \times \bbo \times \bbo \to \bbo, \;\:
(x,y,z) \mapsto x \times y \times z = \half(x(\bar{y}z) - z(\bar{y}x)) .
\end{equation*}
The $4$-form $\psi_{0}$ can be written
in terms of the triple cross product and the inner product as
\begin{equation*}
\psi_{0}(x,y,z,w) = \; \inner{x \times y \times z, w} .
\end{equation*}
Hence $\psi_{0}$ is invariant under the action of $Spin(7)$.
To show that its stabiliser is exactly $Spin(7)$, note that the stabiliser
in $Spin(7)$ of a non-zero vector in $\bbr^{8}$ contains $G_{2}$ because of
the decomposition (\ref{gtspineq}). On the other hand, if we let $t$ be the
first $\bbr$-coordinate in $\bbr^{8} = \bbr \oplus \bbr^{7}$ then comparing
the expressions (\ref{gtformeq}) and (\ref{sp7formeq}) shows that
up to equivalence
\begin{equation}
\label{sppointmodeleq}
\psi_{0} \cong *_{\varphi_{0}}\varphi_{0} + dt \wedge \varphi_{0} .
\end{equation}
Therefore the stabiliser of $\psi_{0}$ together with a non-zero vector is
contained in (a conjugate of) $G_{2}$. Hence the stabiliser of $\psi_{0}$ is
exactly $Spin(7)$, and the stabiliser in $Spin(7)$ of a non-zero vector in
$\bbr^{8}$ is conjugate to exactly $G_{2}$.

For an oriented vector space $V$ of dimension $8$ let
$\lmsp V^{*} \subseteq \Lambda^{4}V^{*}$ be the subset of forms equivalent to
$\psi_{0}$ under some oriented linear isomorphism
$V \cong \bbr^{8}$.
Since the stabiliser of $\psi_{0}$ is $Spin(7)$ we can think of
$\psi \in \lmsp V^{*}$ as defining a \spstr, including an inner product.
Note that the $4$-form $\psi$ is self-dual with respect to
the inner product it defines.

The action of $Spin(7)$ on $\bbr^{8}$ is irreducible.
We make a note of the decomposition of $\Lambda^{*}\bbr^{8}$ into irreducible
components. We let
$\Lambda^{m}_{d}\bbr^{8}$ denote an irreducible component of rank~$d$, and
note first that $\Lambda^{4}\bbr^{8}$ splits into the self-dual and
anti-self-dual parts $\Lambda^{4}_{\pm}\bbr^{8}$.
Further (cf.~\cite[Proposition $10.5.4$]{joyce00})
\begin{equation}
\label{sp7decompeq}
\begin{aligned}
\Lambda^{4}_{+}\bbr^{8} &= \Lambda^{4}_{1}\bbr^{8} \oplus
\Lambda^{4}_{7}\bbr^{8} \oplus \Lambda^{4}_{27}\bbr^{8} , \\
\Lambda^{4}_{-}\bbr^{8} &= \Lambda^{4}_{35}\bbr^{8} , \\
\Lambda^{3}\bbr^{8} &=
\Lambda^{3}_{8}\bbr^{8} \oplus \Lambda^{3}_{48}\bbr^{8} , \\
\Lambda^{2}\bbr^{8} &=
\Lambda^{2}_{7}\bbr^{8} \oplus \Lambda^{3}_{21}\bbr^{8} .
\end{aligned}
\end{equation}
Each of the rank $8$ representations is isomorphic to $\bbr^{8}$,
while each of the rank $7$ representations is isomorphic to the vector
representation $\bbr^{7}$.

$\lmsp V^{*}$ is not an open subset of $\Lambda^{4}V^{*}$. It is easy to see
that
\begin{equation*}
T_{\psi}\lmsp V^{*} = \Lambda^{4}_{1\oplus7\oplus35}V^{*} .
\end{equation*}


\subsection{$Spin(7)$-manifolds}
\label{sp7strsub}

\begin{defn}
A \emph{\spstr} on an oriented manifold $M^{8}$ is a section of the subbundle
$\spspace \subseteq \Lambda^{4}T^{*}M$.
\end{defn}

Since $Spin(7) \subset SO(8)$ a \spstr{} $\psi$ naturally
defines a Riemannian metric $g_{\psi}$ on $M$, 
and hence also a Levi-Civita connection $\nabla_{\psi}$, a Hodge star
$*_{\psi}$ and a codifferential~$d^{*}_{\psi}$.
%Note that $\psi$ is self-dual with respect to $g_{\psi}$.

\begin{defn}
\label{sp7torsionfreedef}
A \spstr{} $\psi$ is \emph{torsion-free}
if $\nabla_{\psi} \psi = 0$. 
\end{defn}

%As an immediate application of proposition \ref{simpleholprop}
%we have that metrics with holonomy contained in $Spin(7)$ are
%equivalent to torsion-free \spstr s.

\begin{cor}
\label{sp7holcor}
Let $(M^{8},g)$ be a Riemannian manifold. Then $Hol(g)$ is a subgroup
of $Spin(7)$ if and only if there is a torsion-free \spstr{}
$\psi$ on $M$ such that $g = g_{\psi}$.
\end{cor}

\begin{defn}
A \emph{\spmfd} is a manifold $M^{8}$ equipped with a torsion-free
\spstr{} $\psi$ and the associated Riemannian metric $g_{\psi}$.
\end{defn}

\begin{thm}[{\cite[Lemma $12.4$]{salamon89}}]
\label{salamonthm}
$\psi \in \spspace$ is torsion-free iff $d\psi = 0$.
\end{thm}

Since $Spin(7) \into Spin(8)$ any manifold $M^{8}$ with a \spstr{}
has a natural spin structure,
which defines spinor bundles $\spinorpm$. The equivalence of representations
(\ref{spspinmeq}) implies that
\begin{equation}
\spinorm \cong T^{*}M .
\end{equation}
Therefore, if $M$ is compact then the dimension of the space of harmonic
negative spinors on $M$ is $b^{1}(M)$.
Also, recall that the difference between the dimensions of harmonic
positive and negative spinors is given by the topological invariant $\ahat(M)$.
Since $M$ is scalar-flat any harmonic spinor is parallel.
The dimension of the space of parallel positive spinors on a compact \spmfd{}
$M$ can therefore be written in terms of topological invariants as
\begin{equation}
\label{cptspindimeq}
\dim \harmsp = b^{1}(M) + \ahat(M) .
\end{equation}
One use for this expression is to give a topological criterion for
when a compact \spmfd{} has holonomy exactly $Spin(7)$; this is because
$Spin(7)$ fixes a unique line in its action on the positive spin representation
$\spinrp{8}$, while any proper subgroup of $Spin(7)$ that can occur as the
holonomy of a simply-connected Riemannian manifold fixes a subspace of
dimension at least $2$. We will discuss this in \S \ref{sptopcondsub}.

\begin{thm}[{cf. \cite[Theorem $10.6.1$]{joyce00}}]
\label{spcpttopcondthm}
A compact \spmfd{} $M^{8}$ has holonomy exactly $Spin(7)$ if and only if
$M$ is simply-connected and $\ahat(M) = 1$.
\end{thm}


\subsection{The group $SU(3)$}
\label{susub}

In the context of the exceptional holonomy groups $SU(3)$ appears as the
stabiliser in $G_{2}$ of a vector. We explain how $SU(3)$-structures
can be defined in terms of forms in a similar way to \gtstr s and \spstr s.

Identify $\bbr^{6}$ with $\bbc^{3}$ by using complex coordinates
$z^{1} = x^{1}+ix^{2}$, $z^{2} = x^{3}+ix^{4}$, $z^{3} = x^{5}+ix^{6}$.
Then $SU(3)$ can be considered as the stabiliser in $GL_{+}(\bbr^{6})$
of the pair
\begin{subequations}
\label{suformeq}
\begin{gather}
\label{omega01eq}
\Omega_{0} = \re (dz^{1} \wedge dz^{2} \wedge dz^{3}) , \\
\omega_{0} = -{\textstyle\frac{i}{2}}(dz^{1} \wedge d \bar z^{1} +
dz^{2} \wedge d \bar z^{2} + dz^{3} \wedge d \bar z^{3}) .
\end{gather}
\end{subequations}
These forms are obviously invariant under $SU(3)$. To see that the stabiliser
is exactly $SU(3)$ consider first the stabiliser of just $\Omega_{0}$.
The standard complex structure on $\bbc^{3}$ can be recovered from
$\Omega_{0}$ and the standard orientation in a similar way to how the standard
Euclidean metric was recovered from $\varphi_{0}$ in \S \ref{gtsub}
(see Hitchin \cite[\S 2.2]{hitchin00} for details).
The stabiliser of $\Omega_{0}$ in $GL_{+}(\bbr^{6})$ must therefore be
contained in $SL(\bbc^{3})$. In particular it preserves also the unique form
\[ \hat\Omega_{0} = \im (dz^{1} \wedge dz^{2} \wedge dz^{3}) \]
such that $\Omega_{0} + i \hat\Omega_{0}$ has type $(3,0)$ with respect to
the standard complex structure.
In turn, $SU(3)$ is the stabiliser of $\omega_{0}$ in $SL(\bbc^{3})$.

For $V$ an oriented real vector space of dimension $6$
let $\Lambda^{3}_{+}V^{*}$ be the set of $\Omega \in \Lambda^{3}V^{*}$
such that $\Omega$ is equivalent to $\Omega_{0}$
%\begin{equation}
%\label{omega02eq}
%\Omega_{0} = dx^{135} - dx^{146} - dx^{236} - dx^{245}
%\end{equation}
under some linear isomorphism $V \cong \bbr^{6}$.
%(cf. (\ref{omega01eq})).
By dimension-counting $\Lambda^{3}_{+}V^{*}$ is open in $\Lambda^{3}V^{*}$,
and following \cite{hitchin00} we call its elements $\Omega$ \emph{stable}.
From the above it follows that
any $\Omega \in \Lambda^{3}_{+}V^{*}$ naturally defines a complex
structure on $V$, and a three-form $\hat\Omega$ such that
$\Omega + i\hat\Omega$ has type $(3,0)$.
(Reversing the orientation of $V$ changes the sign of both $J$
and~$\hat \Omega$.)

Furthermore, let $\lmsu V^{*} \subset \Lambda^{3}V^{*} \oplus \Lambda^{2}V^{*}$
be the set of pairs $(\Omega, \omega)$ equivalent to
$(\Omega_{0}, \omega_{0})$ under some oriented linear isomorphism
$V \cong \bbr^{6}$.
Such a pair $(\Omega,\omega)$ naturally defines a complex structure on $V$
together with a Hermitian inner product. With respect to this inner product
$\hat\Omega = *\Omega$.

Any $(\Omega,\omega) \in \lmsu V^{*}$ must satisfy
the algebraic relations $\Omega \wedge \omega = 0$
and $\quart \Omega \wedge \hat \Omega = \sixth \omega^{3}$. Any tangent
$(\sigma,\tau)$ to $\lmsu V^{*}$ at $(\Omega,\omega)$ must therefore satisfy
the linearisation of these conditions. We can use Schur's lemma
(and the decomposition (\ref{sudecompeq}) below) to check
that the derivative of the homogenous map
$\Omega \mapsto \Omega \wedge \hat\Omega$ is $\cdot \wedge 2\hat\Omega$
(cf. \cite[p. 10]{hitchin00}), so the linearisations are
\begin{subequations}
\label{leqs}
\begin{gather}
L_{1}(\sigma,\tau) = \sigma \wedge \hat\Omega - \tau \wedge \omega^{2} \;= 0,\\
L_{2}(\sigma,\tau) = \sigma \wedge \omega + \Omega \wedge \tau = 0 .
\end{gather}
\end{subequations}
Indeed the tangent space is precisely
\begin{equation}
T_{(\Omega,\omega)}\lmsu V^{*} = \{ (\sigma,\tau) \in \Lambda^{3\oplus2}V^{*}
: L_{1}(\sigma,\tau) = L_{2}(\sigma,\tau) = 0 \} .
\end{equation}

\noindent
Comparing the expressions (\ref{gtformeq}) and (\ref{suformeq}) we observe
that if we let $t$ denote the first coordinate in
$\bbr^{7} = \bbr \oplus \bbr^{6}$ then
$\varphi_{0}$ is equivalent to
\begin{equation}
\label{supointmodeleq}
\varphi_{0} = \Omega_{0} + dt \wedge \omega_{0} .
\end{equation}
It follows that the stabiliser in $G_{2}$ of a non-zero vector in $\bbr^{7}$
is conjugate to $SU(3)$.

Let us also make a note of how the exterior powers of $\bbr^{6}$ split into
irreducible $SU(3)$-representations.
\begin{equation}
\label{sudecompeq}
\begin{aligned}
\Lambda^{2}\bbr^{6} &= \Lambda^{2}_{1}\bbr^{6}
\oplus \Lambda^{2}_{6}\bbr^{6} \oplus \Lambda^{2}_{8}\bbr^{6} , \\
\Lambda^{3}\bbr^{6} &= \Lambda^{3}_{1\oplus1}\bbr^{6}
\oplus \Lambda^{3}_{6}\bbr^{6} \oplus \Lambda^{3}_{12}\bbr^{6} , \\
\Lambda^{4}\bbr^{6} &= \Lambda^{4}_{1}\bbr^{6}
\oplus \Lambda^{4}_{6}\bbr^{6} \oplus \Lambda^{4}_{8}\bbr^{6} .
\end{aligned}
\end{equation}
Each of the subrepresentations $\Lambda^{m}_{d}\bbr^{6}$ is irreducible,
but $\Lambda^{3}_{1\oplus1}\bbr^{6}$ is trivial of rank $2$.
These decompositions are related to the decompositions of the exterior powers
of $\bbc^{3}$ by type, e.g.
$\Lambda^{2}_{1\oplus8}(\bbr^{6})^{*} \otimes \bbc =
\Lambda^{1,1}(\bbc^{3})^{*}$,
%(shorthand for)
%$\Lambda^{2}_{1}T^{*}X \oplus \Lambda^{2}_{8}T^{*}X$
while
$\Lambda^{2}_{6}(\bbr^{6})^{*} \otimes \bbc =
(\Lambda^{2,0} \oplus \Lambda^{0,2})(\bbc^{3})^{*}$.


\subsection{Calabi-Yau manifolds}
\label{cydefsub}

There are many commonly used definitions of what
it means for a manifold $X$ of real dimension $2n$ to be Calabi-Yau.
The following alternative definitions are all
non-equivalent:
% (cf. \cite[p. ??]{joyce00}):

\begin{itemize}
\item $X$ is a Riemannian manifold with $Hol(X) \subseteq SU(n)$.
\item $X$ is a Riemannian manifold with $Hol(X) = SU(n)$.
\item $X$ is a Ricci-flat Kähler manifold.
\item $X$ is a Kähler manifold with $c_{1}(X) = 0$.
\end{itemize}

To see how they are related first recall that $Hol(X) \subseteq U(n)$ if and
only if $X$ is Kähler, i.e. when the metric is Hermitian with respect to a
parallel complex structure.
Equivalently the Kähler form $\omega = g(J\cdot, \cdot)$ is closed.
If $Hol(X) \subseteq SU(n)$ then there is additionally a parallel
$(n,0)$-form $\phi$.
It is elementary to show that there is a linear relation between
the Ricci curvature and the curvature of the canonical bundle
$\Lambda^{n,0}T^{*}X$ on a Kähler manifold~$X$. Therefore if
there is a parallel section of $\Lambda^{n,0}T^{*}X$ then $X$ is Ricci-flat.
The linear relation implies also that if $X$ is Ricci-flat Kähler then
the restricted holonomy $Hol^{0}(X) \subseteq SU(n)$, but there need not be
a \emph{global} $(n,0)$-form.

A parallel $(n,0)$-form on a Kähler manifold $X$ is holomorphic, so if there
is one then the canonical bundle $\Lambda^{n,0}T^{*}X$ is trivial, and 
the first Chern class $c_{1}(X) \in H^{2}(X, \bbz)$ vanishes.
By Chern-Weil theory $c_{1}(X) \in H^{2}(X, \bbr)$ is represented
by a multiple of the curvature of the canonical bundle, so for
$c_{1}(X)$ to vanish in the de Rham cohomology it suffices that $X$ is
Ricci-flat Kähler.
Yau \cite{yau78} proved the Calabi conjecture, which provides a converse.

\begin{thm}
\label{cythm}
Let $X$ be a compact Kähler manifold. If $c_{1}(X) \in H^{2}(X, \bbr)$ vanishes
then for any Kähler form
$\omega$ on $X$ there is a unique Kähler form $\omega'$ in the cohomology
class of $\omega$ such that $\omega'$ defines a Ricci-flat metric.
\end{thm}

A special feature of the case $n=3$ -- which is the case of interest for
us -- is that the almost complex structure $J$
on a Calabi-Yau manifold can be recovered from the real part $\Omega$ of the
holomorphic $3$-form together with the orientation.
This is a consequence of the point-wise linear algebra discussed in
\S \ref{susub}.
We therefore find it convenient to define a \cystr{} on a manifold $X$ of real
dimension $6$ in terms of a pair of forms $(\Omega,\omega) \in
\Omega^{3}(X) \oplus \Omega^{2}(X)$.

\begin{defn}
\label{cy3def}
An \emph{$SU(3)$-structure} on a manifold $X^{6}$
is a section $(\Omega,\omega)$ of $\lmsu T^{*}X$.

$(\Omega, \omega)$ is said to be a \emph{\cystr} if
$\nabla\Omega = 0, \nabla\omega = 0$ with respect to the metric induced by
$(\Omega,\omega)$. $X$ equipped with the structure $(\Omega,\omega)$ and
the associated Riemannian metric is called a \emph{Calabi-Yau 3-fold}.
\end{defn}

\begin{lem}
\label{cy3charlem}
Let $(\Omega,\omega)$ be an $SU(3)$-structure on a manifold $X^{6}$.
Then $(\Omega,\omega)$ is a \cystr{} if and only if  
$d\Omega = d \hat \Omega = 0$, $d\omega = 0$.
\end{lem}

\begin{proof}[Sketch proof.]
$d(\Omega + i \hat \Omega) = 0$ ensures that the almost complex structure
defined by $\Omega$ is integrable.
$\omega$ defines a Kähler metric, so $J$ and $\omega$ are parallel.
$\Omega + i \hat \Omega$ is a holomorphic section of a Hermitian line bundle
and has constant norm, so must be parallel with respect to the Chern
connection.
\end{proof}

\noindent
One can also consider the structure defined by the $3$-form $\Omega$ on its own.

\begin{defn}
An \emph{\slstr} on an oriented manifold $X^{6}$ is a section $\Omega$
of~$\Lambda^{3}_{+}T^{*}X$.
$\Omega$ is \emph{torsion-free} if $d\Omega = d\hat\Omega = 0$.
\end{defn}

If $\Omega$ is torsion-free then so is the almost complex structure $J$ it
defines, and $\Omega + i\hat\Omega$ is a global holomorphic $(3,0)$-form.
A torsion-free \slstr{} is therefore equivalent to a complex structure
with trivial canonical bundle, together with a choice of trivialisation.


\subsection{Cylindrical $G$-structures}

Let $X^{6}$ be a compact manifold, and denote by $t$ the $\bbr$-coordinate
on the cylinder $X \times \bbr$.

\begin{defn}
\label{exactcylg2def}
A \gtstr{} $\varphi$ on $X \times \bbr$ is \emph{cylindrical} if it is
translation-invariant and the associated metric is a product metric
$g_{\varphi} = g_{X} + dt^{2}$.
\end{defn}

We find that torsion-free
cylindrical \gtstr s on $X \times \bbr$ correspond to Calabi-Yau structures
on $X$.
Looking at the point-wise model (\ref{supointmodeleq}) we see that
$(\Omega, \omega)$ is an $SU(3)$-structure on $X$ with metric $g_{X}$
if and only if the translation-invariant
\gtstr{} $\varphi = \Omega + dt \wedge \omega$ on $X \times \bbr$
defines the product metric $g_{X} + dt^{2}$.
$Hol(g_{X} + dt^{2}) \subseteq G_{2}$ if and only if
$Hol(g_{X}) \subseteq SU(3)$, so
$(\Omega,\omega)$ is torsion-free if and only if $\Omega + dt \wedge \omega$ is.
Hence

\begin{prop}
\label{cylg2prop}
Let $X$ be a manifold of dimension 6.
$(\Omega,\omega)$ is a \cystr{} on $X$ if and only if
$\Omega +dt \wedge \omega$ is a torsion-free cylindrical \gtstr{} on
$X \times \bbr$.
\end{prop}

\begin{rmk}
\label{cyldualrmk}
If $\varphi = \Omega + dt \wedge \omega$ is a cylindrical \gtstr{} then
\begin{equation*}
*_{\varphi}\varphi = \half \omega^{2} - dt \wedge \hat\Omega .
\end{equation*}
\end{rmk}

%That $\contrat$ is orthogonal
%to $TX$ corresponds to the condition that $\Omega \wedge \omega = 0$, while
%$\norm{\contrat} = 1$ corresponds to $\Omega \wedge \hat\Omega = \omega^{3}$.

%By rescaling it follows that $\Omega + dt \wedge \omega$ is a torsion-free
%cylindrical \gtstr{} if and only if $(\lambda\Omega, \lambda\omega)$ is
%a Calabi-Yau structure for some $\lambda \in \bbrp$.

\noindent
Similarly contemplation of the expression (\ref{sppointmodeleq}) leads to

\begin{prop}
\label{cylspprop}
Let $X$ be a manifold of dimension 7.
$\varphi$ is a torsion-free \gtstr{} on $X$ if and only if
$*_{\varphi}\varphi + dt \wedge \varphi$ is a torsion-free cylindrical
\spstr{} on $X \times \bbr$.
\end{prop}



\section{Asymptotically cylindrical manifolds}
\label{cylsec}

In this section we define what an asymptotically cylindrical manifold is
and collect some results about analysis and Hodge theory on such manifolds.


\subsection{Manifolds with cylindrical ends}

We define manifolds with cylindrical ends and
exponentially asymptotically cylindrical (EAC) metrics.
% and \gstr s.
%and their long exact sequence
%for cohomology relative to the boundary.

\begin{defn}
\label{cylenddef}
A manifold $M$ is said to have \emph{cylindrical ends} if it is a 
union of two pieces $M_{0}$ and $M_{\infty}$ with common boundary $X$, where
$M_{0}$ is compact, and $M_{\infty}$ is identified with $X \times \bbrp$
by a diffeomorphism (identifying $\partial M_{\infty}$ with $X \times \{0\}$).
$X$~is called the \emph{cross-section} of~$M$.

A \emph{cylindrical coordinate} on $M$ is a smooth function $t : M \to \bbr$
which is equal to the $\bbrp$-coordinate on $M_{\infty}$ and is negative
in the interior of~$M_{0}$.
\end{defn}

If $M$ has cylindrical ends it can be compactified by
inclusion in $\overline M = M_{0} \cup (X \times [0,\infty])$, i.e.
by `adding a copy of $X$ at infinity'.
Conversely the interior of any compact manifold with boundary can be
considered as a manifold with cylindrical ends
by the collar neighbourhood theorem.
The choice of diffeomorphism $M_{\infty} \to X \times \bbrp$ can
be regarded to determine a cylindrical-end structure on~$M$.

Cylindrical ends allow us to define a notion of asymptotic
translation-invariance.

\begin{defn}
A tensor field or differential operator
on $X \times \bbr$ is called
\emph{translation-invariant} if it is invariant under the obvious
$\bbr$-action on $X \times \bbr$.
\end{defn}

\begin{defn}
Let $M$ be a manifold with cylindrical ends. Call a smooth function
$\rho : M \to \bbr$ a \emph{cut-off function for the cylinder}
if it is $0$ on the compact piece $M_{0}$ and $1$ outside a compact subset
of $M$.
\end{defn}

If $s_{\infty}$ is a section of a vector bundle associated to the tangent
bundle on the cylinder $X \times \bbr$ and $\rho$ is a cut-off function for
the cylinder of $M$ then $\rho s_{\infty}$ can be considered
to be a section of the corresponding vector bundle over $M$.

\begin{defn}
\label{asytensordef}
Let $M$ be a manifold with cylindrical ends and cross-section $X$.
Pick an arbitrary product metric $g_{X} + dt^{2}$ on $X \times \bbr$, and
a cut-off function $\rho$ for the cylinder.
A section $s$ of a vector bundle associated to $TM$ is said to
be \emph{decaying} if $\norm{\nabla^{k}s} \to 0$ uniformly on $X$
as $t \to \infty$ for all $k \geq 0$.
$s$ is said to be \emph{asymptotic} to a translation-invariant section
$s_{\infty}$ of the corresponding bundle on $X \times \bbr$ if
$s - \rho s_{\infty}$ decays.

\label{expasytensordef}
Similarly $s$ is said to be \emph{exponentially decaying} with
rate $\delta > 0$ if  $e^{\delta t} \norm{\nabla^{k}s}$ is bounded
on $M_{\infty}$ for all $k \geq 0$, and \emph{exponentially asymptotic}
to a translation-invariant section $s_{\infty}$ if
$s - \rho s_{\infty}$ decays exponentially.
Denote by $C^{\infty}_{\delta}(E)$ the space of sections of $E$
which decay exponentially with rate~$\delta$.
\end{defn}

%We topologise $C^{\infty}_{\delta}(E)$ by requiring the linear isomorphism
%$C^{\infty}_{\delta}(E) \to C^{\infty}(E), s \mapsto e^{\delta t}s$
%to be a homeomorphism.

To develop the analysis for elliptic operators on a manifold $M$ with
cylindrical ends in subsection \ref{cylanalysissub} it will suffice to
equip $M$ with an asymptotically translation-invariant metric, but for the
Hodge theory developed in subsection \ref{hodgesub} we will need to assume
that $M$ is EAC (exponentially asymptotically cylindrical).

\begin{defn}
A metric $g$ on a manifold $M$ with cylindrical ends is said to be \emph{EAC}
%\emph{[exponentially] asymptotically cylindrical}
if it is exponentially asymptotic to a product metric
$g_{X} + dt^{2}$ on $X \times \bbrp$.
%An [exponentially] cylindrical
An EAC manifold is a manifold with cylindrical ends
equipped with an EAC metric.
%[exponentially] cylindrical metric.
\end{defn}

\begin{defn}
\label{cyldiffdef}
Let $M$ be a manifold with cylindrical ends and cross-section $X$.
A diffeomorphism $\Psi_{\infty}$ of the cylinder $X \times \bbr$ is said to be
cylindrical if it is of the form
\[ \Psi_{\infty}(x,t) = (\Xi(x), t + h) , \]
where $\Xi$ is a diffeomorphism of $X$ and $h \in \bbr$.
%
A diffeomorphism $\Psi$ of $M$ is said to be \emph{EAC} with rate $\delta > 0$
%\emph{[exponentially] asymptotically cylindrical}
if there is a cylindrical diffeomorphism
$\Psi_{\infty}$ of $X \times \bbr$, a real $T > 0$ and an exponentially
decaying vector field $V$ on $M$ such that on $X \times (T, \infty)$
\[ \Psi = (\exp V) \circ \Psi_{\infty} . \]
\end{defn}

%asymptotic expansions: see mod2U.tex

We can think of two different choices of diffeomorphism
$M_{\infty} \to X \times \bbrp$ in definition \ref{cylenddef} as defining
`$\delta$-equivalent' cylindrical-end structures
if they differ by a rate $\delta$ EAC diffeomorphism -- then they define
equivalent notions of exponential translation-invariance.
Nevertheless it is convenient to fix the identification
$M_{\infty} \to X \times \bbrp$.

Theorem \ref{palaisthm} states that any isometry of smooth Riemannian
manifolds is smooth. In a sense this means that the smooth structure
can be recovered from the metric (cf. Palais \cite{palais57}).
We wish to generalise this, and show that isometries of EAC metrics are EAC.
%(in the sense of definition \ref{cyldiffdef}).

\begin{prop}
\label{eacisomprop}
Any isometry of EAC (rate $\delta > 0$) manifolds is $C^{\infty}$
and EAC with rate~$\delta$.
\end{prop}

\begin{proof}[Sketch proof.]
By \cite[Theorem 8]{myers39} the isometries are $C^{\infty}$,
so we just need to prove that they are also EAC.

Let $M$ be a manifold with a Riemannian metric $g$. We need to show that
if for $i = 1$,~$2$, $M_{i,\infty} \subseteq M$ have compact complements,
and $\Psi_{i} : M_{i,\infty} \to X_{i} \times \bbrp$ are diffeomorphisms
defining cylindrical-end structures with respect to which $g$ is EAC, then
$\Psi_{1} \circ \Psi_{2}^{-1}$ is EAC.

Consider the space $R$ of half-lines in $M$, i.e. equivalence classes of unit
speed globally distance-minimising geodesic rays $\gamma : [0,\infty) \to M$,
where two rays are equivalent if one is a subset of the other.
We can define a distance function on $R$ by
\begin{equation}
d([\gamma],[\sigma]) = \lim_{u \to \infty} \inf_{v} d(\gamma(u), \sigma(v)) .
\end{equation}
$g$ is pushed forward to an EAC metric on $X_{i} \times \bbrp$ by $\Psi_{i}$.
It is straight-forward to solve the geodesic equation in local coordinates
to show that for each $x \in X_{i}$ there is a unique half-line $[\gamma]$
such that the $X$-component of $\gamma(u)$ approaches $x$ as $u \to \infty$.
$\Psi_{i}$ therefore induce isometries $\Xi_{i} : R \to X_{i}$.
Then $\Xi_{1} \circ \Xi_{2}^{-1}$ is smooth by \cite[Theorem 8]{myers39}.
If $t_{i}$ is the $\bbrp$-coordinate on $X_{i} \times \bbrp$ then
$grad (t_{i}) - \frac{d\gamma}{du}$ decays exponentially as $u \to \infty$ for
each half-line $[\gamma]$, so $\Psi_{1} \circ \Psi_{2}^{-1}$
is exponentially asymptotic to
$(x,t) \mapsto (\Xi_{1} \circ \Xi_{2}^{-1}(x), t+h)$ for some $h \in \bbr$.
\end{proof}

It is a classical result that the isometry group of a compact Riemannian
manifold is compact (cf. Myers and Steenrod \cite{myers39}).
While we will not use it in the main arguments, it is worth noting the
following generalisation.

\begin{lem}[see e.g. {\cite[Lemma $3.6$]{kovalev05}}]
\label{eaccptstablem}
The isometry group of an EAC manifold with a single end is compact.
\end{lem}


\subsection{Analysis on manifolds with cylindrical ends}
\label{cylanalysissub}

Here we collect some results that we will need about analysis on manifolds
$M$ with cylindrical ends. The results are
mainly taken from Lockhart and McOwen \cite{lockhart85}
and Maz'ya and Plamenevski\u{\i} \cite{mazya78}.

%mod3O.tex : exactly t-i metrics for definition of sob norm

\begin{defn}
Let $M^{n}$ be a manifold with cylindrical ends. If $g$ is an asymptotically
translation-invariant metric $g$ on $M$, $E$ is a vector bundle
associated to~$TM$, $\delta$ is a real number and $s$ is a section of $E$,
we define the \emph{Hölder norm with weight $\delta$}
(or $\holdad{k}$-norm) of $s$ in terms of the Hölder norm by
\begin{equation}
\norm{s}_{\holdad{k}(g)} = \norm{e^{\delta t}s}_{\holda{k}(g)} ,
\end{equation}
where $t$ is the cylindrical coordinate on $M$.
Denote the space of sections of $E$ with finite $\holdad{k}$-norm
by $\holdad{k}(E)$.
%
%Also, let the $C^{\infty}_{\delta}(E)$ be the space of sections of $E$
%which decay exponentially with rate $\delta$, with the topology pulled
%back by the map
%$C^{\infty}_{\delta}(E) \to C^{\infty}(E), s \mapsto e^{\delta t}s$.
\end{defn}

Up to Lipschitz equivalence the weighted norms are independent of the choice
of asymptotically translation-invariant metric $g$,
and of the choice of $t$ on the compact piece $M_{0}$. In particular,
the topological vector spaces $\holdad{k}(E)$ are independent of these choices.
\[ \bigcap_{k > 0} \holdad{k}(E) = C^{\infty}_{\delta}(E) , \]
and the natural choice of topology on $C^{\infty}_{\delta}(E)$ is the inverse
limit topology.

\begin{rmk}
There is a similar definition of weighted Sobolev norms $\sobd{k}$.
Aubin \cite[Theorem 2.21]{aubin82} shows that
Sobolev embedding holds on any complete manifold which has bounded curvature
and injectivity radius bounded away from 0, conditions that are clearly
satisfied for a manifold with cylindrical ends and an
asymptotically translation-invariant metric.
From that can deduce that on a manifold $M^{n}$ with cylindrical ends
$\sobd{l}$ embeds continuously in $\holdad{k}$ for any
$l > k + \alpha + \frac{n}{2}$.
Because of this Sobolev embedding result one could use weighted Hölder norms
and weighted Sobolev norms interchangeably in many arguments.
However $\bigcap_{k} \holda{k} = C^{\infty}$, while the same is not true for
Sobolev spaces on non-compact manifolds.
We choose to use Hölder spaces for convenience
in the regularity arguments of \S \ref{cylregsub}.
\end{rmk}

\begin{defn}
\label{asyopdef}
Let $M$ be a manifold with cylindrical ends, $E$ and $F$ vector bundles
associated to $TM$, $A$ a smooth linear differential operator
$\Gamma(E) \to \Gamma(F)$ of order $m$, and $A_{\infty}$ a
translation-invariant operator
on the corresponding bundles over $X \times \bbr$.

The restriction of $A$ to the cylindrical end $M_{\infty}$ can be written in
terms of the Levi-Civita connection of an arbitrary product metric
on $X \times \bbr$ as
$A = \sum_{i = 0}^{m}a_{i}\nabla^{i}$, with coefficients
$a_{i} \in C^{\infty}((TM)^{i} \otimes E^{*} \otimes F)$.
$A$ is said to be \emph{\{exponentially\} asymptotic} to $A_{\infty}$
if the coefficients in the expression for $A$ are \{exponentially\} asymptotic
to those of $A_{\infty}$.
\end{defn}

One of the main results of \cite{lockhart85} is Theorem 6.2, which states

\begin{thm}
\label{ellfredthm}
Let $M$ be Riemannian manifold with cylindrical ends and asymptotically
translation-invariant metric. Let $A : \Gamma(E) \to \Gamma(F)$ be
a smooth linear elliptic differential operator of order $r$ asymptotic to a
translation-invariant operator $A_{\infty}$.

For $\lambda \in \bbc$ let $S_{A}(\lambda)$ be the space of solutions $s$ to
$A_{\infty}s = 0$ on the cylinder $X \times \bbr$ of the form
$s = e^{i\lambda t}p(x,t)$, where $p$ is a section of $E$ over $X \times \bbr$
that is smooth in $x$ and polynomial in $t$.
Let $d_{A}(\lambda) = \dim_{\bbc} S_{A}(\lambda)$, and
\begin{equation}
\label{badefeq}
\mathcal{B}_{A} = \{\im \lambda : \lambda \in \bbc, d_{A}(\lambda) \not= 0\} .
\end{equation}

Then $A$ extends to a bounded map $\holdad{k+r}(E) \to \holdad{k}(F)$ for
all $k$ and $\delta$, which is Fredholm if $\delta \not \in \mathcal{B}_{A}$.
In the latter case let
$i_{\delta}(A) = \ind \left(A: \holdad{k+r}(E) \to \holdad{k}(F) \right)$.
If $\delta_{1}, \delta_{2} \not \in \mathcal{B}_{A}$ and
$\delta_{1} \leq \delta_{2}$ then
\[ i_{\delta_{1}}(A) - i_{\delta_{2}}(A) =
\sum_{\delta_{1} < \im \lambda < \delta_{2}} d_{A}(\lambda) . \]
\end{thm}

\begin{rmk}
Strictly speaking, the results in \cite{lockhart85} use weighted Sobolev
spaces rather than weighted Hölder spaces, but the arguments are the same
in both cases. See also \cite[Theorem $6.4$]{mazya78}.
\end{rmk}

\begin{prop}[{\cite[Lemma 7.1]{lockhart85}}]
\label{miniasyexpprop}
If the interval $[\delta_{1}, \delta_{2}]$ contains no elements of $\calb_{A}$
then the kernels of $A$ in $\holdax{k}{\delta_{1}}(E)$ and 
$\holdax{k}{\delta_{2}}(E)$ are equal.
\end{prop}

We can use integration by parts arguments, provided that the rates of decay
are fast enough to ensure the integrals converge.

\begin{lem}
\label{ibplem}
Let $M$ be a Riemannian manifold with cylindrical ends and
asymptotically
translation-invariant metric.
Let $A : \Gamma(E) \to \Gamma(F)$ be a smooth linear asymptotically
translation-invariant differential operator of order $r$ with formal adjoint
$A^{*} : \Gamma(F) \to \Gamma(E)$.
Suppose that $\alpha \in \holdax{r}{\delta_{1}}(E)$,
$\beta \in \holdax{r}{\delta_{2}}(F)$ with
$\delta_{1} + \delta_{2} > 0$. Then
\[ \linner{A\alpha,\beta}\,=\, \linner{\alpha,A^{*}\beta} . \]
\end{lem}

\begin{proof}
This holds for compactly supported $\alpha, \beta$ by definition.
The condition $\delta_{1} + \delta_{2} > 0$ ensures that
\[ \holdax{r}{\delta_{1}}(E) \times \holdax{r}{\delta_{2}}(F), \;\;
(\alpha,\beta) \mapsto \; \linner{A\alpha,\beta} - \linner{\alpha,A^{*}\beta} \]
is continuous. As the compactly supported forms are dense in the Hölder
spaces the result follows.
\end{proof}

Once we know that the maps $A : \holdad{k+r}(E) \to \holdad{k}(F)$ are Fredholm
we can phrase the `Fredholm alternative' in the following way.

\begin{prop}
\label{fredaltprop}
Let $M$ be a manifold with cylindrical ends and an asymptotically
translation-invariant metric. Let $A : \Gamma(E) \to \Gamma(F)$ be
a smooth linear elliptic asymptotically translation-invariant 
differential operator of order $r$, and
$A^{*} : \Gamma(F) \to \Gamma(E)$ its formal adjoint.

%and $K^{*}_{-\delta}$ the kernel of $A^{*}$ extended to act as a map
%$\holdax{k+r}{-\delta}(F) \to \holdax{k+r}{-\delta}(E)$.
If $\delta$ is a real number not in $\mathcal{B}_{A}$
then the image of $A : \holdad{k+r}(E) \to \holdad{k}(F)$ is precisely the
\ltwoorth{} complement to the kernel of
$A^{*} : \holdax{k+r}{-\delta}(F) \to \holdax{k+r}{-\delta}(E)$
in $\holdad{k}(F)$ .
\end{prop}

\begin{proof}
Let $I$ be the image of $A$ in $\holdad{k+r}(E)$, and $K$ the kernel of
$A^{*}$ in $\holdax{k+r}{-\delta}(F)$.
Then $K$ is contained in $\holdax{k+r}{-\delta'}(F)$ for some
$\delta' > \delta$.
Integration by parts shows that $K$ contains the \ltwoorth{} complement to $I$
in $\holdax{k+r}{-\delta'}(F)$, and that $I$ is contained in the \ltwoorth{}
complement to $K$ in $\holdad{k+r}(E)$. Since $I$ is closed the result follows.
\end{proof}

%Lockhart
\cite[Theorem $7.4$]{lockhart85} gives a method for computing the
index of a formally self-adjoint elliptic asymptotically translation-invariant
differential operator acting on a weighted Hölder space with small weight.

\begin{thm}
\label{indthm}
Let $M$ be a manifold with cylindrical ends and with an asymptotically
translation-invariant metric.
Let $A : \Gamma(E) \to \Gamma(E)$ be a smooth linear elliptic formally
self-adjoint asymptotically translation-invariant differential operator of
order $r$.
%asymptotic to a translation-invariant
%operator $A_{\infty} : \Gamma(E) \to \Gamma(E)$.

Let $\epsilon > 0$ be the smallest positive element of $\mathcal{B}_{A}$.
Then for $0 < \delta < \epsilon$ the index of 
\begin{equation} 
\label{indeq}
A : \holdax{k+r}{\pm\delta}(E) \to \holdax{k}{\pm\delta}(E)
\end{equation}
(which is Fredholm by theorem \ref{ellfredthm}) is
\[ \mp \half \sum_{\lambda \in \bbr} d(\lambda) . \]
\end{thm}

\begin{proof}
It follows from self-adjointness and proposition
\ref{fredaltprop} that $i_{\delta}(A) = -i_{-\delta}(A)$, so using theorem
\ref{ellfredthm} we obtain
\[ -2i_{\delta}(A) = i_{-\delta}(A) - i_{\delta}(A) = 
\sum_{-\delta < \im \lambda < \delta} d_{A}(\lambda) =
\sum_{\lambda \in \bbr} d_{A}(\lambda) . \qedhere \]
\end{proof}

%asymptotic expansion: see mod2U.tex


\subsection{The Laplacian on EAC manifolds}
\label{lapsub}

In this subsection we use results from the previous subsection to study
the Hodge Laplacian and its kernel on an EAC (exponentially asymptotically
cylindrical) manifold. The condition that the metric is EAC (rather than
just asymptotically translation-invariant as in \S \ref{cylanalysissub})
allows us to prove a Hodge decomposition result for exponentially decaying
forms, and is needed to develop the Hodge theory in the next subsection.

Theorem \ref{ellfredthm} about elliptic operators being Fredholm on the
weighted Hölder spaces for all but a discrete set of weights applies in
particular to the Hodge Laplacian $\triangle$ of an asymptotically
translation-invariant metric, and we can define

\begin{defn}
\label{epsionedef}
Given an asymptotically cylindrical manifold $M^{n}$ let $\epsilon_{1} > 0$
be the smallest positive element of $\mathcal{B}_{\triangle}$ (where
$\mathcal{B}_{\triangle}$ is defined as in (\ref{badefeq}), and $\triangle$
is considered to act on sections of $\bigoplus^{n}_{m = 0} \Lambda^{m}$).
\end{defn}

Throughout this subsection we will use weights $\delta$ such that
$0 < \delta < \epsilon_{1}$, so that the Laplacian is Fredholm on weighted
spaces for all positive weights less than or equal to $\delta$.
It is obvious from the definition of $\calb_{\triangle}$ that 
$\epsilon_{1}$ depends only on the asymptotic model $g_{X}+dt^{2}$
for the metric on $M$.

\begin{lem}
\label{epsionectslem}
$\epsilon_{1}$ is a lower semi-continuous
function of $g_{X}$ with respect to the $C^{1}$-norm.
\end{lem}

\begin{proof}
$\epsilon_{1}^{2}$ is in fact the smallest positive eigenvalue $\lambda_{1}$ of
the Hodge Laplacian $\triangle_{X}$ defined by $g_{X}$ on $\Omega^{*}(X)$
(cf. (\ref{topdegeq})).
To prove the proposition it therefore suffices to show that $\lambda_{1}$ is
lower semi-continuous in $g_{X}$.

Let $g, g'$ be smooth Riemannian metrics on $X$, $\triangle, \triangle'$
their Laplacians and $\lambda_{1}, \lambda_{1}'$ the smallest positive
eigenvalues of the Laplacians.
%
Let $T$ be the $L^{2}(g)$-orthogonal complement to $\ker \triangle$ in
$\holda{2}(\Lambda^{*}T^{*}X)$.
Then for any $\beta \in T$ with unit $L^{2}(g)$-norm
\[ \lambda_{1} \leq \; \lginner{\triangle \beta, \beta} \; =
\lgnorm{d\beta}^{2}+\lgnorm{d^{*}\beta}^{2} . \]
Since $d+d^{*}$ is an elliptic operator it gives a Fredholm map
$\sob{1}(\Lambda^{*}T^{*}X) \to L^{2}(\Lambda^{*}T^{*}X)$, so it is
bounded below transverse to its kernel. In other words, there is
a constant $C_{1}$ such that
$\norm{\beta}^{2}_{\sob{1}(g)} \leq C_{1}
\left(\lgnorm{d\beta}^{2}+\lgnorm{d^{*}\beta}^{2}\right)$
for any $\beta \in T \cap \sob{1}(\Lambda^{*}T^{*}X)$.

Let $e_{1}$ be an eigenvector of $\triangle'$ with eigenvalue $\lambda_{1}'$.
By Hodge theory for compact manifolds $\ker \triangle$ and $\ker \triangle'$
have the same dimension, so $(\ker \triangle' \oplus \bbr e_{1}) \cap T$ is
non-trivial. Hence
\begin{equation*}
\lambda_{1}' \geq
\frac{\lgpinner{\triangle' \beta, \beta}}
{\lgpinner{\beta,\beta}} =
\frac{\lgpnorm{d\beta}^{2} + \lgpnorm{d^{*'}\beta}^{2}}
{\lgpnorm{\beta}^{2}}
\end{equation*}
for some $\beta \in T$ with unit $L^{2}(g)$-norm. The RHS
depends differentiably on $g'$ (with respect to the $C^{1}(g)$-norm) and the
derivative at $g' = g$
can be estimated in terms of $\norm{\beta}^{2}_{\sob{1}(g)}$.
Therefore there is a constant $C_{2}$ (independent of $\beta$)
such that for any $g'$ close to $g$
\begin{multline*}
\lambda_{1}' \geq
\frac{\lgpnorm{d\beta}^{2} + \lgpnorm{d^{*'}\beta}^{2}}
{\lgpnorm{\beta}^{2}} \\ \geq 
\lgnorm{d\beta}^{2}+\lgnorm{d^{*}\beta}^{2}
- C_{2}\norm{g'-g}_{C^{1}(g)}\norm{\beta}^{2}_{\sob{1}(g)}
\geq \left(1 - \norm{g'-g}_{C^{1}(g)}C_{1}C_{2}\right)\lambda_{1} . \qedhere
\end{multline*}
\end{proof}

Now let $M$ be an EAC manifold with rate $\delta_{0}$ and cross-section $X$,
and assume that $0 < \delta < \min\{\epsilon_{1}, \delta_{0}\}$.
We fix some notation for various spaces of harmonic forms.

\begin{defn}
\label{harmdef}
Denote by
\begin{enumerate}
\item $\harm^{m}_{\pm}$ the space of harmonic $m$-forms in
$\holdax{k}{\pm\delta}(\Lambda^{m}T^{*}M)$,
\item $\harm^{m}_{0}$ the space of bounded harmonic $m$-forms on $M$,
\item $\harm^{m}_{\infty}$ the space of translation-invariant
harmonic $m$-forms on $X \times \bbr$,
\item $\harm^{m}_{X}$ the space of harmonic $m$-forms on $X$.
\end{enumerate}
\end{defn}

\noindent
By elliptic regularity $\harm^{m}_{\pm}$ consists of smooth forms, and is
independent of $k$ for $k \geq 2$.

We can use theorem \ref{indthm} to compute the index of the Laplacian.
The result can also be found in \cite[\S 3]{lockhart87} and 
\cite[\S 6.4]{melrose94}.

\begin{prop}
\label{lapindprop}
Let $M$ be an asymptotically cylindrical manifold.
%
For $0 < \delta < \epsilon_{1}$
\[ \ind \left(\triangle : \holdax{k+2}{\pm\delta}(\Lambda^{m}) \to
\holdax{k}{\pm\delta}(\Lambda^{m}) \right) =
\mp (b^{m-1}(X) + b^{m}(X)) . \]
\end{prop}

\begin{proof}
$\triangle$ is asymptotic to $\triangle_{\infty}$, the Laplacian of the
product metric $g_{\infty} = g_{X} + dt^{2}$ on the cylinder $X \times \bbr$.
According to theorem \ref{indthm} we can find the desired index by computing
the dimensions $d_{\triangle}(\lambda)$ of the spaces of solutions
$S_{\triangle}(\lambda)$ for $\lambda \in \bbr$.

Any section of $\Lambda^{m}T^{*}(X\times\bbr)$ on $X \times \bbr$
can be written uniquely as
$\psi + dt \wedge \tau$, where $\psi$ and $\tau$ are sections of
$\Lambda^{m}T^{*}X$ and $\Lambda^{m-1}T^{*}X$ respectively. We can express
$\triangle_{\infty}$ in terms of the Laplacian $\triangle_{X}$ on $X$ by
\begin{equation}
\label{prodlapeq}
\triangle_{\infty} (\psi + dt \wedge \tau) = 
-\frac{\partial^{2} \psi}{\partial t^{2}} + \triangle_{X}\psi + %(-1)^{m+1}
\left( -\frac{\partial^{2} \tau}{\partial t^{2}} + \triangle_{X} \tau \right)
\wedge dt .
\end{equation}

Now suppose that
$\triangle_{\infty}\left(e^{i\lambda t}(\psi + dt \wedge \tau)\right) = 0$,
where $\psi+\tau\wedge dt$ is a nontrivial polynomial of degree $r$ in $t$,
and $\lambda \in \bbr$.
Denote the degree $s$ coefficients by $\psi_{s}$ and~$\tau_{s}$
(these are independent of $t$, so can be considered as forms on $X$).
Using (\ref{prodlapeq}) the condition that the degree $r$ component of
$\triangle_{\infty}\left(e^{i\lambda t}(\psi + dt \wedge \tau)\right)$ vanishes
becomes
\begin{equation}
\label{topdegeq}
\lambda^{2}\psi_{r} + \triangle_{X}\psi_{r} = 0,\;
\lambda^{2}\tau_{r} + \triangle_{X}\tau_{r} = 0 .
\end{equation}
Since $\triangle_{X}$ is semi-positive-definite it follows that $\lambda = 0$
(in particular $d_{\triangle}(\lambda) = 0$ for
$\lambda \in \bbr\setminus\{0\}$).
It follows from (\ref{topdegeq}) that
$\psi_{r} \in \harm^{m}_{X}, \tau_{r} \in \harm^{m-1}_{X}$.

Suppose that the degree $r \geq 2$. Then, since $\lambda = 0$, the condition
that the degree $r-2$ component of 
$\triangle_{\infty}(e^{i\lambda t}(\psi + dt \wedge \tau))$ vanishes becomes
\[ -r(r-1)\psi_{r} + \triangle_{X}\psi_{r-2} = 0, \;
-r(r-1)\tau_{r} + \triangle_{X}\tau_{r-2} = 0 . \]
This is impossible since $\psi_{r}$ and $\tau_{r}$ are harmonic (not both zero)
forms on $X$, and the image of $\triangle_{X}$ is \ltwoorth{} to the
kernel.
Hence any solution of $\triangle_{\infty}(\psi + dt \wedge \tau) = 0$ that is
polynomial in $t$ is actually linear. The space of such solutions
is precisely
\begin{equation}
\label{linharmeq}
S_{\triangle}(0) =
\{ t(\psi_{1} + \tau_{1} \wedge dt) + (\psi_{0} + \tau_{0} \wedge dt) :
\psi_{s} \in \harm^{m}_{X}, \tau_{s} \in \harm^{m-1}_{X} \} ,
\end{equation}
and its dimension is
\[ d_{\triangle}(0) = 2b^{m}(X) + 2b^{m-1}(X) . \]
The result now follows from theorem \ref{indthm}.
\end{proof}


\begin{rmk}
\label{harminftyrmk}
It follows from (\ref{prodlapeq}) that
\[ \harm^{m}_{\infty} = \{\psi + dt \wedge \tau : \psi \in \harm^{m}_{X},
\tau \in \harm^{m-1}_{X} \} . \]
In particular the dimension of $\harm^{m}_{\infty}$ is $-\ind
\left(\triangle : \holdad{k+2}(\Lambda^{m}) \to \holdad{k}(\Lambda^{m}) \right)$
for small $\delta > 0$. 
\end{rmk}

Now that we have computed the index of the Laplacian on weighted Hölder spaces
we use an index-counting argument to deduce results about the kernel and
a Hodge decomposition theorem.
Let $M$ be an EAC manifold with rate $\delta_{0}$, $k \geq 0$ and
$0 < \delta < \min\{\epsilon_{1},\delta_{0}\}$.

Let $i = b^{m}(X) + b^{m-1}(X)$. $\harm^{m}_{\infty}$ has dimension $i$
and the index of
\[ \triangle : \holdad{k+2}(\Lambda^{m}) \to \holdad{k}(\Lambda^{m}) \]
is $-i$.
$\triangle(\rho\harm^{m}_{\infty})$ and $\triangle(\rho t \harm^{m}_{\infty})$
consist of exponentially decaying forms.
Therefore
\begin{equation}
\label{padeq}
\triangle : \holdad{k+2}(\Lambda^{m}) \oplus \rho\harm^{m}_{\infty}
\oplus \rho t \harm^{m}_{\infty} \to \holdad{k}(\Lambda^{m})
\end{equation}
is well-defined, and its index is $+i$. The map
\begin{equation}
\label{nonpadeq}
\triangle : \holdax{k+2}{-\delta}(\Lambda^{m}) \to \holdax{k}{-\delta}(\Lambda^{m})
\end{equation}
also has index $+i$, and its kernel $\harm^{m}_{-}$ contains that of
(\ref{padeq}).
By proposition \ref{fredaltprop} the image of (\ref{nonpadeq}) is exactly the
orthogonal complement of $\harm^{m}_{+}$ in $\holdax{k}{-\delta}(\Lambda^{m})$.
The image of (\ref{padeq}) is a subset, so it is contained in the orthogonal
complement of
$\harm^{m}_{+}$ in $\holdad{k}(\Lambda^{m})$.
Thus the cokernel of (\ref{padeq}) is
at least a large as the cokernel of (\ref{nonpadeq}).

It follows by dimension-counting that the kernels of
(\ref{padeq}) and (\ref{nonpadeq}) are equal and that the cokernels have
the same dimension. In particular

\begin{prop}
\label{explodeprop}
Let $M$ be EAC with rate $\delta_{0}$,
$k \geq 0$ and $0 < \delta < \min\{\epsilon_{1}, \delta_{0}\}$.
Then any harmonic form in
$\holdax{2}{-\delta}(\Lambda^{m})$ lies in
\[ \holdad{k}(\Lambda^{m}) \oplus \rho\harm^{m}_{\infty}
\oplus \rho t \harm^{m}_{\infty} . \]
\end{prop}

\noindent
Since $\harm^{m}_{-}$ contains both the $L^{2}$ harmonic forms and
the bounded harmonic forms $\harm^{m}_{0}$ it follows that

\begin{prop}
\label{decayprop}
Let $M$ be EAC with rate $\delta_{0}$, $k \geq 0$
and $0 < \delta < \min \{\epsilon_{1}, \delta_{0}\}$.
Then the space of $L^{2}$-integrable harmonic $m$-forms on $M$ is exactly
$\harm^{m}_{+}$, and
\begin{equation*}
\harm^{m}_{0} \subset \holdad{k}(\Lambda^{m}) \oplus \rho\harm^{m}_{\infty} .
\end{equation*}
\end{prop}

\begin{rmk}
Results like propositions \ref{explodeprop} and \ref{decayprop} about the
asymptotic behaviour of solutions of elliptic equations are less a consequence
of theorem \ref{ellfredthm} than a part of its proof. The ordering here
is for presentational convenience.
\end{rmk}

% obsolete: So $\harm^{m} = \harm^{m}_{+}$.
\noindent
We can also use the index-counting argument
to prove a Hodge decomposition result.
We need to distinguish $d\holdad{k}(\Lambda^{m-1})$, the space of exterior
derivatives of decaying $(m-1)$-forms, from the space of decaying exact
$m$-forms.

\begin{defn}
For $0 < \delta < \epsilon_{1}$ let
%\begin{subequations}
\begin{align*}
\holdad{k}[d\Lambda^{m-1}] &= d\Omega^{m-1}(M) \cap \holdad{k}(\Lambda^{m}), \\
\holdad{k}[d^{*}\Lambda^{m+1}] &=
d^{*}\Omega^{m+1}(M) \cap \holdad{k}(\Lambda^{m}) .
\end{align*}
%\end{subequations}
%
%\begin{equation} \begin{split}
%\holdad{k}[d\Lambda^{m-1}] &= d\left(\holdax{k+1}{-\delta/2}(\Lambda^{m-1})\right)
%\cap \holdad{k}(\Lambda^{m}) \\
%\holdad{k}[d^{*}\Lambda^{m+1}] &= d^{*}\left(\holdax{k+1}{-\delta/2}(\Lambda^{m+1})
%\right) \cap \holdad{k}(\Lambda^{m})
%\end{split} \end{equation}
\end{defn}

\begin{thm}
\label{hodgedecompthm}
%flag
Let $M^{n}$ be EAC with rate $\delta_{0}$, $k \geq 0$, and
$0 < \delta < \min\{\epsilon_{1}, \delta_{0}$\}.
Then there is an \ltwoorth{} direct sum decomposition
\begin{equation}
\label{noncpthodgeeq}
\holdad{k}(\Lambda^{m}) = \harm^{m}_{+} \oplus \holdad{k}[d\Lambda^{m-1}]
\oplus \holdad{k}[d^{*}\Lambda^{m+1}] ,
\end{equation}
and the projections onto these summands are bounded.
%
Furthermore, if $\rho$ is a cut-off function for the cylindrical ends of $M$
then any element of $\holdad{k}[d\Lambda^{m-1}]$ can be written as $d\phi$
for some coexact
$\phi \in \holdad{k+1}(\Lambda^{m-1}) \oplus \rho\harm^{m-1}_{\infty}$, and
any element of $\holdad{k}[d^{*}\Lambda^{m-1}]$ can be written as $d^{*}\phi$
for some exact
$\phi \in \holdad{k+1}(\Lambda^{m+1}) \oplus \rho\harm^{m+1}_{\infty}$.
%, where $\rho$ is a cut-off function for the cylinder on $M$.
\end{thm}

\begin{proof}
Theorem \ref{absderhamthm} below implies that
$\harm^{m}_{+} \cap \holdad{k}[d\Lambda^{m-1}] = 0$.
Applying the Hodge star gives
$\harm^{m}_{+} \cap \holdad{k}[d^{*}\Lambda^{m+1}] = 0$.
Since
$\holdad{k}[d\Lambda^{m-1}] \cap \holdad{k}[d^{*}\Lambda^{m+1}]
\subseteq \harm^{m}_{+}$
it follows that the sum is direct.

Since the cokernels of (\ref{padeq}) and (\ref{nonpadeq}) have
equal dimension
the image of (\ref{padeq}) is precisely the orthogonal complement of
$\harm^{m}_{+}$ in $\holdad{k}(\Lambda^{m})$, i.e.
\[ \holdad{k}(\Lambda^{m}) = \harm^{m}_{+} \oplus
\triangle \left( \holdad{k+2}(\Lambda^{m}) \oplus \rho\harm^{m}_{\infty}
\oplus \rho t \harm^{m}_{\infty} \right) . \]
Since $\harm^{m}_{+}$ is finite-dimensional the projections are bounded.

Now let $G$ be a Fredholm inverse for (\ref{padeq}) and for
$\beta \in \holdad{k}(\Lambda^{m})$ define
\begin{gather*}
P_{E} \beta = dd^{*}G\beta, \\
P_{E^{*}} \beta = d^{*}dG\beta .
\end{gather*}
$P_{E}$ and $P_{E^{*}}$ map into
$\holdad{k}[d\Lambda^{m-1}]$ and $\holdad{k}[d^{*}\Lambda^{m+1}]$ respectively,
and $\beta = P_{E}(\beta) + P_{E^{*}}(\beta)$ for any
$\beta \in \triangle \left( \holdad{k}(\Lambda^{m}) \oplus
\rho\harm^{m}_{\infty} \oplus \rho t \harm^{m}_{\infty} \right)$.
Therefore
\[ \triangle \left( \holdad{k+2}(\Lambda^{m}) \oplus \rho\harm^{m}_{\infty}
\oplus \rho t \harm^{m}_{\infty} \right) = 
\holdad{k}[d\Lambda^{m-1}] \oplus \holdad{k}[d^{*}\Lambda^{m+1}] \]
with projections $P_{E}$ and $P_{E^{*}}$. The result follows.
\end{proof}


\subsection{Hodge theory on EAC manifolds}
\label{hodgesub}

In this subsection we study the correspondence between harmonic forms
and the de Rham cohomology on an EAC Riemannian manifold $M^{n}$ with
cross-section~$X^{n-1}$.

We first describe what form the long exact sequence for relative cohomology
takes for an EAC manifold.
Following Melrose \cite[\S $6.4$]{melrose94} we then define subspaces
$\harm^{m}_{abs}, \harm^{m}_{rel}$ of the bounded harmonic forms
$\harm^{m}_{0}$ and describe their relation to the long exact sequence.
Hodge theory on classes of non-compact manifolds, including asymptotically
cylindrical ones, is also discussed in Lockhart \cite{lockhart87}.

\begin{rmk}
\label{nonorientrmk}
While we will usually apply the Hodge theory results to orientable
EAC manifolds $M$, we can still prove them in the non-orientable case.
Some arguments rely on using the Hodge star on $M$. These results we prove
first for $M$ orientable.
In the case when $M$ is not orientable let $\tilde M$ be the orientable double
cover of $M$. $\tilde M$ is EAC, and $M$ is a quotient of $\tilde M$ by
an orientation-reversing involution $a$. The harmonic forms of $M$ can be
identified with the $+1$ eigenspace of $a$ on the harmonic forms of $\tilde M$,
and the result for $M$ can be deduced from that for~$\tilde M$.
\end{rmk}

%The subsection is ended by a summary of the notation and results. Not. %xxx

\hfuzz=1pt
\begin{defn}
Let $H^{*}_{cpt}(M)$ be the compactly supported cohomology of~$M$, i.e.
the cohomology of the cochain complex
\[ \Omega^{0}_{cpt}(M) \stackrel{d}{\longrightarrow}\Omega^{1}_{cpt}(M)
\stackrel{d}{\longrightarrow} \Omega^{2}_{cpt}(M) 
\stackrel{d}{\longrightarrow} \cdots \]
of compactly supported smooth forms on $M$.
\end{defn}
\hfuzz=0.1pt

Recall that when $M$ has cylindrical ends then we can compactify $M$ by
`adding a copy of $X$ at infinity', i.e. by
including it in $\overline M = M_{0} \cup (X \times [0,\infty])$.
The cohomology (with coefficients $\bbr$) of $\overline M$ relative to its
boundary can be identified with $H^{*}_{cpt}(M)$.
The long exact sequence for relative cohomology of $\overline M$
can be written as
\begin{equation}
\label{relexacteq}
\cdots \longrightarrow H^{m-1}(X) \stackrel{\partial}{\longrightarrow}
H^{m}_{cpt}(M) \stackrel{\inc}{\longrightarrow} H^{m}(M)
\stackrel{j^{*}}{\longrightarrow} H^{m}(X) \longrightarrow \cdots
\end{equation}
$\inc : H^{m}_{cpt}(M) \to H^{m}(M)$ is induced
by the inclusion $\Omega^{*}_{cpt}(M) \hookrightarrow \Omega^{*}(M)$.
The image of $e$ is the subspace of de Rham cohomology classes with compact
representatives.

\begin{defn}
Let
$H^{m}_{0}(M) = \im \left(\inc : H^{m}_{cpt}(M) \to H^{m}(M) \right)$.
%and $b^{m}_{0}(M) = \dim H^{m}_{0}(M)$.
\end{defn}

In the asymptotically cylindrical setting the map
$j^{*} : H^{m}(M) \to H^{m}(X)$ can be described as follows:
For $s \in \bbrp$ let $j_{s} : X \hookrightarrow M$ be
the inclusion $x \mapsto (x,s) \in M_{\infty}$. The maps $j_{s}$ are
homotopic, so they all give the same map 
$j^{*} : H^{m}(M) \to H^{m}(X)$.

If $\alpha$ is an asymptotically translation-invariant $m$-form let
$B(\alpha)$ denote its asymptotic limit, which is a
translation-invariant form on the cylinder $X \times \bbr$.
$B(\alpha)$ can be written as
$B_{a}(\alpha) + dt \wedge B_{e}(\alpha)$, where $B_{a}(\alpha)$,
$B_{e}(\alpha)$ are forms on $X$ of degree $m$ and $m-1$ respectively.

If $\alpha$ is a closed asymptotically translation-invariant $m$-form then
for any $m$-cycle $C$ in~$X$
\[ \int_{C}j^{*}_{s}([\alpha]) \to \int_{C}[B_{a}(\alpha)] \]
as $s \to \infty$, so
\begin{equation}
\label{jbaeq}
j^{*}([\alpha]) = [B_{a}(\alpha)] .
\end{equation}
%
By proposition \ref{decayprop} any $\alpha \in \harm^{m}_{0}$ is
exponentially asymptotically translation-invariant, and
$B({\alpha}) \in \harm^{m}_{\infty}$. By remark \ref{harminftyrmk}
\[ \harm^{m}_{\infty} \cong \harm^{m}_{X} \oplus dt \wedge \harm^{m-1}_{X} , \]
so we get maps $B_{a} : \harm^{m}_{0} \to \harm^{m}_{X}$,
$B_{e} : \harm^{m}_{0} \to \harm^{m-1}_{X}$.

\begin{defn}
Let
\begin{gather*}
\harm^{m}_{abs} = \ker B_{e} \subseteq \harm^{m}_{0} , \\
\harm^{m}_{rel} = \ker B_{a} \subseteq \harm^{m}_{0} .
\end{gather*}
Further let $\harm^{m}_{E}, \harm^{m}_{E^{*}} \subseteq \harm^{m}_{0}$
denote the spaces of bounded exact and coexact harmonic forms respectively.
\end{defn}

It follows immediately from (\ref{jbaeq}) that
$\harm^{m}_{E} \subseteq \harm^{m}_{rel}$. It follows by applying the Hodge
star that $\harm^{m}_{E^{*}} \subseteq \harm^{m}_{abs}$
(using remark \ref{nonorientrmk} if $M$ is not orientable).

Let $\Omega^{*}_{-\delta}(M)$ be the cochain complex of forms $\beta$ such
that $e^{-\delta t}\beta$ is uniformly bounded with all derivatives,
and let $H^{*}_{-\delta}(M)$ be its cohomology.

\begin{lem}
\label{bdhlem}
For $\delta > 0$ the natural map $H^{*}_{-\delta}(M) \to H^{*}(M)$ is an
isomorphism.
\end{lem}

\begin{proof}
The inclusion $i : \Omega^{*}_{-\delta}(M) \hookrightarrow \Omega^{*}(M)$
is a chain map, so induces a well-defined map
$i : H^{*}_{-\delta}(M) \to H^{*}(M)$.

Let $\rho : \bbr \to [0,1]$ be a smooth function with $\rho(t) = 0$
for $t \leq 0$ and $\rho(t) = 1$ for $t \geq 1$. Define a map
$c : M \to M$ which is the identity on the compact piece $M_{0}$ and
$(x,t) \mapsto (x, (1-\rho(t))t + \rho(t)\arctan t)$ on the cylindrical part.
$c$ is smooth, and $c^{*}$ maps $\Omega^{*}(M)$ to $\Omega^{*}_{-\delta}(M)$.
It is a chain map, so induces a map $c^{*} : H^{*}(M) \to H^{*}_{-\delta}(M)$.
We deduce that $i$ and $c^{*}$ are inverses from the fact that $c$ is
homotopic to the identity.
\end{proof}

The next theorem is part of \cite[Theorem $6.18$]{melrose94}.

\begin{thm}
\label{absderhamthm}
Let $M$ be an EAC manifold. The natural %projection 
map $\harm^{m}_{abs} \to H^{m}(M)$ is an isomorphism.
\end{thm}

\begin{proof}
Note that by integration by parts the elements of
$\harm^{m}_{0}$ are closed, so represent cohomology classes.
Pick some $0 < \delta < \epsilon_{1}$. 
By the previous lemma it suffices to check that
$\harm^{m}_{abs} \to  H^{m}_{-\delta}(M)$ is an isomorphism.

We first prove injectivity. If
$\alpha \in \harm^{m}_{abs}$ and $\alpha = d\beta$ for some
$\beta \in \Omega^{m}_{-\delta}(M)$ then $\alpha \in \harm^{m}_{+}$, since
$\harm^{m}_{E} \subseteq \harm^{m}_{rel}$ and
$\harm^{m}_{abs} \cap \harm^{m}_{rel} = \harm^{m}_{+}$.
As $\alpha$ is exponentially decaying with rate $\delta$ we can
integrate by parts to deduce $\alpha = 0$. Thus
$\harm^{m}_{abs} \to H^{m}_{-\delta}(M)$ is an injection.
% and, by lemma
%\ref{bdhlem}, $\harm^{m}_{abs}$ maps injectively to $H^{m}(M)$.

It remains to show that %\harm^{m}_{abs} \to H^{m}(M)$ is surjective.
%By lemma \ref{bdhlem} it suffices to show that
if $\gamma \in \Omega^{m}_{-\delta}(M)$ is closed then $\gamma$ is cohomologous
to an element of $\harm^{m}_{abs}$. 
$\holdax{0}{-\delta}(\Lambda^{m}) =
\harm^{m}_{+} \oplus \triangle \holdax{2}{-\delta}(\Lambda^{m})$
by proposition \ref{fredaltprop}. Thus $\gamma$ can be written as
\[ \gamma = \phi + \triangle \psi \]
with $\phi \in \harm^{m}_{+}$ and $\psi \in \holdax{2}{-\delta}(\Lambda^{m})$.
Let $\chi = d\psi$. Then
\[ d\gamma = 0 \Rightarrow dd^{*}\chi = 0 \Rightarrow \triangle \chi = 0 . \]
By proposition \ref{explodeprop},
$\chi \in \holdad{2}(\Lambda^{m+1}) \oplus \rho \harm^{m+1}_{\infty}
\oplus \rho t \harm^{m+1}_{\infty}$ so
$d^{*}\chi \in \harm^{m}_{E^{*}} \subseteq \harm^{m}_{abs}$.
Therefore
\begin{equation}
\label{gammadecompeq}
\gamma - dd^{*}\psi = \phi + d^{*}\chi \in \harm^{m}_{abs} 
\end{equation}
represents the same cohomology class as $\gamma$.
Hence $\harm^{m}_{abs} \to H^{m}_{-\delta}(M)$ is surjective.
\end{proof}

\begin{prop}
\label{hsplitprop}
$\harm^{m}_{abs} = \harm^{m}_{+} \oplus \harm^{m}_{E^{*}}$, 
$\harm^{m}_{rel} = \harm^{m}_{+} \oplus \harm^{m}_{E}$.

Furthermore
$\harm^{m}_{E} = d\harm^{m-1}_{-}$,
$\harm^{m}_{E^{*}} = d^{*}\harm^{m+1}_{-}$.
\end{prop}

\begin{proof}
Theorem \ref{absderhamthm} implies that
$\harm^{m}_{E} \cap \harm^{m}_{+} = 0$. Similarly
$\harm^{m}_{E^{*}} \cap \harm^{m}_{+} = 0$, so the sums are direct.
%
If $\gamma \in \harm^{m}_{0}$ then, since $\gamma$ is closed,
(\ref{gammadecompeq}) gives
\[ \gamma  = \phi + d^{*}\chi + dd^{*}\psi , \]
where $\phi \in \harm^{m}_{+}$, $d^{*}\chi \in d^{*}\harm^{m+1}_{-}$.
Analogously, since $\gamma$ is also coclosed,
$dd^{*}\psi \in d\harm^{m-1}_{-}$. Hence
\[ \harm^{m}_{0} =
\harm^{m}_{+} \oplus d^{*}\harm^{m+1}_{-} \oplus  d\harm^{m-1}_{-} . \]
Since $d\harm^{m-1}_{-} \subseteq \harm^{m}_{E} \subseteq \harm^{m}_{rel}$ and
$d^{*}\harm^{m+1}_{-} \subseteq \harm^{m}_{E^{*}} \subseteq \harm^{m}_{abs}$
the result follows.
\end{proof}

As a corollary of theorem \ref{absderhamthm} we can determine that
the image of the space $\harm^{m}_{+}$ of $L^{2}$ harmonic forms in the
de Rham cohomology $H^{m}(M)$ is precisely the subspace $H^{m}_{0}(M)$ of
compactly supported classes. Similar results (with more general hypotheses)
appears in e.g. \cite[Proposition $4.9$]{atiyah75} and
\cite[Theorems $7.6$ and $7.9$]{lockhart87}.

\begin{thm}
\label{derhamthm}
Let $M$ be an EAC manifold.
The natural map $\harm^{m}_{+} \to H^{m}(M)$ is an isomorphism onto
$H^{m}_{0}(M)$.
\end{thm}

\begin{proof}
$\harm^{m}_{+}$ is $\ker B_{a}$ in $\harm^{m}_{abs}$, and it follows from
theorem \ref{absderhamthm} it is mapped isomorphically to
$H^{m}_{0}(M) = \ker j^{*} \subseteq H^{m}(M)$.
\end{proof}

The fact that the image of $\harm^{m}_{+}$ is contained in $H^{m}_{0}(M)$
could be seen more explicitly by applying the following lemma,
which is useful for other purposes.

If $\alpha$ is a closed exponentially asymptotically translation-invariant $m$-form on $M$ we can write it as
$\alpha_{\infty} + \beta_{t} + dt \wedge \gamma_{t}$ on the
cylindrical part $X \times \bbrp$,
where $\alpha_{\infty}$ is translation-invariant, and
$\beta_{t}, \gamma_{t}$ are sections of
$\Lambda^{m}T^{*}X$, $\Lambda^{m-1}T^{*}X$ respectively, and are
exponentially decaying in $t$. Let
\begin{equation}
\label{etaeq}
\eta(\alpha) = \rho \int_{t}^{\infty}\gamma_{s}ds ,
\end{equation}
where $\rho$ is a smooth cut-off function for the cylindrical end of $M$,
equal to $1$ for $t > t_{0}$. $\eta(\alpha)$ is a well-defined exponentially
decaying $(m-1)$-form on $M$.

\begin{lem}
\label{cylfylem}
Let $M$ be a manifold with cylindrical ends, and $\alpha$ a closed
exponentially asymptotically translation-invariant $m$-form on $M$.
Then $\alpha + d\eta(\alpha)$ is translation-invariant
on $\{ y \in M : t > t_{0} \}$.
\end{lem}

\begin{proof}
Closure of $\alpha$ means that $\contrat \beta_{t} + \dsubx \gamma_{t} = 0$
where $\dsubx$ is the exterior derivative on $X$. Thus
$\beta_{t} = -\int_{t}^{\infty}\dsubx\gamma_{s}ds$. Since $X$ is
compact the dominated convergence theorem ensures that for $t > t_{0}$
\[ d\eta = \dsubx\eta + dt \wedge \contrat \eta =
-\beta_{t} - dt \wedge \gamma_{t} . \qedhere \]
\end{proof}

\begin{defn}
Let $\mcal{A}{m} = B_{a}(\harm^{m}_{0}) \subseteq \harm^{m}_{X}, \, \:
\mcal{E}{m} = B_{e}(\harm^{m+1}_{0}) \subseteq \harm^{m}_{X}$, and
let $A^{m}, E^{m}$ be the subspaces of $H^{m}(X)$ that they represent
($A^{m}$ is of course just $j^{*}(H^{m}(M)) \subseteq H^{m}(X)$).
\end{defn}


When $M^{n}$ is oriented the Hodge star on $M$ identifies $\harm^{m}_{abs}$
and $\harm^{m-n}_{rel}$. If $\beta \in \harm^{m}_{0}$ then
$B_{e}(*\beta) = *B_{a}(\beta)$, so the Hodge star on $X$ identifies
$\mathcal{A}^{m}$ with $\mathcal{E}^{n-m-1}$
(and hence $A^{m}$ with $E^{n-m-1}$).

By proposition \ref{explodeprop} any $\psi \in \harm^{m}_{-}$ can be
written as $\chi + \alpha + t\beta + dt \wedge \gamma + tdt \wedge \delta$
with $\chi$ exponentially decaying,
$\alpha, \beta \in \harm^{m}_{X}$ and
$\gamma, \delta \in \harm^{m-1}_{X}$. Thus we can define a `boundary data' map
\[ BD : \harm^{m}_{-} \to (\harm^{m}_{X})^{2} \oplus (\harm^{m-1}_{X})^{2},
\quad \psi \mapsto \tbtmatrix{\alpha}{\beta}{\gamma}{\delta} . \]

Let $BD_{a}$, $BD_{e}$ be the composition of $BD$ with the projection to
$(\harm^{m}_{X})^{2}$ and $(\harm^{m-1}_{X})^{2}$ respectively.
Let $\tcal{A}^{m} = BD_{a}(\harm^{m}_{-})$ and
$\tcal{E}^{m-1} = BD_{e}(\harm^{m}_{-})$. The following proposition is
a refinement of \cite[Lemma $6.15$]{melrose94}.

\begin{prop}
\label{bdlagprop}
Let $M$ be an EAC manifold.
\begin{enumerate}
\item \label{adjointitem}
If $\psi_{1}, \psi_{2} \in \harm^{m}_{-}$ with 
$BD(\psi_{i}) = \tbtmatrix{\alpha_{i}}{\beta_{i}}{\gamma_{i}}{\delta_{i}}$ then
in the $L^{2}$ inner product on $X$
\begin{subequations}
\begin{gather}
\label{adjoint1eq}
\linner{\alpha_{1}, \beta_{2}} \;=\; \linner{\alpha_{2}, \beta_{1}}, \\
\label{adjoint2eq}
\linner{\gamma_{1}, \delta_{2}} \;=\; \linner{\gamma_{2}, \delta_{1}} .
\end{gather}
\end{subequations}
\item \label{lagitem}
$BD(\harm^{m}_{-}) = \tcal{A}^{m} \oplus \tcal{E}^{m}$, and
$\tcal{A}^{m} \subset (\harm^{m}_{X})^{2}$ and
$\tcal{E}^{m-1} \subset (\harm^{m-1}_{X})^{2}$ are Lagrangian subspaces.
\end{enumerate}
\end{prop}

\begin{proof}
First assume that $M$ is oriented.
Then $\partial : H^{m}(X) \to H^{m+1}_{cpt}(M)$ is the Poincar\'e dual of
$j^{*} : H^{n-m+1}(M) \to H^{n-m}(X)$, so
\[ \linner{\alpha_{i}, \beta_{j}} \;= ([\alpha_{i}], [*\beta_{j}])_{X} = 
([\alpha_{i}], j^{*}[*d\psi_{j}])_{X} =
(\partial[\alpha_{i}], [*d\psi_{j}])_{M} . \]
Note that $\partial[\alpha_{i}] =
[d(\psi_{i} + \eta(d\psi_{i}) -\rho t \beta_{i})] \in H^{m+1}_{cpt}(M)$.
Hence
\begin{multline}
\label{intedeq}
\linner{\alpha_{1}, \beta_{2}} - \linner{\alpha_{2}, \beta_{1}}
\;= \int_{M} (d\psi_{1} - d(\rho t \beta_{1})) \wedge *d\psi_{2}
- (d\psi_{2} - d(\rho t \beta_{2})) \wedge *d\psi_{1} \\ =
\int_{M} d(\rho t (\beta_{2} \wedge *d\psi_{1} - \beta_{1} \wedge *d\psi_{2})) .
\end{multline}
Since
%\[ BD(\rho\beta_{2} \wedge *\psi_{1} - \rho\beta_{1} \wedge *\psi_{2}) = 
\[ \beta_{2} \wedge B(*d\psi_{1}) - \beta_{1} \wedge B(*d\psi_{2}) =
\beta_{2} \wedge * \beta_{1} - \beta_{1} \wedge * \beta_{2} = 0 \]
the integrand in the RHS of (\ref{intedeq}) is the exterior derivative of an
exponentially decaying form.
The vanishing of the integral proves (\ref{adjoint1eq}),
and (\ref{adjoint2eq}) follows by applying $*$.

This proves \ref{adjointitem} in the oriented case.
When $M$ is not orientable we use remark \ref{nonorientrmk}.

\ref{adjointitem} implies that $\tcal{A}^{m} \subset (\harm^{m}_{X})^{2}$ and
$\tcal{E}^{m-1} \subset (\harm^{m-1}_{X})^{2}$ are null spaces.
In particular
\begin{equation*}
\dim \tcal{A}^{m} \leq b^{m}(X), \,\: \dim \tcal{E}^{m-1} \leq b^{m-1}(X) .
\end{equation*}
Since $BD(\harm^{m}_{-}) \subseteq \tcal{A}^{m} \oplus \tcal{E}^{m-1}$ and
has dimension $b^{m-1}(X) + b^{m}(X)$ equality must hold, so
$\tcal{A}^{m}$ and $\tcal{E}^{m-1}$ are Lagrangian.
\end{proof}

\begin{prop}
\label{aeprop}
If $M^{n}$ is an EAC manifold with cross-section $X$ then
\[ \harm^{m}_{X} = \mathcal{A}^{m} \oplus \mathcal{E}^{m} \]
is an orthogonal direct sum.
\end{prop}

\begin{proof}
If $\psi \in \harm^{m}_{-}$ with
$BD_{a}(\psi) = \tbomatrix{\alpha_{1}}{\beta_{1}} \in \tcal{A}^{m}$ then
$\beta_{1} = B_{e}(d\psi) \in \mathcal{E}^{m}$. Thus the second projection
$\tcal{A}^{m} \to \harm^{m}_{X}$ has image $\mathcal{E}^{m}$, and
kernel $\mathcal{A}^{m}$, so
\[\dim \mathcal{A}^{m} + \dim \mathcal{E}^{m} = \dim \tcal{A}^{m} =
\dim \harm^{m}_{X} \]
Furthermore if $\alpha_{2} \in \mathcal{A}^{m}$ then
$\tbomatrix{\alpha_{2}}{0} \in \tcal{A}^{m}$,
and (\ref{adjoint2eq}) implies that
$\inner{\beta_{1}, \alpha_{2}} \;= 0$, so $\mathcal{A}^{m}$ and
$\mathcal{E}^{m}$ are orthogonal.
\end{proof}

It follows from proposition \ref{aeprop} that we can define an isomorphism
\[   
\harm^{m}_{rel} \to H^{m}_{cpt}(M), \;\alpha \mapsto
\left\{
\begin{array}{ll}
[\alpha + d\eta(\alpha)] & \textrm{for } \alpha \in \harm^{m}_{+} \\
\partial([B_{e}(\alpha)]) & \textrm{for } \alpha \in \harm^{m}_{E}
\end{array} \right. . \]


\begin{cor}
\label{degonehodgecor}
Let $M^{n}$ be an asymptotically cylindrical manifold which has a single end
(i.e. the cross-section $X$ is connected). Then
$\inc : H^{1}_{cpt}(M) \to H^{1}(M)$ is injective.

In particular $\harm^{1}_{E} = 0$, and $\harm^{1}_{0} \to H^{1}(M)$ is
an isomorphism.
\end{cor}

\begin{proof}
Consider the start of the long exact sequence for relative cohomology
\[ H^{0}_{cpt}(M) \to H^{0}(M) \to H^{0}(X) \stackrel{\partial}{\to}
H^{1}_{cpt}(M) \stackrel{\inc}{\to} H^{1}(M) . \]
The dimensions of the first three terms are $0, 1$, and $1$, so $\partial = 0$,
and thus $\inc$ is injective.
%
$\harm^{1}_{rel} \leftrightarrow H^{1}_{cpt}(M)$ identifies $\harm^{1}_{E}$
with $\ker \inc$, so the result follows.
\end{proof}

% mod3C: cylindrical example

Finally we make two simple observations for the case when $M$ is Ricci-flat,
as it is when the holonomy is $G_{2}$ or $Spin(7)$.

\begin{prop}
\label{bochnerprop}
If $M$ is a Ricci-flat EAC manifold then $\harm^{1}_{0}$ is the space of
parallel $1$-forms on $M$. In particular $\harm^{1}_{+} = 0$, and
$j^{*} : H^{1}(M) \to H^{1}(X)$ is injective.
\end{prop}

\begin{proof}
This is proved by the same standard `Bochner argument' as corollary
\ref{bochnercor}, adapted to the EAC setting.
If $\phi$ is a 1-form then by proposition \ref{weitzenprop}
\begin{equation}
\label{weitzeneq}
\triangle \phi = \nabla^{*}\nabla \phi .
\end{equation}
It follows that any parallel $1$-form $\phi$ is harmonic, and
parallel forms are of course bounded.

To show that any bounded harmonic form is parallel we use (\ref{weitzeneq})
together with an integration by parts argument, justified by lemma \ref{ibplem}.

$\harm^{1}_{+} = 0$ since it consists of parallel decaying forms. By theorem
\ref{derhamthm}, the kernel $H^{1}_{0}(M)$ of $j^{*} : H^{1}(M) \to H^{1}(X)$ is
represented by $\harm^{1}_{+}$.
\end{proof}

\begin{cor}
\label{onecptcor}
If $M$ is a Ricci-flat EAC manifold with a single end then $H^{1}_{cpt}(M) = 0$.
\end{cor}

\begin{proof}
Follows from proposition \ref{bochnerprop} and corollary \ref{degonehodgecor}.
\end{proof}


\subsection{The Dirac operator on EAC manifolds}
\label{eacdiracsub}

We apply the ideas of \S \ref{lapsub} to study the Dirac operator on
an EAC manifold. This will later be used in slice constructions on EAC
\gtmfd s and \spmfd s.

Let $M^{n}$ be an EAC spin manifold with cross-section $X$, and (complex)
spinor bundle~$\spinor$. Let
$\harms_{0}$ and $\harms_{+}$ denote the spaces of bounded and decaying
harmonic spinors on $M$ with respect to the Dirac Laplacian $\dirac^{2}$,
and let $\harms_{\infty}$ be the translation-invariant
spinors on the cylinder $X \times \bbr$.
Let $\spinorx$ be the bundle of spinors on the cross-section, and
$\harms_{X}$ the harmonic spinors on $X$.

The Dirac operator on $M^{n}$ is asymptotically translation-invariant.
If $n$ is even then each of the positive and negative spinor bundles on
the cylinder $X \times \bbr$ are isomorphic to the pull-back of $\spinorx$
from $X$, and the asymptotic limit can be written as
\begin{equation*}
\dirac_{\infty} = \tbtmatrixl{0}{\contrat + \diracx}{-\contrat + \diracx}{0}
\; : \; \Gamma(\spinorx \oplus \spinorx) \to \Gamma(\spinorx \oplus \spinorx) .
\end{equation*}
Similarly, if $n$ is odd then the spinor bundle on the cylinder is isomorphic to
$\spinorx$, and the asymptotic limit is
\begin{equation*}
 \dirac_{\infty} = \tbtmatrixl{i\contrat}{0}{0}{-i\contrat} + \diracx \; :
\; \Gamma(\spinorxp \oplus \spinorxm) \to \Gamma(\spinorxp \oplus \spinorxm) .
\end{equation*}
Since the Dirac operator $\dirac$ is formally self-adjoint the index of
\begin{equation*}
\dirac : \holdax{k+1}{\pm \delta}(\spinor) \to \holdax{k}{\pm \delta}(\spinor)
\end{equation*}
is $\mp \half \dim \harms_{\infty}$ by theorem \ref{indthm}.

Integration by parts shows that harmonic asymptotically translation-invariant
spinors are solutions of the Dirac equation $\dirac\psi = 0$.
If $\rho$ is a cut-off function for the cylindrical end on $M$ then
$\rho \harms_{\infty}$ can be identified with a space of
asymptotically translation-invariant spinors on $M$.
It is easy to use index-counting arguments as in \S \ref{lapsub} to
show that
\begin{equation}
\label{eatdiraceq}
\dirac :
\holdad{k+1}(\spinor) \oplus \rho \harms_{\infty} \to \holdad{k}(\spinor)
\end{equation}
has kernel precisely $\harms_{0}$ and that the image is the \ltwoorth{}
complement to $\harms_{+}$. In particular, spinors in $\harms_{0}$
are asymptotic to elements of $\harms_{\infty}$, which gives a boundary
map
\begin{equation*}
B : \harms_{0} \to \harms_{\infty}
\end{equation*}
By the above the image has dimension $\half \dim \harms_{\infty}$.

\begin{rmk}
We can check case by case that $\dim \harms_{\infty}$ is even
\textit{a priori}.
If $n$ is even then %the spinor bundle on the cylinder $X \times \bbr$ is
%isomorphic to $2\spinorx$, so
$\harms_{\infty} \cong 2\harms_{X}$.
%, where $\harms_{X}$ is the space of harmonic spinors on $X$. Similarly
If $n$ is odd then
%the spinor bundle on the cylinder is isomorphic to $\spinorx$, and
$\harms_{\infty} \cong \harms_{X}$ splits as
$\harmsp_{X} \oplus \harmsm_{X}$. If $n-1 \equiv 2 \mod 4$ then
$\harmsp_{X} \cong \overline{\harm^{\spinorm}_{X}}$ for any
spin manifold $X$ of dimension $n-1$.
If $n-1 \equiv 0 \mod 4$ then the index
$\ahat(X)$ of the Dirac operator $\dirac_{+}$ on $X$ must be $0$
because it is a spin cobordism invariant
(see \cite[(7.9) and Theorem 7.10]{lawson89}),
so $\dim \harmsp_{X} = \harmsm_{X}$.
\end{rmk}

If the dimension of $M$ is even then $\harms_{0}$ and $\harms_{\infty}$ split
by chirality and 
$\harmspm_{\infty} \cong \harms_{X}$ because $\spinorpm|_{X} \cong \spinorx$.
Therefore the boundary maps on $\harmspm_{0}$ can be considered as
mapping
\begin{equation*}
B : \harmspm_{0} \to \harms_{X} .
\end{equation*}

\begin{prop}
\label{ltspinsplitprop}
Let $M^{2n}$ be an EAC spin manifold with cross-section $X^{2n-1}$. Then
$\harms_{X}$ can be written as an \ltwoorth{} sum
\begin{equation*}
\harms_{X} = B(\harmsp_{0}) \oplus B(\harmsm_{0}) .
\end{equation*}
\end{prop}

\begin{proof}
We saw above that $\dim B(\harms_{0}) = \half \dim \harms_{\infty} =
\dim \harms_{X}$, so we just need to check \ltwoorth ity.
Define a map
\begin{equation*}
\spinorp \times \spinorm \to TM, \;\: (\psi_{+}, \psi_{-}) \mapsto
\gen{\psi_{+},\psi_{-}}
\end{equation*}
by taking $\gen{\psi_{+}, \psi_{-}}$ to be the metric dual of
the $1$-form $v \mapsto \; \inner{v \psi_{+},\psi_{-}}$, where
$v \psi_{+}$ denotes the Clifford product.
Then
\begin{equation*}
\inner{\gen{\psi_{+},\psi_{-}}, v} \;=\; \inner{v \psi_{+},\psi_{-}}
\; = - \inner{\psi_{+}, v \psi_{-}}
\end{equation*}
for all $v \in TM$, $\psi_{\pm} \in \spinorpm$.
When we identify both of $\spinorpm$ with $\spinorx$ on the cylinder,
Clifford multiplication by $\contrat$ is identified with $\pm id_{\spinorx}$.
Hence for %any exponentially asymptotically translation-invariant
$\psi_{\pm} \in \harmspm_{0}$
\begin{multline*}
\linner{B(\psi_{+}), B(\psi_{-})} \; =
\int_{X} dt(\gen{\psi_{+},\psi_{-}}) \vol_{X} =
\int_{M} \divergence \gen{\psi_{+},\psi_{-}} \vol_{M} \\ = 
\int_{M} \inner{\dirac\psi_{+},\psi_{-}} -
\inner{\psi_{+},\dirac\psi_{-}} \vol_{M}  = 0 . \qedhere
\end{multline*}
\end{proof}

\begin{rmk}
When $\dim M = 8$ the map $\gen{,}$ is non-degenerate by the triality
discussed in \S \ref{sp7sub}.
\end{rmk}




\chapter{Deformations of compact $G$-manifolds}
\label{cptchap}

In this chapter we will consider the deformation theory of compact
\gmfd s, where $G = Spin(7)$, $G_{2}$ or $SU(3)$. We will prove in each case
that the moduli space of torsion-free \gstr s is a smooth manifold.
These results were known previously, due to Joyce in the cases $Spin(7)$ and
$G_{2}$ and Tian and Todorov for $SU(3)$ (and more generally $SU(n)$).
Goto \cite{goto04} also proved smoothness of the moduli
space on compact \gmfd s for any of the Ricci-flat holonomy groups
$G = SU(n)$, $Sp(n)$, $Spin(7)$ and $G_{2}$ in a fairly uniform way.
%Nevertheless some case by case checking is required, so it is
%not a definite simplification.
The arguments used here are geared to make it easier to generalise to
the asymptotically cylindrical case in chapter \ref{eacgchap}.

One of the key ingredients for the proofs is to describe a neighbourhood
in the moduli space by using a local `slice' in the space of torsion-free
\gstr s, containing representatives for all the diffeomorphism classes
in the neighbourhood.
Before going into the details of the deformation problems we will discuss
an elementary slice argument.

\S \ref{cptsec} is a detailed account of an elementary construction
of the moduli space of torsion-free \gtstr s on a compact $7$-manifold
based on an outline by Hitchin \cite{hitchin00}.
In \S \ref{spmodsec} and \S \ref{cymodsec} these arguments
are adapted to construct moduli spaces of torsion-free \spstr s and
$SU(3)$-structures.


\section{Moduli space constructions}
\label{modconstrsec}

A standard technique in constructing moduli spaces is to use slice arguments.
%In this section we consider different approaches to slices.
A model example where slice arguments are explained very carefully
is Ebin \cite{ebin70}, which studies the moduli space of Riemannian metrics
on a compact manifold.
%
In the deformation problems considered in this thesis the moduli spaces are
-- unlike the moduli space of Riemannian metrics -- finite-dimensional.
There is therefore some scope for simplifying Ebin's arguments.
We prove a slice theorem whose hypotheses are less general than Ebin's,
but easier to verify in the problems to which we want to apply it.
This will also make it easier to generalise the slice arguments to
the asymptotically cylindrical case.


\subsection{Functional analysis}

In the technical arguments we will use spaces of Hölder sections rather
than smooth sections. One reason is that this allows application of
Fredholm results for elliptic operators, as used in \S \ref{cylsec}.

The space of $\holda{k}$ Hölder sections of a vector bundle
is a complete normed vector space, i.e. a Banach space.
The inverse function theorem holds for Banach spaces and we will make use
of two of its corollaries: the implicit function theorem and the submersion
theorem.

\begin{thm}[{see e.g. \cite[Theorem XIV $2.1$]{lang93}}]
\label{impfunthm}
Let $X$, $Y$ and $Z$ be Banach spaces, and $U$, $V$ open neighbourhoods of 0
in $X$ and~$Y$. Suppose that $F : U \times V \to Z$ is a smooth function,
$F(0,0) = 0$, and
that $DF_{(0,0)} \vert _{Y} : Y \to Z$ is an isomorphism of $Y$, $Z$ as
Banach spaces. Then there exist connected open neighbourhoods
$U' \subseteq U$ of $0$ in $X$ and $V' \subseteq V$ of $0$ in $Y$ such that
\begin{itemize}
\item there is a smooth map $G : U' \to V'$ such that the set of zeros of
$F$ in $U' \times V'$ is precisely the graph $\{(x,G(x)) : x \in U'\}$,
\item $F$ is an open map on a neighbourhood of $0$ in $X \times Y$.
\end{itemize}
\end{thm}

To ensure that the functions to which we wish to apply the implicit function
theorem are really smooth we use a chain rule result:
composing $\holda{k}$-sections with a smooth fibre-wise map
gives a smooth mapping of Hölder spaces, with the obvious derivative.

%This is fairly intuitive, and we give a direct but dull proof.

\begin{prop}
\label{fibremapprop}
Let $E$, $F$ be normed vector bundles over a compact manifold $M^{n}$,
$U$ an open subbundle of $E$, and $\Psi : U \to F$ a smooth map
that preserves fibres.
Let  $\holda{k}(U) \subseteq \holda{k}(E)$ denote the set of sections of $E$
which are bounded in $\holda{k}$ Hölder norm and take values in $U$.
Then for $k \geq 0$, $\Psi$ defines a map
$\Psi_{k} : \holda{k}(U) \to \holda{k}(F)$ by composition:
if $s \in \holda{k}(U)$ and $p \in M$ then $(\Psi_{k}(s))(p) = \Psi(s(p))$.

Moreover, $\Psi_{k}$ is smooth and its derivative at $r \in \holda{k}(U)$ is
obtained by
applying the `vertical differential' of $\Psi$ at $r$. In symbols
\begin{equation}
\label{adereq}
(D\Psi_{k})_{r}s(p) = (D\Psi_{p})_{r(p)}(s(p)) ,
\end{equation}
where $r \in U_{k}$, $s \in E_{k}$ and $\Psi_{p} : U_{p} \to F_{p}$ is the
restriction of $\Psi$ to the fibre of at $p$.
\end{prop}

%mod3K.tex : proof

%\begin{proof}
%Theorems $9.10$ and $11.3$ of \cite{palais68}. (What about Hölder??)
%\end{proof}

We can define a Banach manifold to be a space with coordinate charts
(mapping to neighbourhoods in a Banach space)
which have smooth transition functions.
Since the inverse function theorem, implicit function theorem and submersion
theorem are local results they have versions for smooth maps between
Banach manifolds.
The space of Hölder sections of a smooth fibre bundle $E$ is a Banach manifold
and the tangent space at a section is the space of Hölder sections of the
bundle of fibre-wise tangents at the section.

We will also need to work with the  is the $\holda{k+1}$
completion $\kpcal{D}$ of the group of diffeomorphisms on a compact
manifold. $\kpcal{D}$ is generated by exponentiated
$\holda{k+1}$ vector fields.
It is a Banach manifold, with $T_{id}\kpcal{D} \cong \holda{k+1}(TM)$.
The product on $\kpcal{D}$ is continuous, but not differentiable.

If $E$ is a smooth fibre bundle on a compact manifold, then
the action of $\kpcal{D}$ on $\holda{k}(E)$ is $C^{0}$ but not $C^{1}$.
The linearisation of the map
\[ \kpcal{D} \times \holda{k}(E) \to \holda{k}(E), \;\:
(\phi, s) \mapsto \phi^{*}s \]
depends on the first derivatives of $s$; the map is therefore $C^{1}$
considered as a map into $\holda{k-1}(E)$, but not into $\holda{k}(E)$.
However

\begin{thm}[cf. {\cite[p. 17]{ebin70}}]
\label{smoothactthm}
Let $E$ be a smooth fibre bundle on a compact manifold.
If $s$ is a smooth section of $E$ then
\[ \kpcal{D} \to \holda{k}(E), \;\:
\phi \mapsto \phi^{*}s \]
is smooth.
\end{thm}


\subsection{Slices}
\label{slicesub}

%In order to construct moduli spaces we will use some standard `slice'
%arguments. A model example where these arguments are explained in
%detail is Ebin \cite{ebin70}. Here we outline a simplified version
%that will be sufficient for our purposes.

%We first explain the simplified slice arguments that we will use
%and then for comparison outline Ebin's approach.

The basic setup in this section will be as follows.
$M^{n}$ is a smooth compact manifold.
$\mathcal{D}$~is the identity component of the group of diffeomorphisms,
i.e. the diffeomorphisms isotopic to the identity.
For concreteness we take $\calc$ to be a space of \gstr s on $M$,
which we consider to be smooth sections of a smooth subbundle
$\gspace \subseteq T^{*}M$, and let $\calx \subseteq \calc$ be the
closed subspace of torsion-free \gstr s.
$\mathcal{C}$~and $\mathcal{X}$ are invariant under the action of
$\mathcal{D}$ by pull-backs.
We wish to describe the moduli space $\mathcal{M} = \mathcal{X}/\mathcal{D}$.

Let $x \in \calx$. The tangent space $T_{x}\calc$ is a space of smooth
sections of the vector bundle of point-wise tangents to $\gspace$ at the
section $x$.
In principle we try to describe a neighbourhood of $x\mathcal{D}$ in
$\mathcal{M}$ in the following way: 
Identify the tangent space $\orbt$ of the
$\mathcal{D}$-orbit at $x$, and find a direct complement $\slt$
for $\orbt$ in~$T_{x}\calc$. We consider a small manifold
$\mathcal{S} \subseteq \calc$ with $T_{x}\mathcal{S} = \slt$
to be a \emph{slice} for the $\mathcal{D}$-action at $x$.
The \emph{pre-moduli space} near $x$ is
$\mathcal{R} = \mathcal{X} \cap \mathcal{S}$.
We then try to show that $\mathcal{R}$
contains a unique representative for each $\mathcal{D}$-orbit near~$x$, and
that the natural map $\mathcal{R} \to \mathcal{M}$ is a homeomorphism
onto a neighbourhood of $x\mathcal{D}$. If so, the problem of
describing $\mathcal{M}$ is reduced to describing~$\mathcal{R}$.

In practice it is hard to implement the argument outlined above working
with just spaces of smooth sections. The space of $C^{\infty}$ sections of a
vector bundle over a compact manifold is a \frch{} space with respect to the
sequence of norms $C^{k}$. $\calc$ can therefore be regarded as a \frch{}
manifold.
The Nash-Moser theorem (see Hamilton \cite{hamilton82})
is a form of the inverse function theorem in the \frch{} setting, but the
hypotheses are too restrictive for it to be useful to us. Moreover, it would be
unwieldy to carry out the technical work of finding the slice working only
in the smooth setting.

We work instead with spaces of $\holda{k}$ Hölder sections
(for some \mbox{$k \geq 1$}, $\alpha \in (0,1)$).
In the cases we consider the Riemannian metric used to define the
$\holda{k}$-norm is defined naturally by the \gstr{} $x \in \mathcal{X}$,
but it could involve some arbitrary choice.

So given $x \in \mathcal{X}$ let $\kcal{C}$, $\kcal{X}$ and $\orbt_{k}$
be the $\holda{k}$ completions of $\calc$, $\mathcal{X}$ and $\orbt$,
respectively. $\kcal{C}$ is a Banach manifold.
Note that $\orbt_{k}$ is the tangent space to the $\kpcal{D}$ orbit at $x$,
where $\kpcal{D}$ is the $\holda{k+1}$
completion of $\mathcal{D}$.
%$\kpcal{D}$ is generated by exponentiated $\holda{k+1}$ vector fields.
%It is a Banach manifold, with $T_{id}\kpcal{D} \cong \holda{k+1}(TM)$.
%The action of $\kpcal{D}$ on $\kcal{C}$ is~$C^{1}$.
%
Usually we take $\slt_{k}$ to be the \ltwoorth{} subspace to $\orbt_{k}$ with
respect to a metric and prove that $T_{x}\kcal{C} = \orbt_{k} \oplus \slt_{k}$
with projections bounded in the $\holda{k}$-norm.
%If this holds for all $k$ then it
%follows that the \ltwoorth{} subspace $\slt$ to $\orbt$ is a direct complement
%of $\orbt$ in $T_{x}\calc$, and that $\slt$ and $\orbt$ are both closed.

If we let $\kcal{S}$ be a neighbourhood of $x$ in the affine space
$x + \slt_{k}$, then the linearisation of
\begin{equation*}
%\label{gkeq}
\kpcal{D} \times \kcal{S} \to \kcal{C}, \;\:
(\phi, s) \mapsto \phi^{*}s
\end{equation*}
at $(id, x)$ is surjective. It would be therefore be convenient to claim that
the submersion theorem ensures that
the image of $\kcal{S}$ in $\kcal{C}/\kpcal{D}$ is open. The map is, however,
continuous but not differentiable near $(id, x)$. Ebin \cite{ebin70} shows
carefully, in the case when $\kcal{C}$ is the space of Riemannian metrics,
that the conclusion that $\kcal{S} \to \kcal{C}/\kpcal{D}$ is open
does hold if one replaces $\kcal{S}$ by another submanifold of $\kcal{C}$
with $T_{x}\kcal{S} = \slt_{k}$, defined as a natural exponentiation of
$\slt_{k}$. Ebin then deduces that $\mathcal{S}$, the smooth elements
of this $\kcal{S}$, has open image in $\mathcal{C}/\mathcal{D}$.
The argument carries over to any situation where the elements of $\mathcal{C}$
define Riemannian metrics.

Since the stabiliser $\cali_{x} \subset \mathcal{D}$ of $x$ is compact
it is easy to arrange for the slice $\cals_{k}$ at $x$ to be invariant under
$\cali_{x}$. One key property of the slice is
that if $\phi^{*}(\kcal{S}) \cap \kcal{S} \not = \emptyset$
for any $\phi \in \kpcal{D}$ then $\phi \in \cali_{x}$
(see \cite[Theorem $7.1$]{ebin70}).
Therefore $\kcal{S}/\cali_{x}$ is in fact homeomorphic to an open set
in $\kcal{C}/\kpcal{D}$, and $\cals/\cali_{x}$ to an open set in
$\mathcal{C}/\mathcal{D}$.

Given $x \in \calx$ let $\calr = \cals \cap \calx$. This is the pre-moduli
space near $x$. It follows immediately that the image of $\calr$ in $\calm$
is open, and that $\calr/\cali_{x}$ is homeomorphic to a neighbourhood of
$x\cald$ in $\calm$.
If one can show that $\mathcal{R}$ is a manifold and that the action
of $\mathcal{I}_{x}$ on $\mathcal{R}$ factors through a suitable finite
group then one may be able to use $\mathcal{R}/\mathcal{I}_{x}$ as a
coordinate chart for $\mathcal{M}$, and deduce that $\mathcal{M}$
is an orbifold.


\subsection{Simplifications}

For the deformation problems considered in this text there are
some simplifying circumstances.

Firstly, the moduli spaces are in a sense `unobstructed'. In many deformation
problems the pre-moduli space $\kcal{R}$ at some $x \in \mathcal{X}$ is
defined as the zero-set in the slice $\cals_{k}$ of some smooth function $F$.
One can then try to apply the implicit function theorem to show
that $\kcal{R}$ is a manifold, with tangent space $\ker DF_{x}$ at $x$.
$\ker DF_{x}$ can be interpreted as `infinitesimal deformations' of $x$.
The implicit function theorem requires the derivative $DF_{x}$ to be
surjective but in general there may be a cokernel. If there is a cokernel
then there could be elements of $\ker DF_{x}$ which
are not tangent to any path in $\kcal{R}$. The cokernel of $DF_{x}$
can therefore be interpreted as obstructions to the integrability of
the infinitesimal deformations.

For the main deformation problems considered in this thesis we are able to
pick the defining function $F$ in such a way that $DF_{x}$ is surjective,
and thus show by direct application of the implicit function theorem that
$\kcal{R}$, as well as a neighbourhood of $x$ in $\kcal{X}$, is a manifold
(a problem where this does not happen is the deformations of Ricci-flat metrics
considered briefly in \S \ref{ricdefsub}).
Finding such $F$ can involve some technical work but provides an essentially
elementary way of showing that the moduli space is unobstructed.

Another simplification is that we will be able to choose the slice $\cals$ so
that the pre-moduli space $\kcal{R}$ consists of smooth elements. In view of
theorem \ref{smoothactthm} this allows us to prove a core slice result
by a naive application of the submersion theorem, without the extra details
in Ebin's setup.

\begin{thm}
\label{slicethm}
Let $x \in \calx$. Choose a Riemannian metric on $M$ and
$k \geq 1$.

Suppose that $T_{x}\kcal{C} = \orbt_{k} \oplus \slt_{k}$ with $\orbt_{k}$ and
$\slt_{k}$ both closed and that $\kcal{S} \subseteq \kcal{C}$ is a submanifold
with $T_{x}\kcal{S} = \slt_{k}$.
Suppose that $x$ has a neighbourhood $\kcal{U}$ in $\kcal{X}$ that is a
submanifold of $\kcal{C}$, and that
$\kcal{R} = \kcal{U} \cap \kcal{S}$ consists of smooth elements.
%, and that $\mathcal{I}_{x}$ is finite-dimensional.
%
%If $\kpcal{V}$ is a sufficiently small neighbourhood of $id$ in $\kpcal{D}$
If $\kcal{S}$ is chosen small enough then
\begin{equation}
\label{gkeq}
%G_{k} :
\kpcal{D} \times \kcal{R} \to \kcal{X}, \;\:
(\phi, s) \mapsto \phi^{*}s
\end{equation}
is a submersion onto a neighbourhood of $x$ in $\kcal{U}$.
\end{thm}

\begin{proof}
Note that $T_{x}\kcal{U}$ must obviously contain $\orbt_{k}$.
Therefore $\kcal{U}$ and $\kcal{S}$ intersect transversely, so $\kcal{R}$ is a
manifold, and $T_{x}\kcal{U} = T_{x}\kcal{R} \oplus \orbt_{k}$.
Because the elements of $\kcal{R}$ are smooth, theorem \ref{smoothactthm}
ensures that (\ref{gkeq}) is a smooth map.
The derivative of (\ref{gkeq}) at $(x, id)$ is
\[ \holda{k+1}(TM) \times T_{x}\kcal{R}, \to T_{x}\kcal{C}, \;\:
(V, y) \mapsto y + \mathcal{L}_{V}x \]
which is %an isomorphism
surjective. %by hypothesis.
The result follows from the %inverse function
submersion theorem for Banach spaces.
\end{proof}

We now add the hypothesis that
\emph{isomorphisms of elements of $\mathcal{X}$ are smooth}, i.e. that if
$x \in \mathcal{X}$ and $\phi^{*}x \in \mathcal{X}$
for a $C^{1}$ diffeomorphism $\phi$ of $M$ then $\phi$ is $C^{\infty}$.
This implies in particular that the stabiliser $\mathcal{I}_{x}$ of
$x \in \mathcal{X}$ in $\kpcal{D}$ consists of smooth maps, i.e. that
$\mathcal{I}_{x} \subseteq \mathcal{D}$.
The assumption is true if elements of $\mathcal{X}$ naturally define a smooth
Riemannian metric.

\begin{thm}[{\cite[Theorem 8]{myers39}}]
\label{palaisthm}
Any isometry of smooth Riemannian manifolds is smooth.
\end{thm}

\noindent
The regularity assumption also holds if elements of $\calx$ define
an almost complex structure, since the Cauchy-Riemann equations are
over-determined elliptic.

\begin{cor}
\label{smoothslicecor}
Let $x \in \calx$. %If $\kcal{R}$ consists of smooth elements
%-- i.e. $\kcal{R} \subseteq \mathcal{X}$ --
If isomorphisms of elements of
$\mathcal{X}$ are smooth then the natural map
$\kcal{R} \to \mathcal{M}$ is open.
\end{cor}

\begin{proof}
If $U \subseteq \kcal{R}$ is open then it follows from theorem \ref{slicethm}
that $U\kpcal{D} \subseteq \kcal{X}$ is open. The assumption that
isomorphisms of elements of $\mathcal{X}$ are smooth implies that
$U\kpcal{D} \cap \mathcal{X} = U\mathcal{D}$. Hence $U\mathcal{D}$ is open
in $\mathcal{X}$, i.e. the image of $U$ in $\mathcal{M}$ is open.
\end{proof}

Like in the previous subsection, if the pre-moduli space $\kcal{R}$ is
$\cali_{x}$-invariant then $\kcal{R}/\cali_{x}$ is homeomorphic to
a neighbourhood of $x$ in $\calm$.

\begin{thm}
\label{sliceinvthm}
Assume the hypotheses of theorem \ref{slicethm}, and that each element of
$\kcal{X}$ naturally defines a Riemannian metric.
If $\kcal{R}$ is $\cali_{x}$-invariant and sufficiently small then
for any $\phi \in \kpcal{D}$
\begin{equation*}
\phi^{*}(\kcal{R}) \cap \kcal{R} \not = \emptyset
\; \Rightarrow \; \phi \in \cali_{x} .
\end{equation*}
\end{thm}

\begin{proof}[Sketch proof.]
See Theorem $7.1(2)$ in Ebin \cite{ebin70} for details.
The key point is to show that the orbit of $x$ under $\kcal{D}$, which
is an immersed submanifold of $\kcal{C}$ by theorem \ref{smoothactthm},
is in fact embedded. This follows from a sequential compactness property:
if $g$ is a Riemannian metric on a compact manifold and $\phi_{i}$ is a
sequence of diffeomorphisms such that $\phi_{i}^{*}g \to g$ then there
is a subsequence which converges to an isometry of $g$.
\end{proof}

\begin{rmk}
If the orbit of $x$ is not embedded then the moduli space $\calm$ is not
Hausdorff.
\end{rmk}

If the action of $\cali_{x}$ on $\kcal{R}$ factors through a finite group
then one can deduce that $\calm$ is an orbifold (smooth local right inverses
of (\ref{gkeq}) can be used to define the transition functions).

Another simplification in the problems considered is that $\calx$ consists
of closed forms. There is therefore a natural projection map
$\pi_{H} : \calx \to H$ to de Rham cohomology,
which is invariant under the action of~$\mathcal{D}$.
In the problems considered
it turns out that the restrictions $\pi_{H} : \kcal{R} \to H$
are embeddings, so the elements of $\kcal{R}$ represent distinct points
in the moduli space. We will therefore not need to worry about ensuring
that the slice at $x$ is invariant under the stabiliser $\mathcal{I}_{x}$,
or about how $\mathcal{I}_{x}$ acts on $\kcal{R}$ (indeed, if the slice is
$\mathcal{I}_{x}$-invariant then $\mathcal{I}_{x}$ acts trivially on
$\kcal{R}$).

\begin{thm}
\label{modmfdthm}
Suppose that $\pi_{H} : \calx \to H$ is a smooth $\mathcal{D}$-invariant
map to a finite-dimensional vector space $H$.
Suppose also for each $x \in \mathcal{X}$ that the hypotheses of theorem
\ref{slicethm} hold, that $\kcal{R}$ is a manifold consisting of smooth
elements and that $\pi_{H} : \kcal{R} \to H$ is an embedding if $\kcal{R}$
is small enough.

Then $\mathcal{M}$ has a unique smooth structure such that
$\pi_{H} : \mathcal{M} \to H$ is an immersion.
\end{thm}

\begin{proof}
Take $x \in \mathcal{X}$.
By corollary \ref{smoothslicecor} the continuous natural map
$i : \kcal{R} \to \mathcal{M}$ is open.
Since $\pi_{H} : \kcal{R} \to H$ is injective so is $i$.
Therefore $i$ is in fact a homeomorphism onto a neighbourhood $U$ of
$x\mathcal{D}$ in $\mathcal{M}$, and $\pi_{H} : U \to H$
is a homeomorphism onto an embedded submanifold.

Since the pre-moduli spaces $\kcal{R}$ are manifolds the maps $i$ 
can be taken as coordinate charts for $\mathcal{M}$, provided that they agree
on overlaps. But on an overlap $U_{1} \cap U_{2}$ both charts define
the unique smooth structure that makes $\pi_{H} : U_{1} \cap U_{2} \to H$
into an embedding, so they agree.

Hence the maps $\kcal{R} \to \mathcal{M}$ can be used to give $\mathcal{M}$
the structure of a smooth manifold,
and it is immediate that $\pi_{H} : \mathcal{M} \to H$ is an immersion.
\end{proof}

Finally, it will be convenient to state the following result about the
stabilisers of points in a $\mathcal{I}_{x}$-invariant slice at $x$.

\begin{prop}
\label{autinvprop}
Assume the hypotheses of theorem \ref{modmfdthm} and
that each element of $\mathcal{X}$ naturally defines a Riemannian metric.
Let $x \in \mathcal{X}$.
If $\calr$ is $\mathcal{I}_{x}$-invariant and small enough then
$\cali_{x}$ acts trivially on $\calr$
and $\mathcal{I}_{y} = \mathcal{I}_{x}$ for all $y \in \mathcal{R}$.
\end{prop}

\begin{proof}
Since $\pi_{H}$ is injective on $\calr$ it is fixed by $\cali_{x}$.
The reverse inclusion $\cali_{y} \subseteq \cali_{x}$ follows from
theorem \ref{sliceinvthm}.
\end{proof}


\section{Deformations of compact $G_{2}$-manifolds}
\label{cptsec}

In this section we show that the moduli space of torsion-free \gtstr s on
a compact manifold $M^{7}$ is smooth.
The result was first published by Joyce \cite{joyce96-I}.
The argument given here is based on Hitchin \cite{hitchin00},
although we provide some more details. 
The argument is adjusted to make it easier to generalise to
the case when $M$ is asymptotically cylindrical, which will be done
in \S \ref{cylgtmodsec}.

Throughout this section $M^{7}$ will denote a compact \gtmfd.
Let $\mathcal{X}$ be the set of torsion-free \gtstr s 
on $M$ and $\mathcal{D}$ the group of diffeomorphisms of $M$ isotopic
to the identity. $\mathcal{D}$~acts on $\mathcal{X}$ by pull-backs,
and the moduli space of torsion-free \gtstr s on $M$ is the quotient
$\mathcal{M} = \mathcal{X}/\mathcal{D}$.
The main theorem of this section is

\begin{thm}[{\cite[Theorem $10.4.4$]{joyce00}}]
\label{maincptthm}
Let $M$ be a compact \gtmfd. Then $\mathcal{M}$ is a smooth manifold
of dimension $b^{3}(M)$, and the natural projection
\begin{equation}
\pi_{H} : \mathcal{M} \to H^{3}(M, \bbr),
\;\: \varphi \mathcal{D} \mapsto [\varphi]
\end{equation}
is a local diffeomorphism.
\end{thm}

%The statement is copied from Theorem $10.4.4$ in \cite{joyce00}.

\begin{rmk}
By the theorem the tangent space of $\mathcal{M}$ at $\varphi\mathcal{D}$ is
identified with $H^{3}(M)$. The torsion-free \gtstr{} $\varphi$ determines
a decomposition (cf. (\ref{hmdecompeq}))
\[ H^{3}(M) = H^{3}_{1}(M) \oplus H^{3}_{7}(M) \oplus H^{3}_{27}(M) . \]
This decomposition is independent of the choice of representative
$\varphi$ for the point $\varphi\mathcal{D} \in \mathcal{M}$ and thus
gives a splitting of the tangent space of $\mathcal{M}$ at $\varphi\mathcal{D}$.
The $H^{3}_{1}(M)$ summand (which has dimension $1$ when $M$ is connected)
corresponds to rescaling the \gtstr.
The $H^{3}_{7}(M)$ summand is the tangent
space to deformations of the \gtstr{} that leave the metric unchanged.
The relationship between the moduli space of \gtstr s and the metrics they
define is elaborated on in \S \ref{bundlesec}.
\end{rmk}

By Poincar\'e duality $H^{3}(M) \cong (H^{4}(M))^{*}$ so
$H^{3}(M) \oplus H^{4}(M)$ is a symplectic vector space in a natural way.

\begin{prop}[{\cite[Proposition $10.4.5$]{joyce00}}]
\label{cptlagembprop}
\begin{equation}
\label{cptlageq}
\defstr \to H^{3}(M) \oplus H^{4}(M),
\;\: \varphi\cald \mapsto ([\varphi], [*_{\varphi}\varphi])
\end{equation}
is a Lagrangian immersion.
\end{prop}

\noindent
This is easy to prove once the deformation theory has been set up.
In \S \ref{cylgtmodsec} we will give a proof for an asymptotically cylindrical
analogue.


\subsection{Plan and notation}

Here we outline the plan of the proof of theorem \ref{maincptthm} and establish
some notation. We use the slice methods explained in \S \ref{modconstrsec}.

Let $\ztre$ be the space of smooth closed 3-forms and
$\mathcal{C}^{3} \subseteq \ztre$ the subset of closed positive 3-forms
(which is open in the uniform topology).
The torsion-free \gtstr s $\calx$ form a closed subset of~$\calc$.
Let $\ztre_{k}$, $\kcal{C}^{3}, \kcal{X}$ be the corresponding spaces of
$\holda{k}$-sections, where $k \geq 1$.

We fix a torsion-free \gtstr{} $\varphi$ on $M$.
We show that $T_{\varphi}\ztre_{k}$ splits as a direct
sum of the tangent space $\orbt_{k}$ of the $\kpcal{D}$-orbit at $\varphi$ and
the \ltwoorth{} complement $\slt_{k}$. We take a small neighbourhood
$\kcal{S}$ of $\varphi$ in the affine space $\varphi + \slt_{k}$ as our slice
for the $\kpcal{D}$-action at $\varphi$.

We apply the implicit function theorem to an appropriate function to deduce
that $\kcal{R} = \kcal{S} \cap \kcal{X}$
is a manifold. Regularity arguments show that the elements of $\kcal{R}$,
which \textit{a~priori} merely have finite $\holda{k}$-norm, are smooth.
Therefore $\kcal{R}$ is the \emph{pre-moduli space $\mathcal{R}$
near $\varphi$} described in \S\ref{modconstrsec}.

The tangent space of $\mathcal{R}$ at $\varphi$ is the space $\harm^{3}$
of harmonic forms. By Hodge theory for compact manifolds
the natural projection map to de Rham cohomology
\begin{equation}
\label{cptderhameq}
\pi_{H} : \harm^{m} \to H^{m}(M), \;\: \alpha \mapsto [\alpha]
\end{equation}
is an isomorphism. Therefore, if $\mathcal{R}$ is taken sufficiently small
then $\mathcal{R} \to H^{3}(M)$ is a diffeomorphism onto its
image. Theorem \ref{maincptthm} then follows by application of
theorem \ref{modmfdthm}.

%For any vector bundle $E$ on $M$ let $\holda{k}(E)$ be the space of
%sections of $E$ which have finite Hölder $\holda{k}$-norm.
Abbreviate $\Lambda^{m}T^{*}M$ to $\Lambda^{m}$ and recall that this splits
into subbundles corresponding to irreducible representations of $G_{2}$.
\begin{equation}
\label{gtformdecompeq}
\begin{aligned}
\Lambda^{2} &= \Lambda^{2}_{7}
\oplus \Lambda^{2}_{14} , \\
\Lambda^{3} &= \Lambda^{3}_{1} \oplus
\Lambda^{3}_{7} \oplus \Lambda^{3}_{27} , \\
%\Lambda^{4} &= \Lambda^{4}_{1} \oplus
%\Lambda^{4}_{7} \oplus \Lambda^{4}_{27} \\
%\Lambda^{5} &= \Lambda^{5}_{7} \oplus
%\Lambda^{5}_{14}
\end{aligned}
\end{equation}
where $\Lambda^{m}_{d}$ is a subbundle of the exterior cotangent bundle
$\Lambda^{m}$ of rank $d$. Its sections $\Omega^{m}_{d}(M)$ are the
`type $d$ $m$-forms'.
These type decomposition, as well as covariant derivatives and Hodge stars,
are defined with respect to the \gtstr{} defined by $\varphi$,
unless otherwise indicated by a subscript (e.g. $\Lambda^{2}_{7,\psi}$
denotes the rank 7 subrepresentation
of $\Lambda^{2}$ defined by the \gtstr{}~$\psi$).


\subsection{The Dirac operator}

We will use Fredholm properties of the Dirac operator associated to a \gtstr{}
$\varphi$ to obtain a direct sum decomposition for the tangent space
to the space of \gtstr s (proposition \ref{ortsumprop}).

Recall that because $G_{2} \subset Spin(7)$ a \gtstr{} on a manifold $M^{7}$
induces a spin structure, a spinor bundle $S$, and the Dirac operator
\[ \dirac : \Gamma(\spinor) \to \Gamma(\spinor) . \]
The point-wise equivalence (\ref{gtspineq}) implies that
\begin{equation}
\label{gtspinorseq}
\spinor \cong \Lambda^{0} \oplus \Lambda^{1} .
\end{equation}
Moreover, we can identify the spin representation $\spinr{7}$ with the
octonions and the natural representation $\bbr^{7}$ of $G_{2}$
with the imaginary octonions in such a way that Clifford multiplication 
is identified with the octonion multiplication.
Since $\varphi$ is the `multiplication table' for the imaginary octonions
(\ref{multtableeq}) we find that,
under the isomorphism (\ref{gtspinorseq}), Clifford multiplication by some
$\alpha \in \Omega^{1}(M)$ corresponds to
%\newsavebox{\boxr}
%\newsavebox{\boxl}
%\newlength{\boxrw}
%\newlength{\boxlw}
%\sbox{\boxr}{$(-\inner{\alpha,\beta}$}
%\sbox{\boxl}{$f \alpha + *(\alpha \wedge \beta \wedge *\varphi)).$}
%\settowidth{\boxrw}{\usebox{\boxr}}
%\settowidth{\boxlw}{\usebox{\boxl}}
\begin{equation*}
\Omega^{0}(M) \oplus \Omega^{1}(M) \to \Omega^{0}(M) \oplus \Omega^{1}(M), \;\:
(f, \beta) \mapsto (-\inner{\alpha,\beta}, \;
f \alpha + *(\alpha \wedge \beta \wedge *\varphi)) . 
%(f, \beta) \mapsto \usebox{\boxr}, \; \usebox{\boxl}
\end{equation*}
Thus the Dirac operator is identified with
\begin{equation}
\label{diracmodeleq}
\Omega^{0}(M) \oplus \Omega^{1}(M) \to \Omega^{0}(M) \oplus \Omega^{1}(M), \;\:
(f, \beta) \mapsto (d^{*}\beta, \: df + *(d\beta \wedge *\varphi)) .
\hspace{28pt} 
%(f, \beta) \mapsto (\usebox{\boxr}, \; \usebox{\boxl}) .
%(f, \beta) \mapsto \makebox[\the\boxrw][r]{$(d^{*}\beta$}, \;
%\makebox[\the\boxlw][l]{$df + *(d\beta \wedge *\varphi)).$}
\end{equation}

We can see directly from (\ref{diracmodeleq}) that the Dirac Laplacian
$\dirac^{2}$ is identified with the Hodge Laplacian on
$\Omega^{0}(M) \oplus \Omega^{1}(M)$. This follows also from remark
\ref{lichrmk}.


\subsection{The slice}
\label{cptslicesub}

$\kpcal{D}$ acts on $\ztre_{k}$ by pull-backs. We want to identify
the tangent space to the orbits of $\kpcal{D}$ and find an appropriate
slice for the action at $\varphi$.

\begin{prop}
\label{cptorbittangprop}
Let $\varphi$ be a \gtstr{} on a compact manifold $M^{7}$ such that
$d\varphi = 0$.
Then the tangent space to the $\kpcal{D}$-orbit at
$\varphi$ is $d\holda{k+1}(\Lambda^{2}_{7})$.
\end{prop}

\begin{proof}
The tangent space to the $\kpcal{D}$-orbit at $\varphi$ consists
of the Lie derivatives of $\varphi$ with respect to $\holda{k+1}$
vector fields $V$.
Since $\varphi$ is closed, $\mathcal{L}_{V}\varphi = d(V\lrcorner \varphi)$.
The forms $V \lrcorner \varphi$ are precisely the sections
of~$\Lambda^{2}_{7}$.
\end{proof}

We find that we can take $\slt_{k} = \harm^{3} \oplus W$
as the complement in $\ztre_{k}$ of the tangent space to the orbit,
where $W$ consists of the exact $3$-forms of type $27$.

\begin{defn}
Let $W = d\holda{k+1}(\Lambda^{2}) \cap \Omega^{3}_{27}(M)$ .
\end{defn}

\begin{prop}
\label{ortsumprop}
Let $M^{7}$ be a compact \gtmfd{} with \gtstr{} $\varphi$, and
$k \geq 0$. Then $d\holda{k+1}(\Lambda^{2})$ can be written as an
\ltwoorth{} direct sum
\begin{equation}
d\holda{k+1}(\Lambda^{2}) = d\holda{k+1}(\Lambda^{2}_{7}) \oplus W
\end{equation}
and the projections are bounded in Hölder norm.
\end{prop}

\begin{proof}
We can identify the spinor bundle $S$ both with
$\Lambda^{0} \oplus \Lambda^{2}_{7}$ and with $\Lambda^{3}_{1\oplus7}$
(shorthand for $\Lambda^{3}_{1} \oplus \Lambda^{3}_{7}$)
so that the Dirac operator
$\dirac : \Gamma(S) \to \Gamma(S)$ is identified with
\begin{equation*}
\Omega^{0}(M) \oplus \Omega^{2}_{7}(M) \to \Omega^{3}_{1 \oplus 7}(M), \;\:
(f, \eta) \mapsto \pi_{1\oplus7}d\eta + *(df \wedge \varphi) .
\end{equation*}
The Dirac operator is elliptic and therefore Fredholm on Hölder spaces
of sections, and its kernel and cokernel consists of harmonic forms.
If $\beta \in d\holda{k+1}(\Lambda^{2})$ then $\pi_{1\oplus7}\beta$
is \ltwoorth{} to $\harm^{3}$, so lies in the image of the Dirac
operator map, i.e.
\[ \pi_{1\oplus7}\beta = \pi_{1\oplus7}d\eta + *(df \wedge \varphi) \]
for some $\eta \in \holda{k+1}(\Lambda^{2}_{7})$,
$f \in \holda{k+1}(\Lambda^{0})$.
Integrating by parts
\begin{equation}
\label{dfzeq}
49\lnorm{df}^{2} = \lnorm{\!*\!d(f\varphi)}^{2} =
\;\linner{*d(f\varphi), \beta-d\eta} \; =
\;\linner{*f\varphi, d(\beta-d\eta)} \; = 0,
\end{equation}
so $df = 0$. Hence the exact form
$\beta - d\eta$ has type $27$, i.e. $\beta - d\eta \in W$.
\end{proof}

So we can indeed take $\slt_{k} = \harm^{3} \oplus W$ as a direct complement
to the tangent space to the diffeomorphism orbit and use a neighbourhood
$\kcal{S}$ of $\varphi$ in the affine space $\varphi + \slt_{k}$ as a slice.

\begin{rmk}
\label{bryantrmk}
For $\beta \in \Omega^{3}_{27}(M)$ there is a linear relation between
$\pi_{7}d\beta$ and $\pi_{7}d^{*}\beta$ (see \cite[Proposition $3$]{bryant03}).
Therefore $\harm^{3} \oplus W$ is precisely the $\holda{k}$ kernel of the
formal adjoint $\pi_{7}d^{*} : \Omega^{3}(M) \to \Omega^{2}_{7}(M)$ of
$d : \Omega^{2}_{7}(M) \to \Omega^{3}(M)$.
\end{rmk}

\begin{defn}
Let $P_{W}$ be the \ltwoorth{} projection $\holda{k}(\Lambda^{3}) \to W$.
\end{defn}

The orthogonal projection
$P_{E} : \holda{k}(\Lambda^{3}) \to d\holda{k+1}(\Lambda^{2})$
is bounded by Hodge decomposition so it follows from proposition
\ref{ortsumprop} that $P_{W}$ is bounded in Hölder norm.


\subsection{The pre-moduli space}

We want to define a function on $\mkcal{C}{3}$ whose zeros near $\varphi$
are precisely the 3-forms giving torsion-free \gtstr s, and
define it in such a way that we can apply the implicit function
theorem to it.

\begin{defn}
For $k \geq 1$ define $F : \mkcal{C}{3} \to W$ by
\begin{equation}
\label{fdefeq}
F(\psi) = P_{W}(*\Theta(\psi)) .
\end{equation}
\end{defn}

It follows from proposition \ref{fibremapprop} and the fact that $P_{W}$ is
a bounded linear map that $F$ is a smooth function between
(open subsets of) Banach spaces.

We wish to show that near $\varphi$ the zeros of $F$ are precisely the
torsion-free \gtstr s.
Recall that, according to theorem \ref{graythm}\ref{torfreeitem},
$\psi \in \mathcal{C}^{3}$ defines a
torsion-free \gtstr{} if and only if $d\Theta(\psi) = 0$.
%$d\Theta(\psi) = 0$ if and only if
Closure of $\Theta(\psi)$ is in turn equivalent to $P_{E}(*\Theta(\psi)) = 0$,
where $P_{E} : \holda{k}(\Lambda^{3}) \to d\holda{k+1}(\Lambda^{2})$ is the
orthogonal projection to the exact part in the Hodge decomposition
\begin{equation*}
%\label{cpthodgeeq}
\holda{k}(\Lambda^{3}) = \harm^{3} \oplus d\holda{k+1}(\Lambda^{2})
\oplus d^{*}\holda{k+1}(\Lambda^{4}) .
\end{equation*}
$W \subseteq d\holda{k+1}(\Lambda^{2})$, so certainly
$P_{E}(*\Theta(\psi)) = 0$ implies that $P_{W}(*\Theta(\psi)) = 0$.
%Thus $d\Theta(\psi) = 0 \Rightarrow F(\psi) = 0$.
It remains to show the converse, so that
we do not `lose any information' by considering zeros of $F$ instead
of $\psi \mapsto d\Theta(\psi)$. 

\begin{prop}
\label{appropriateprop}
For $k \geq 1$ and $\psi \in \mkcal{C}{3}$ sufficiently close to $\varphi$
\[ d\Theta(\psi) = 0 \iff F(\psi) = 0 . \]
\end{prop}

\begin{proof}
If $F(\psi) = 0$ then $P_{E}(*\Theta(\psi))$ is \ltwoorth{} to $W$.
At the same time, theorem \ref{graythm}\ref{pi7item} implies that for
any \gtstr{} $\psi$ with $d\psi = 0$
\[ d\psi = 0 \Rightarrow \pi_{7,\psi}d^{*_{\psi}}\psi = 0 . \]
It follows that $d^{*}(*\Theta(\psi))$ is point-wise orthogonal to
$\Lambda^{2}_{7,\psi}$, so that, integrating by parts,
$P_{E}(*\Theta(\psi))$ is \ltwoorth{} to the tangent space
$T_{\psi} = d\holda{1}(\Lambda^{2}_{7,\psi})$
to the $\cald_{1}$-orbit at~$\psi$. Now the linear map
\begin{equation*}
W \oplus \holda{1}(TM) \to d\holda{1}(\Lambda^{2}), \;\:
(w, V) \mapsto w + d(V \lrcorner \psi)
\end{equation*}
is surjective at $\psi = \varphi$ by proposition \ref{ortsumprop}.
Since it depends continuously on $\psi$,
\[ d\holda{1}(\Lambda^{2}_{7}) = W + T_{\psi} \]
%d\holda{k+1}(\Lambda^{2}_{7,\psi}) \]
for any $\psi$ sufficiently close to $\varphi$. Now the fact
that  $P_{E}(*\Theta(\psi))$
\ltwoorth{} to both $W$ and $T_{\psi}$ %$d\holda{k+1}(\Lambda^{2}_{7,\psi})$
implies $P_{E}(*\Theta(\psi)) = 0$.
\end{proof}

\begin{rmk}
The argument above is a somewhat drier version of Hitchin
\cite{hitchin00}. Hitchin interprets $P_{E}(*\Theta(\psi))$ as the derivative
of the volume functional (i.e. the total volume of $M$ with respect to
the volume form defined by~$\psi$) restricted to the cohomology class
of $\psi$. %\linebreak
The fact that the volume functional is
$\kpcal{D}$-invariant implies that $P_{E}(*\Theta(\psi))$ is
\ltwoorth{} to $d\holda{k}(\Lambda^{2}_{7,\psi})$

In the case when $M$ is asymptotically cylindrical rather than compact
the total volume of a \gtstr{} and $L^{2}$-inner product of asymptotically
translation-invariant forms do not converge, but the proof of
proposition \ref{appropriateprop} works with only notational changes.
\end{rmk}

Now we compute the derivate of $F : \mkcal{C}{3} \to W$.
$\mkcal{C}{3}$ is an open subset of~$\ztre_{k}$, so the tangent space at
$\varphi$ is
$\ztre_{k} = \harm^{3} \oplus d\holda{k+1}(\Lambda^{2}_{7}) \oplus W$.

\begin{prop}
\label{fderprop}
For $k \geq 1$ the derivative
$DF_{\varphi} : \harm^{3} \oplus d\holda{k+1}(\Lambda^{2}_{7}) \oplus W \to W$
is 0 on $\harm^{3} \oplus d\holda{k+1}(\Lambda^{2}_{7})$ and $-id$ on $W$.
\end{prop}

\begin{proof}
Since all the forms in the $\kpcal{D}$-orbit of $\varphi$ define torsion-free
\gtstr s $F$ is always 0 on the orbit. Therefore $DF_{\varphi}$ must be
0 on the tangent space to the orbit, which is $d\holda{k+1}(\Lambda^{2}_{7})$
by \ref{cptorbittangprop}.

Using the expression (\ref{derthetaeq}) and the chain rule (\ref{adereq})
to obtain the derivative of the non-linear map
$\Theta : \mkcal{C}{3} \to \holda{k}(\Lambda^{4})$
we find 
\[ DF_{\varphi} =
P_{W} \circ ({\textstyle\frac{4}{3}}\pi_{1} + \pi_{7} - \pi_{27}) . \]
This is 0 on $\harm^{3}$ since type components of harmonic forms are harmonic
and therefore have no exact part.
On $W$ 
\[ DF_{\varphi} =
P_{W} \circ ({\textstyle\frac{4}{3}}\pi_{1} + \pi_{7} - \pi_{27}) =
P_{W} \circ (-id) = -id . \qedhere \]
\end{proof}

We can now apply the implicit function theorem to $F$ to deduce that its
zero set near $\varphi$ in $\kcal{S}$ is a manifold with tangent space
$\harm^{3}$ at $\varphi$. Hence if $\kcal{R}$ is a sufficiently small
neighbourhood of $\varphi$ in $\kcal{S} \cap \kcal{X}$ then
$\kcal{R}$ is a manifold and $\pi_{H} : \kcal{R} \to H^{3}(M)$
is a diffeomorphism onto its image.
The implicit function theorem shows also that $\varphi$ has a neighbourhood
in $\kcal{X}$ that is a submanifold of $\kcal{C}$.
In order to deduce theorem \ref{maincptthm}
from theorem \ref{modmfdthm} it remains only to show that $\kcal{R}$
consists of smooth forms.


\subsection{Regularity}
\label{cptregsub}

% discuss Hitchin's "proof(?)" ?

We prove that elements of $\kcal{R}$ are smooth
by a boot-strapping method for non-linear PDEs
(for a similar solution to a similar problem see \cite[p. $303$]{joyce00}).

\begin{prop}
\label{cptregprop}
Let $k \geq 1$. If $\kcal{R}$ is taken sufficiently small then its elements
are smooth.
\end{prop}

\begin{proof}
If $\psi = \varphi + \beta$ with $\beta \in \slt_{k}$ then we can write
$d\!*\!d\Theta(\psi)$ as
\[ d\!*\!d\Theta(\psi) =
-dd^{*}({\textstyle\frac{4}{3}}\pi_{1}\beta+\pi_{7}\beta-\pi_{27}\beta)
+ P(\beta, \nabla \beta, \nabla^{2}\beta) + R(\beta, \nabla \beta) , \]
where $-dd^{*}({\textstyle\frac{4}{3}}\pi_{1}+\pi_{7}-\pi_{27})$ is the
linearisation at $\varphi$, $P$ consists of the quadratic terms that involve
the second derivative $\nabla^{2}\beta$ and $R$ consists of the remaining
quadratic terms.
$P$ and $R$ depend only point-wise on their arguments, and furthermore
$P$ is linear in $\nabla^{2}\beta$.

Now note that
$-dd^{*} \circ ({\textstyle\frac{4}{3}}\pi_{1}+\pi_{7}-\pi_{27}) = \triangle$
on $\harm^{3} \oplus W$: on $\harm^{3}$ both vanish, while
$W$ consists of closed sections of $\Lambda^{3}_{27}$.
If $\psi$ is a zero of $F$ near $\varphi$ in
$(\harm^{3} \oplus W) \cap \mkcal{C}{3}$ then $d\Theta(\psi) = 0$, so
\begin{equation}
\label{nonlineareq}
\triangle \beta + P(\beta, \nabla\beta, \nabla^{2}\beta) =
-R(\beta, \nabla\beta) .
\end{equation}
Considering $\beta$ and $\nabla \beta$ to be fixed we can define a second-order
linear differential operator
$A : \zeta \mapsto P(\beta, \nabla \beta, \nabla^{2} \zeta)$. Then 
(\ref{nonlineareq}) can be rephrased as saying that $\zeta = \beta$ is a
solution of the linear second-order PDE
\begin{equation}
\label{lineareq}
(\triangle + A)\zeta = -R .
\end{equation}

If $\psi = \varphi$ then $\triangle + A = \triangle$, which is elliptic.
Ellipticity is open condition so if we take $\kcal{R}$ sufficiently small then
$\triangle+A$ is elliptic for all $\psi \in \kcal{R}$.

Now suppose that $\beta$ is in the Hölder space $\holda{l+1}$ for some
integer $l \geq 1$.
The coefficients of $\triangle+A$ depend algebraically on $\beta$ and
$\nabla \beta$, so have regularity $\holda{l}$. The same goes for the RHS of
(\ref{lineareq}). Therefore by standard regularity results
about linear elliptic operators on compact manifolds
(e.g. Theorem $40$ in the appendix of \cite{besse87})
$\beta$ is $C^{l+2,\alpha}$.

By induction the elements of $\kcal{R}$ are $\holda{l}$ for all $l$,
so they are smooth.
\end{proof}

This concludes the proof of theorem \ref{maincptthm}.


\section{Deformations of compact $Spin(7)$-manifolds}
\label{spmodsec}

In this section we show that the moduli space of torsion-free \spstr s on
a compact \spmfd{} $M^{8}$ is smooth.
This was first proved by Joyce \cite{joyce96-III} and the proof here is
essentially the same. We use the same slice arguments as for the deformations
of compact \gtmfd s, which will carry over without too much trouble to the
asymptotically cylindrical case in \S \ref{eacgchap}.
We will describe in detail only how to set up the slice construction.

Throughout this section $M^{8}$ will denote a compact \spmfd.
Let $\mathcal{X}$ be the set of torsion-free \spstr s 
on $M$ and $\mathcal{D}$ the group of diffeomorphisms of $M$ isotopic
to the identity. $\mathcal{D}$~acts on $\mathcal{X}$ by pull-backs,
and the moduli space of torsion-free \spstr s on $M$ is the quotient
$\mathcal{M} = \mathcal{X}/\mathcal{D}$.
The main theorem of this section is

\begin{thm}
\label{sp7cptthm}
Let $M^{8}$ be a compact \spmfd. Then $\mathcal{M}$ is a smooth manifold
of dimension
\begin{equation}
\dim \defstr = b^{4}_{-}(M) + b^{1}(M) + \ahat(M) ,
\end{equation}
and the natural projection
\begin{equation}
\pi_{H} : \mathcal{M} \to H^{4}(M, \bbr), \;\: \psi \mathcal{D} \mapsto [\psi]
\end{equation}
is an immersion.
\end{thm}

\begin{rmk}
\label{sptermrmk}
As explained in (\ref{cptspindimeq}),
the term $b^{1}(M) + \ahat(M)$ is just a way of writing
the dimension of the space of positive harmonic spinors on the \spmfd{}
in terms of topological invariants. In \S \ref{bundlesec} we will see that this
corresponds to deformations of the \spstr{} that rescale the metric
or leave it unchanged.
\end{rmk}

\subsection{The Dirac operator on $Spin(7)$-manifolds}
\label{sp7diracsub}

Let $\psi$ be a fixed torsion-free \spstr.
Recall from \S \ref{sp7sub} that we consider $Spin(7)$ to act irreducibly
on $\bbr^{8}$ and that (considered a subgroup of $Spin(8)$)
its spin representations are $\spinrm{8} \cong \bbr^{8}$ and
$\spinrp{8} \cong \bbr \oplus \bbr^{7}$ (where $\bbr^{7}$ is the irreducible
vector representation of $Spin(7)$, factoring through $SO(7)$).
Thus the negative spinor bundle $\spinorm \cong TM$.

We abbreviate $\Lambda^{m}T^{*}M$ to $\Lambda^{m}$.
As explained in \S \ref{harmholsub} the exterior forms decompose into
subbundles corresponding to irreducible representations of $Spin(7)$.
\begin{equation}
\label{sp7formdecompeq}
\begin{aligned}
\Lambda^{4}_{+} &= \Lambda^{4}_{1} \oplus
\Lambda^{4}_{7} \oplus \Lambda^{4}_{27} , \\
\Lambda^{4}_{-} &= \Lambda^{4}_{35} , \\
\Lambda^{3} &= \Lambda^{3}_{8} \oplus \Lambda^{3}_{48} , \\
\Lambda^{2} &= \Lambda^{2}_{7} \oplus \Lambda^{3}_{21} .
\end{aligned}
\end{equation}
We can identify subbundles isomorphic to the spinor bundles
\begin{equation}
\label{sp7spinisoseq}
S_{+} \cong \Lambda^{4}_{1\oplus7}, \;\; S_{-} \cong \Lambda^{3}_{8} .
\end{equation}
It is therefore natural to compare the Dirac operator
$\dirac_{-} : \Gamma(S_{-}) \to \Gamma(S_{+})$ with
\begin{equation}
\label{sp7diraceq}
\pi_{1\oplus7} d : \Omega^{3}_{8}(M) \to \Omega^{4}_{1\oplus7}(M) ,
\end{equation}
and if the isomorphisms (\ref{sp7spinisoseq}) are chosen appropriately
then they are in fact equal (cf. \cite[p. 260]{joyce00}).
The easiest way to check this is to note that the values of both
$\diracm$ and $\pi_{1\oplus7}d$ on
$V \in \Gamma(TM) \cong \Gamma(\spinorm) \cong \Omega^{3}_{8}(M)$
depend algebraically on $\nabla V \in \Gamma(End(TM))$ and that
as a $Spin(7)$-representation $End(\bbr^{8})$ contains a unique irreducible
component of rank $1$ and $7$ respectively.

\subsection{The pre-moduli space}

Let $M^{8}$ be a compact \spmfd{} and fix a torsion-free \spstr{} $\psi$
on $M$.
We explain how to choose a slice for the diffeomorphism action at $\psi$
and identify a submanifold that we use as the pre-moduli space of
torsion-free \spstr{} near $\psi$.
Type decompositions of forms, covariant derivatives etc. will be defined
with respect to the \spstr{} $\psi$.

Let $\calcf$ be the space of smooth \spstr s on $M$. Let $\calcfk$ be its
$\holda{k}$ completion for some $k \geq 1$, $\alpha \in (0,1)$,
and let $\kpcal{D}$ be the $\holda{k+1}$ completion of
the group of diffeomorphisms isotopic to the identity. The tangent space
to the $\kpcal{D}$-orbit at $\psi$ is given by the Lie derivatives of $\psi$.
Since $\psi$ is closed, $\call_{V}\psi = d(V \lrcorner \psi)$ for any
vector field $V$, and the forms $V \lrcorner \psi$ are precisely the
sections of $\Lambda^{3}_{8}$. Thus
\[ T_{\psi}(\psi\kpcal{D}) = d\holda{k+1}(\Lambda^{3}_{8}) . \]
Recall from \S \ref{sp7sub} that the tangent space to the space of
\spstr s consists of $4$-forms of type $1$, $7$ and $35$, i.e.
\[ T_{\psi}\calcfk = \holda{k}(\Lambda^{4}_{1 \oplus 7 \oplus 35}) . \]
We use the interpretation of the Dirac operator above to identify the
\ltwoorth{} complement of the tangent space to the orbit.
Let $\harm^{4}_{1\oplus7\oplus35}$ denote the harmonic forms of type
$1$, $7$ and $35$, and let $W$ be the \ltwoorth{} complement
of $\harm^{4}_{35}$ in $\holda{k}(\Lambda^{4}_{35})$.

\begin{prop}
\label{sportsumprop}
Let $M$ be a compact \spmfd{} with \spstr{} $\psi$.
Then
\begin{equation}
\holda{k}(\Lambda^{4}_{1 \oplus 7 \oplus 35}) = 
\harm^{4}_{1 \oplus 7 \oplus 35} \oplus d\holda{k+1}(\Lambda^{3}_{8})
\oplus W.
\end{equation}
\end{prop}

\begin{proof}
Suppose $\beta \in \holda{k}(\Lambda^{4}_{1 \oplus 7 \oplus 35})$ is
\ltwoorth{} to $\harm^{4}_{1 \oplus 7 \oplus 35}$. Then
$\pi_{1\oplus7}\beta$ is \ltwoorth{} to $\harm^{4}_{1 \oplus 7}$, so it lies
in the image of the Dirac operator map (\ref{sp7diraceq}), i.e.
$\beta = \pi_{1\oplus7}d\eta$ for some
$\eta \in \holda{k+1}(\Lambda^{4}_{8})$. Then $\beta - d\eta \in W$.
\end{proof}

This means that we may use a small neighbourhood $\cals$ of $\psi$ in
$\harm^{4}_{1\oplus7\oplus35} \oplus W$ as a slice for the diffeomorphism
action at $\psi$.
Let $\kcal{X} \subset \kcal{C}^{4}$ be the set of torsion-free $\holda{k}$
\spstr s, and
$\calr_{k} = \kcal{X} \cap \cals$.
%Let $\calr_{k} \subseteq \cals$ be the subset of torsion-free \spstr s.
This is the pre-moduli space of torsion-free
\spstr s near $\psi$.

\begin{prop}[{cf. \cite[Proposition 10.7.3]{joyce00}}]
$\psi$ has a neighbourhood in $\kcal{X}$ that is a submanifold of
$\kcal{C}^{4}$.
If $\cals$ is taken small enough then $\calr_{k} \subseteq \cals$
is a submanifold,
with
\[ T_{\psi}\calr_{k} = \harm^{4}_{1\oplus7\oplus35} . \]
\end{prop}

\begin{proof}
By theorem \ref{salamonthm}, $\calr_{k}$ is the zero set of
\begin{equation}
\label{sp7funceq}
d : \cals \to d\holda{k+1}(\Lambda^{4}) .
\end{equation}
The RHS is the space of exact Hölder $5$-forms, which is a closed subspace
of $\holda{k}(\Lambda^{5})$ by Hodge decomposition. Any element can be written
as $d\eta$ for some unique coexact $\eta \in \holda{k+1}(\Lambda^{4})$.
Then $\eta - *\eta$ is anti-self-dual and \ltwoorth{} to the harmonic forms,
i.e. $\eta - *\eta \in W$. Since $W \subseteq T_{\psi}\cals$ it follows that
(\ref{sp7funceq}) has surjective derivative at $\psi$.
If $\cals$ is taken to be small then $\calr_{k}$ is a submanifold by the
implicit function theorem.

On the other hand any closed anti-self-dual form is harmonic, so the
space of closed forms in $T_{\psi}\cals$ is exactly
$\harm^{4}_{1\oplus7\oplus35}$.

Applying the implicit function theorem to 
$d : \kcal{C}^{4} \to d\holda{k+1}(\Lambda^{4})$ shows that a small
neighbourhood of $\psi$ in $\kcal{X}$ is a manifold too.
\end{proof}

In particular the projection to de Rham cohomology
$\pi_{H} : \calr_{k} \to H^{4}(M)$ is an embedding.
Analogously to \S \ref{cptregsub} we can show that $\calr_{k}$
consists of smooth elements, and then apply theorem \ref{modmfdthm}
to deduce that the pre-moduli spaces can be used as coordinate charts
to give the moduli space $\defstr$ a smooth structure.

It remains only to compute
the dimension of the moduli space. This is just the dimension of the
tangent space $\harm^{4}_{1\oplus7\oplus35}$ to the pre-moduli space.
$\dim \harm^{4}_{35} = b^{4}_{-}(M)$, while the discussion in
\S \ref{sp7diracsub} identifies $\harm^{4}_{1\oplus7}$ with the kernel
of the Dirac operator $\dirac_{+} : \Gamma(S_{+}) \to \Gamma(S_{-})$.
Therefore (\ref{cptspindimeq}) yields
\[ \dim \harm^{4}_{1\oplus7} = \ahat(M) + b^{1}(M) . \]
This completes the proof of theorem \ref{sp7cptthm}.


\section{Deformations of compact Calabi-Yau 3-folds}
\label{cymodsec}

In this section we develop some deformation theory for compact Calabi-Yau
$3$-folds, which we need because the cross-sections of asymptotically
cylindrical \gtmfd s are Calabi-Yau $3$-folds.
It is a well-known result, independently due to Tian \cite{tian87} and
Todorov \cite{todorov89}, that the moduli space of Calabi-Yau structures on a
compact irreducible Calabi-Yau $n$-fold is smooth for $n \geq 3$.
Tian and Todorov's arguments are based on Kodaira's deformation theory for
complex structures, but in the case $n = 3$ it is straight-forward to
give a simple proof based on the method of Hitchin \cite{hitchin00}, which we
also employed in \S \ref{cptsec} to deal with deformations of compact \gtmfd s.
This approach is in terms of differential forms and exhibits pre-moduli spaces
that provide useful coordinate charts. Such pre-moduli spaces have also been
obtained by Goto \cite{goto04}.

Let $X^{6}$ be a compact manifold, and $\mathcal{Y}$ the set of
\cystr s $(\Omega, \omega)$ on $X$ in the sense of definition \ref{cy3def}.
Let $\mathcal{D}$ be the group of smooth diffeomorphisms isotopic
to the identity. The moduli space of Calabi-Yau structures on $X$ is
$\mathcal{N} = \mathcal{Y}/\cald$, and there is a natural
projection to the de Rham cohomology
\begin{equation}
\pi_{\mathcal{N}} : \mathcal{N} \to H^{3}(X) \times H^{2}(X), \;\:
(\Omega,\omega)\cald \mapsto ([\Omega], [\omega]) .
\end{equation}

\begin{thm}
\label{cymodthm}
Let $X^{6}$ be a compact connected Calabi-Yau $3$-fold.
The moduli space $\mathcal{N}$ of \cystr s on $X$ is a manifold
of dimension
\begin{equation}
\dim \mathcal{N} = b^{3}(X)+b^{2}(X)-b^{1}(X)-1 ,
\end{equation}
and $\pi_{\mathcal{N}} : \mathcal{N} \to H^{3}(X) \times H^{2}(X)$
is an immersion.
\end{thm}

\begin{rmk}
The definition of a Calabi-Yau 3-fold $X^{6}$ used here allows
$Hol(X)$ to be a proper subgroup of $SU(3)$. If $Hol(X)$ is exactly $SU(3)$
(so $X$ is irreducible as a Riemannian manifold)
then $b^{1}(X) = 0$, and the formula for the dimension simplifies to
$b^{3}(X)+b^{2}(X)-1$.

If $X$ is an irreducible Calabi-Yau manifold then for any Calabi-Yau structure
$(\Omega,\omega)$ on $X$ and $\lambda \in \bbrp$ we can define a torsion-free
product \gtstr{} $\varphi = \Omega + \lambda d\theta \wedge \omega$ on
$X \times S^{1}$. The metric defined by $\varphi$ is the product of the
Calabi-Yau metric on $X$ and the metric on $S^{1}$ with radius $\lambda$
(cf. proposition \ref{cylg2prop}). The moduli space
of such torsion-free product \gtstr s has dimension
\[ \dim \mathcal{N} + 1 = b^{3}(X) + b^{2}(X) = b^{3}(X \times S^{1}) , \]
which equals the dimension of the moduli space of torsion-free \gtstr s
on $X \times S^{1}$ by theorem \ref{maincptthm}.
\end{rmk}


%The theorem is proved in the next subsection.

\subsection{Comparison with known results}
\label{tiansub}

Theorem \ref{cymodthm} agrees with the results of Tian \cite{tian87} and
Todorov \cite{todorov89}. 
The deformation theory for complex manifolds of Kodaira-Spencer shows that
the moduli space of complex structures on $X$ is a manifold provided that
a certain `obstruction space' vanishes. In that case the moduli space is
locally diffeomorphic to~$H^{1}(\mathcal{T})$, the first cohomology of
the sheaf of holomorphic sections of the tangent bundle.

Tian and Todorov show that when $X$ is
a closed connected Calabi-Yau manifold of complex dimension $n \geq 3$
(with holonomy exactly $SU(n)$) then the deformations of the complex structures
are unobstructed. For a compact Calabi-Yau manifold
$H^{1}(\mathcal{T}) \cong (H^{n-1}(\Omega^{1}))^{*}$ by Serre duality,
where $\Omega^{1}$ is the sheaf of holomorphic $1$-forms.
$H^{n-1}(\Omega^{1}) \cong H^{1, n-1}(X)$ by the Dolbeault theorem, so
$\dim_{\bbc} H^{1}(\mathcal{T}) = h^{1,n-1}(X)$.

On the other hand, Yau's solution to the Calabi conjecture shows that for
a fixed complex structure there is a unique Calabi-Yau metric $\omega$
in each Kähler class, and the Kähler classes form an open subset of
$H^{1,1}_{\bbr}(X)$.
Our definition of a Calabi-Yau structure also involves a choice of
normalised holomorphic \mbox{$(n,0)$-form}.
This is determined up to phase by the complex structure and the metric.
Hence the dimension of the moduli space of
Calabi-Yau structures for a fixed complex structure is $h^{1,1}(X) + 1$.

A result of Kodaira \cite[Theorem 15]{kodaira60} shows that given a family of
deformations of the complex structure on a Kähler manifold there is a
corresponding family of deformations of the metric which are Kähler with
respect to the deformed complex structures.  This can be used together with the
above to deduce that the moduli space of Calabi-Yau structures on a complex
$n$-fold is a manifold of dimension
\begin{equation}
\label{tianmodeq}
2h^{1,n-1}(X) + h^{1,1}(X) + 1
\end{equation}
(cf. \cite[\S $6.8$]{joyce00}). On a compact Calabi-Yau $3$-fold $X$ Serre
duality gives $h^{0,2}(X) = h^{0,1}(X)$, and the Hodge decomposition
of $H^{*}(X)$ that
\begin{gather*}
b^{3}(X) = 2h^{1,2}(X) + 2, \\
b^{2}(X) = h^{1,1}(X) + 2h^{0,2}(X) = h^{1,1}(X) + b^{1}(X) .
\end{gather*}
Hence for $n = 3$ the formula (\ref{tianmodeq})
for the dimension of the moduli space agrees with that stated in
theorem \ref{cymodthm}.

More recently, Goto \cite{goto04} gave a proof of the smoothness of the moduli
space of Calabi-Yau structures that does not rely on Kodaira-Spencer theory.
Theorem \ref{cymodthm} is a special case of
\cite[Theorems 4.2.5 and 4.2.6]{goto04} for complex dimension $n = 3$.
Goto studies the problem in terms of differential forms and finds pre-moduli
spaces of the form that we will need later.
By specialising to the case $n = 3$ and employing the slice methods from
\S \ref{modconstrsec} we obtain a simpler proof.

\begin{rmk}
Hitchin \cite{hitchin00} uses elementary methods to construct the moduli
space of torsion-free \slstr s on a compact manifold $X^{6}$.
In \cite[§$4$]{jn1} this is extended to another alternative proof of
theorem \ref{cymodthm}. Hitchin's result implies that the Hölder
completion of the space $\mathcal{Y}_{1}$ of torsion-free \slstr s is a
manifold.
If we let $\ztwo$ be the space of closed Hölder $2$-forms then an implicit
function theorem argument can be set up to prove essentially that the set of
$(\Omega, \omega) \in \mathcal{Y}_{1} \times \ztwo$ satisfying the point-wise
algebraic conditions
$\quart \Omega \wedge \hat\Omega = \sixth \omega^{3}$,
$\Omega \wedge \omega = 0$ is a manifold. Together with the open condition
that $\omega$ defines a positive-definite metric this ensures that 
$(\Omega, \omega)$ is an $SU(3)$-structure.
Theorem \ref{cymodthm} then
follows by slice arguments like those described in section \ref{modconstrsec}.
\end{rmk} 


\subsection{The slice}

Let $X^{6}$ be a compact Calabi-Yau $3$-fold, and $\mathcal{Y}$ the space of
\cystr s on $X$.
To prove theorem \ref{cymodthm} we first identify a slice for the action
of the diffeomorphism group at each $(\Omega,\omega) \in \mathcal{Y}$.

Recall from \S \ref{cydefsub} that we define an $SU(3)$-structure on $X$
in terms of a pair of differential forms $(\Omega,\omega)$, which is a section
of a subbundle
$\lmsu T^{*}X \subset \Lambda^{3}T^{*}X \oplus \Lambda^{2}T^{*}X$.
An  $SU(3)$-structure $(\Omega,\omega)$ defines a metric, and it is
a \cystr{} if and only if it is parallel with respect to the Levi-Civita
connection.

We fix a \cystr{} $(\Omega,\omega)$ on $X$ and study a neighbourhood of
$(\Omega,\omega)$ in the moduli space. Abbreviate $\Lambda^{m}T^{*}X$ to
$\Lambda^{m}$. The \cystr{} determines a type decomposition of $\Lambda^{m}$
modelled on (\ref{sudecompeq}).
\begin{equation}
\label{cydecompeq}
\begin{aligned}
\Lambda^{2} &= \Lambda^{2}_{1} \oplus \Lambda^{2}_{6} \oplus \Lambda^{2}_{8} ,\\
\Lambda^{3} &= \Lambda^{3}_{1\oplus1} \oplus \Lambda^{3}_{6}
\oplus \Lambda^{3}_{12} .
%\\ \Lambda^{4}\bbr^{6} &= \Lambda^{4}_{1}\bbr^{6}
%\oplus \Lambda^{4}_{6}\bbr^{6} \oplus \Lambda^{4}_{8}\bbr^{6}
\end{aligned}
\end{equation}
If $(\sigma,\tau) \in \Lambda^{3} \oplus \Lambda^{2}$ is tangent at
$(\Omega,\omega)$ to the space of $SU(3)$-structures on $X$ then (\ref{leqs})
yields a relation between the type $1$ and $6$ components of $\sigma$
and $\tau$.
\begin{subequations}
\label{leqs2}
\begin{gather}
L_{1}(\sigma,\tau) = \sigma \wedge \hat\Omega - \tau \wedge \omega^{2} \;= 0,\\
L_{2}(\sigma,\tau) = \sigma \wedge \omega + \Omega \wedge \tau = 0 .
\end{gather}
\end{subequations}

Pick $k \geq 1$ and $\alpha \in (0,1)$, and let
\begin{equation*}
\kcal{C} = \{ (\Omega,\omega) \in \holda{k}(\lmsu) : d\omega = 0 \} .
\end{equation*}
Then $\kcal{C}$ is a Banach manifold, and $\mathcal{Y}$ embeds continuously in
$\kcal{C}$. The tangent space at $(\Omega,\omega)$ is
\begin{equation*}
T_{(\Omega,\omega)}\kcal{C} = \{(\sigma,\tau) \in \holda{k}(\Lambda^{3\oplus2})
\; : \; L_{1}(\sigma,\tau) = L_{2}(\sigma,\tau) = 0, \; d\tau = 0 \} .
\end{equation*}
Let $\kpcal{D}$ be the group of $\holda{k+1}$ diffeomorphisms of $X$
isotopic to the identity.
The tangent space to the $\kpcal{D}$-orbit at $(\Omega, \omega)$ is given
by Lie derivatives
\begin{equation*}
\orbt_{k} =
\{ (d(V\lrcorner\Omega), d(V\lrcorner \omega)) : V \in \holda{k+1}(TM) \} .
\end{equation*}

To find a direct complement for $\orbt_{k}$ in $T_{(\Omega,\omega)}\kcal{C}$
we make use of the Dirac operator $\dirac : \spinorm \to \spinorp$.
Considered as real vector bundles $\spinorpm$ are both isomorphic to
$\bbr^{2} \oplus TX$. We can choose isomorphisms
$\spinorp \cong \Lambda^{3}_{1\oplus1\oplus6}$ and
$\spinorm \cong \Lambda^{2}_{6} \oplus \Lambda^{0} \oplus \Lambda^{0}$
such that $\dirac$ is identified with
\begin{equation}
\label{cydiraceq}
\Omega^{2}_{6}(X) \oplus \Omega^{0}(X) \oplus \Omega^{0}(X) \to
\Omega^{3}_{1\oplus1\oplus6}(X), \;\:
(\beta, f, g) \mapsto \pi_{1\oplus1\oplus6}d\beta + d(f\omega) + *d(g\omega) .
\end{equation}
To see that this is possible, observe that $\dirac$ and (\ref{cydiraceq})
are both obtained algebraically from a covariant derivative taking values
in $T^{*}X \otimes (T^{*}X \oplus \bbr^{2})$. The vector bundle $T^{*}X$ is
associated to the $SU(3)$-structure by the standard representation $\bbc^{3}$
of $SU(3)$. Considering $\bbc^{3}$ as a real representation,
$\bbc^{3} \otimes \bbc^{3}$ contains a unique irreducible component isomorphic
to~$\bbc^{3}$. We can therefore make the identifications so that
the $\Omega^{2}_{6}(X) \to \Omega^{3}_{6}(X)$ parts of $\dirac$ and
(\ref{cydiraceq}) agree. It is then easy to ensure that the remaining
components agree too.

\begin{defn}
Let $W \subseteq \holda{k}(\Lambda^{3}_{12})$ be the \ltwoorth{} subspace to the
harmonic forms.
Let $W_{2} \subset W$ be the subspace of exact forms, and $W_{1} \subset W$
its \ltwoorth{} complement.
\end{defn}

\begin{lem}
\label{cysplitlem}
\begin{subequations}
\begin{gather}
\label{cysplit1eq}
\holda{k}(\Lambda^{3}) = \harm^{3} \oplus d\holda{k+1}(\Lambda^{2}_{6}) 
\oplus d\holda{k+1}(\Lambda^{2}_{1}) \oplus *d\holda{k+1}(\Lambda^{2}_{1}) 
\oplus W , \\
\label{cysplit2eq}
d\holda{k+1}(\Lambda^{2}) = d\holda{k+1}(\Lambda^{2}_{6}) 
\oplus d\holda{k+1}(\Lambda^{2}_{1}) \oplus W_{2} .
\end{gather}
\end{subequations}
\end{lem}

\begin{proof}
The Dirac operator is elliptic, so the image of
$\dirac : \holda{k+1}(\spinorm) \to \holda{k}(\spinorp)$ is the \ltwoorth{}
complement to the kernel in $\holda{k}(\spinorp)$. The isomorphism
$\spinorp \cong \Lambda^{3}_{1\oplus1\oplus6}$ identifies the kernel
of the Dirac operator with the harmonic forms in
$\Omega^{3}_{1\oplus1\oplus6}$ (cf. remark \ref{lichrmk}).
Therefore, if $\gamma \in \holda{k}(\Lambda^{3})$ is \ltwoorth{} to $\harm^{3}$
then $\pi_{1\oplus1\oplus6}\gamma$ is in the image of %the Dirac operator
(\ref{cydiraceq}), i.e.
\begin{equation*}
\pi_{1\oplus1\oplus6}\gamma =
\pi_{1\oplus1\oplus6}d\beta + d(f\omega) + *d(g\omega)
\end{equation*}
for some $\beta \in \holda{k+1}(\Lambda^{2}_{6})$,
$f, g \in \holda{k+1}(\Lambda^{0})$.
Then $\gamma - d\beta  - d(f\omega) - *d(g\omega) \in W$.
If $\gamma$ is exact then integrating by parts shows that $dg = 0$.
\end{proof}

The tangent space to the pre-moduli space of \cystr s will turn out to be
the space of harmonic tangents to the space of $SU(3)$-structures.

\begin{defn}
Let
\[ \harm_{SU} = \{ (\sigma,\tau) \in \harm^{3} \times \harm^{2} :
L_{1}(\sigma,\tau) = L_{2}(\sigma,\tau) = 0 \} . \]
\end{defn}

\noindent
The map $\harm^{3} \times \harm^{2} \to \harm^{6} \times \harm^{5},
\;\: (\sigma, \tau) \mapsto (L_{1}, L_{2})$ is surjective, so
\begin{equation}
\label{harmsudimeq}
\dim \harm_{SU} = b^{3}(X)+b^{2}(X)-b^{1}(X)-b^{0}(X).
\end{equation}

\begin{prop}
\begin{equation*}
T_{(\Omega,\omega)}\kcal{C} =
\harm_{SU} \oplus \orbt_{k} \oplus (W \times \{0\}) .
\end{equation*}
\end{prop}

\begin{proof}
Suppose $(\sigma, \tau) \in T_{(\Omega,\omega)}\kcal{C}$.
We wish to show that $(\sigma, \tau)$ lies in the RHS. By subtracting an
element of $\harm_{SU}$ we may assume that $\tau$ is exact and that $\sigma$ is
\ltwoorth{} to~$\harm^{3}$. Using (\ref{cysplit1eq})
and the fact that $d\holda{k+1}(\Lambda^{2}_{6})$ is the space of Lie
derivatives of $\Omega$, we can also subtract an element of $W$ from $\sigma$
to ensure that
\begin{equation*}
\sigma = d(f\omega) + *d(g\omega)
\end{equation*}
for some $f, g \in \holda{k+1}(\Lambda^{0})$.
The conditions (\ref{leqs2}) determine the $\Lambda^{2}_{1}$ and
$\Lambda^{2}_{6}$ components of $\tau$ from $\sigma$, giving
\begin{equation*}
\tau = d^{*}(f\Omega + g\hat\Omega) + \beta ,
\end{equation*}
with $\beta \in \holda{k}(\Lambda^{2}_{8})$. Integrating by parts shows
that $df = dg = 0$. Thus $\beta$ is exact.
But $d\!*\!\beta = -d(\omega \wedge \beta) = 0$, so $\beta = 0$.
\end{proof}

Hence $\slt = \harm_{SU} \oplus (W \times \{0\})$ is a direct complement
in $T_{(\Omega,\omega)}\kcal{C}$ to the tangent space to the diffeomorphism
orbit at $(\Omega,\omega)$.
On a neighbourhood $U$ of the zero section in $T_{(\Omega,\omega)}\kcal{C}$
we may define a bundle map $\exp : U \to \kcal{C}$ such that
$D\exp_{(\Omega,\omega)} = id$. Then $\cals = \exp \slt$ is a smooth submanifold
of $\kcal{C}$ with $T_{(\Omega,\omega)}\cals = \slt$ and can be used
as a slice for the diffeomorphism action at $(\Omega,\omega)$.


\subsection{The pre-moduli space}

Let $(\Omega, \omega) \in \mathcal{Y}$, and let $\calq \subset \cals$ be the
subset of \cystr s in the slice at $(\Omega,\omega)$.
This is the pre-moduli space of \cystr s near $(\Omega,\omega)$.
We will show that $\calq$ is a smooth manifold with
$T_{(\Omega,\omega)}\calq = \harm_{SU}$ and then claim that $\calq$ is
homeomorphic to a neighbourhood in the moduli space $\caln$,

Let $(\Omega',\omega') \in \kcal{C}$. Then $d\omega' = 0$ \textit{a priori},
so lemma \ref{cy3charlem} implies that $(\Omega',\omega')$ is a \cystr{} if
and only if
\[ d\Omega' = d\hat\Omega' = 0 . \]
By lemma \ref{cysplitlem} there are unique
$\beta_{1} \in *d\holda{k+1}(\Lambda^{2}_{1}) \oplus W_{1}$ and
$\beta_{2} \in d\holda{k+1}(\Lambda^{2}) = d\holda{k+1}(\Lambda^{2}_{6})
\oplus d\holda{k+1}(\Lambda^{2}_{1}) \oplus W_{2}$ such that
\[ d\Omega' = d\beta_{1}, \;\: d\hat\Omega' = d\!*\!\beta_{2} . \]
Let $F_{i}(\Omega')$ be the projection of $\beta_{i}$ to $W_{i}$.
We prove that $\calq$ is a smooth manifold by applying the implicit
function theorem to
\[ F : \cals \to W_{1} \oplus W_{2},
\;\: (\Omega', \omega') \mapsto (F_{1}(\Omega'), F_{2}(\Omega')) . \]

\begin{prop}
If $(\Omega',\omega') \in \kcal{C}$ is sufficiently close to $(\Omega,\omega)$
then $(\Omega',\omega')$ is a \cystr{} if and only if $F(\Omega') = 0$.
\end{prop}

\begin{proof}
Clearly $d\Omega' = 0 \Rightarrow F_{1}(\Omega') = 0$ and
$d\hat\Omega' = 0 \Rightarrow F_{2}(\Omega') = 0$.
Conversely, if $F_{1}(\Omega') = 0$ then $d\Omega' = d\beta_{1}$ with
$\beta_{1} = *d(f\omega')$ %\in *d\holda{k+1}(\Lambda^{2}_{1})$.
for some $f \in \holda{k+1}(\Lambda^{0})$. But
%But for any $f \in \holda{k+1}(\Lambda^{0})$
\begin{equation}
\label{omegaortheq}
\linner{\beta_{1}, *d(f\omega')} \; =
\int_{X} \beta_{1} \wedge d(f\omega') =
\int_{X} d\Omega' \wedge f\omega' =
\int_{X} fd(\Omega' \wedge \omega') = 0 .
\end{equation}
If $\omega'$ is sufficiently close to $\omega$ then it follows that
$\beta_{1} = 0$, so $d\Omega' = 0$.

Similarly, $d\hat\Omega' = d\!*\!\beta_{2}$ for some
$\beta_{2} \in d\holda{k+1}(\Lambda^{2})$, and $F_{2}(\Omega') = 0$ implies
that $\beta_{2}$ is \ltwoorth{} to $W_{2}$.
A calculation analogous to (\ref{omegaortheq}) shows that
$\beta_{2}$ is also \ltwoorth{} to $d(f\omega')$
for any $f \in \holda{k+1}(\Lambda^{0})$.

By definition, $\Omega' + i\hat\Omega'$ has type $(3,0)$ with respect to the
almost complex structure defined by $\Omega'$, so
$d\Omega' + id\hat\Omega'$ has no $(1,3)$-component. As $d\Omega' = 0$
it follows that $d\hat\Omega'$ has type $(2,2)$.
Thus for any $V \in \holda{k+1}(TX)$
\begin{equation*}
\linner{\beta_{1}, \call_{V}\Omega'} \; =
\int_{X} \beta_{1} \wedge d(V \lrcorner \Omega') =
\int_{X} d\hat\Omega' \wedge (V \lrcorner \Omega') = 0 .
\end{equation*}
If $(\Omega',\omega')$ is sufficiently close to $(\Omega,\omega)$
then
(cf. \ref{cysplit2eq})
\[ d\holda{k+1}(\Lambda^{2}) = \{ \call_{V}\Omega' : V \in \holda{k+1}(TX) \}
\oplus \{ d(f\omega') : f \in \holda{k+1}(\Lambda^{0}) \} \oplus W_{2} . \]
Hence $\beta_{1}$ is \ltwoorth{} to all of $d\holda{k+1}(\Lambda^{2})$
and therefore vanishes, so $d\hat\Omega' = 0$.
\end{proof}

\begin{prop}
The derivative
$DF_{(\Omega,\omega)} : T_{(\Omega,\omega)}\kcal{C} \to W_{1} \oplus W_{2}$
is $0$ on $\harm_{SU} \oplus \orbt_{k}$ and bijective on $W \times \{0\}$.
\end{prop}

\begin{proof}
The derivative is obviously $0$ on $\harm_{SU}$ and on the tangent space to the
diffeomorphism orbit of $(\Omega,\omega)$. 
It is the identity on $W_{1} \times \{0\}$, so the only non-trivial
part is to check that the derivative is scalar on $W_{2} \times \{0\}$.
To see this it suffices to verify that if $\beta \in \Lambda^{3}_{12}$
then the derivative at $\Omega$ of $\Omega' \mapsto \hat\Omega'$ maps
$\beta$ to $-\!*\!\beta$. This is a point-wise statement, and can be deduced
from proposition \ref{derthetaprop} and remark \ref{cyldualrmk}.
\end{proof} 

Now we can apply the implicit function theorem to $F$ to show

\begin{prop}
\label{cypremodprop}
If the slice $\cals$ is taken sufficiently small then
$\mathcal{Q} \subset \cals$ is a submanifold.
Its tangent space at $(\Omega,\omega)$ is $\harm_{SU}$.
\end{prop}

The implicit function theorem shows also that a small neighbourhood of
$(\Omega, \omega)$ in $\kcal{X}$ is a submanifold of $\kcal{C}$.
Regularity arguments analogous to those in \S \ref{cptregsub} show that
the elements of $\calq$ are smooth.
Theorem \ref{cymodthm} then follows from the slice theorem \ref{modmfdthm}.


\chapter{EAC $G$-manifolds}
\label{eacgchap}


In this chapter we discuss some elementary properties of EAC manifolds with
exceptional holonomy, and their deformation theory.
For the $G_{2}$ case the material is largely covered in~\cite{jn1}.


\section{EAC $G$-structures}

\begin{defn}
\label{cylg2def}
Let $M^{7}$ be a connected oriented manifold with cylindrical ends and
cross-section $X^{6}$.
A \gtstr{} $\varphi$ on $M$ is said to be \emph{exponentially
asymptotically cylindrical} (EAC) if it is exponentially asymptotic
(see definition \ref{expasytensordef}) to a
cylindrical \gtstr{} on $X \times \bbr$ (see definition \ref{exactcylg2def}).
$M$ equipped with a torsion-free EAC
\gtstr{} and the associated metric is called an
\emph{EAC \gtmfd}.
\end{defn}

Recall that by proposition \ref{cylg2prop} a torsion-free cylindrical \gtstr{}
on $X^{6} \times \bbr$ corresponds to a Calabi-Yau structure on $X$. Similarly

\begin{defn}
\label{cylsp7def}
Let $M^{8}$ be a connected oriented manifold with cylindrical ends and
cross-section $X^{7}$.
A \spstr{} $\psi$ on $M$ is said to be EAC if it is exponentially asymptotic
to a cylindrical \spstr{} on $X \times \bbr$.
$M$ equipped with a torsion-free EAC
\spstr{} and the associated metric is called an
\emph{EAC \spmfd}.
\end{defn}

The next theorem implies that an EAC \gtmfd{} or \spmfd{} is not interesting
unless it has a single end.
The result can be proved using the Cheeger-Gromoll line splitting
theorem \cite{cheeger71}, or by more elementary methods
(cf. Salur \cite{salur05}).

\begin{thm}
\label{salurthm}
Let $M$ be an orientable connected asymptotically 
cylindrical Ricci-flat manifold. Then either
$M$ has a single end, i.e. its cross-section $X$ is connected,
or $M$ is a cylinder $X \times \bbr$ with a product metric.
\end{thm}


\subsection{Summary of EAC Hodge theory}
\label{hodgesummary}

For convenience we summarise the notation and key properties of Hodge theory on
an EAC manifold $M^{n}$ with cross-section $X^{n-1}$ that we developed in
\S \ref{lapsub}--\ref{hodgesub}.

\smallskip
\noindent \emph{Harmonic forms.}
The following denote spaces of harmonic $m$-forms on $M$.
\smallskip

\begin{tabular}{ll}
$\harm^{m}_{+}$ & $L^{2}$-integrable \\
$\harm^{m}_{0}$ & bounded \\
$\harm^{m}_{-}$ & not exponentially growing \\
$\harm^{m}_{E}$ & bounded and exact \\
$\harm^{m}_{E^{*}}$ & bounded and coexact \\
$\harm^{m}_{abs}$ & $\harm^{m}_{+} \oplus \harm^{m}_{E^{*}}$ \\
$\harm^{m}_{rel}$ & $\harm^{m}_{+} \oplus \harm^{m}_{E}$
\end{tabular}
\smallskip

In addition $\harm^{m}_{X}$ is the space of harmonic $m$-forms on $X$ and
$\harm^{m}_{\infty}$ the translation-invariant harmonic $m$-forms on
$X \times \bbr$.
\[ \harm^{m}_{\infty} = \harm^{m}_{X} \oplus dt \wedge \harm^{m-1}_{X} . \]

By elliptic regularity any harmonic form on $M$ that is $\holdad{k}$
(for some $k \geq 1$, $\alpha \in (0,1)$ and $\delta \in \bbr$) is smooth.
If $0 < \delta < \epsilon_{1}$ then the harmonic forms in
$\holdax{k}{\pm\delta}(\Lambda^{m})$ lie in $\harm^{m}_{\pm}$.
%
In~$\harm^{m}_{-}$
\[ \begin{array}{ccc}
\harm^{m}_{-} = \ker d + \ker d^{*} & &
\harm^{m}_{0} = \harm^{m}_{abs} + \harm^{m}_{rel} \\
\harm^{m}_{0} = \ker d \cap \ker d^{*} & &
\harm^{m}_{+} = \harm^{m}_{abs} \cap \harm^{m}_{rel} .
\end{array} \]

\noindent \emph{Boundary maps.}
Any bounded harmonic form on $M$ is asymptotic to a translation-invariant form
on the cylinder $X$. This gives a map
\[ B : \harm^{m}_{0} \to \harm^{m}_{\infty} . \]
We may write $B = B_{a} + dt \wedge B_{e}$, where
\[ B_{a} : \harm^{m}_{0} \to \harm^{m}_{X},
\;\: B_{e} : \harm^{m}_{0} \to \harm^{m-1}_{X} . \]
$B_{a}$ is injective on $\harm^{m}_{E^{*}}$ and $0$ on $\harm^{m}_{rel}$,
while $B_{e}$ is injective on $\harm^{m}_{E}$ and $0$ on $\harm^{m}_{abs}$.
Let
\[ \mathcal{A}^{m} = \im B_{a}, \;\: \mathcal{E}^{m-1} = \im B_{e} . \]
Then
\begin{equation}
\label{ae2eq}
\harm^{m}_{X} = \mathcal{A}^{m} \oplus \mathcal{E}^{m}
\end{equation}
is an orthogonal direct sum.

Any $\psi \in \harm^{m}_{-}$ can be
written as $\chi + \alpha + t\beta + dt \wedge \gamma + tdt \wedge \delta$
with $\chi$ exponentially decaying, $\alpha, \beta \in \harm^{m}_{X}$ and
$\gamma, \delta \in \harm^{m-1}_{X}$. Thus we can define a `boundary data' map
\[ BD : \harm^{m}_{-} \to (\harm^{m}_{X})^{2} \oplus (\harm^{m-1}_{X})^{2},
\quad \psi \mapsto \tbtmatrix{\alpha}{\beta}{\gamma}{\delta} . \]
The top row of the $2 \times 2$ array for $BD(\psi)$ represents the linearly
growing part of $\psi$ while the bottom row is the translation-invariant part.
Let
\begin{equation*}
BD_{a} : \harm^{m}_{-} \to (\harm^{m}_{X})^{2},
\; \psi \mapsto \tbomatrix{\alpha}{\beta}, \qquad
BD_{e} : \harm^{m}_{-} \to (\harm^{m-1}_{X})^{2},
\; \psi \mapsto \tbomatrix{\gamma}{\delta} ,
\end{equation*}
be the projections of $BD$ to
$(\harm^{m}_{X})^{2}$ and $(\harm^{m-1}_{X})^{2}$ respectively.
Let $\tcal{A}^{m} = \im BD_{a}$ and $\tcal{E}^{m-1} = \im BD_{e}$. Then
\[ \im BD = \tcal{A}^{m} \oplus \tcal{E}^{m-1} , \]
and $\tcal{A}^{m} \subseteq (\harm^{m}_{X})^{2}$ and
$\tcal{E}^{m-1} \subseteq (\harm^{m-1}_{X})^{2}$ are Lagrangian subspaces.

\ignore{
The boundary data corresponding to various spaces of harmonic forms on $M$
can be summarised in the table below.
\[ \begin{array}{lclc}
\harm^{m}_{-} &
\tbtmatrix{*}{\mathcal{E}^{m}}{*}{\mathcal{A}^{m-1}} &
\harm^{m}_{+} & \tbtmatrix{0}{0}{0}{0} \\ 
\begin{array}{l}\ker d \\\subseteq \harm^{m}_{-} \end{array}& 
\tbtmatrix{\mathcal{A}^{m}}{0}{*}{\mathcal{A}^{m-1}} &
\begin{array}{l}\ker d^{*} \\\subseteq \harm^{m}_{-} \end{array}& 
\tbtmatrix{*}{\mathcal{E}^{m}}{\mathcal{E}^{m-1}}{0} \\
\harm^{m}_{0} &
\tbtmatrix{\mathcal{A}^{m}}{0}{\mathcal{E}^{m-1}}{0} \\
\harm^{m}_{abs} &
\tbtmatrix{\mathcal{A}^{m}}{0}{0}{0} &
\harm^{m}_{rel} &
\tbtmatrix{0}{0}{\mathcal{E}^{m-1}}{0}
\end{array} \]
} % end ignore

%\label{endhodgesummary}

\noindent \emph{Hodge theory.}
The natural map
\[ \pi_{H} : \harm^{m}_{abs} \to H^{m}(M) \]
is an isomorphism. The subset $\harm^{m}_{+} \subseteq \harm^{m}_{abs}$ is
mapped to the compactly supported subspace $H^{m}_{0}(M) \subseteq H^{m}(M)$.
There is also an isomorphism 
\[   
\harm^{m}_{rel} \to H^{m}_{cpt}(M), \;\: \alpha \mapsto
\left\{
\begin{array}{ll}
[\alpha + d\eta(\alpha)] & \textrm{for } \alpha \in \harm^{m}_{+} \\
\partial([B_{e}(\alpha)]) & \textrm{for } \alpha \in \harm^{m}_{E}
\end{array} \right. . \]
Let
\[ A^{m}, E^{m} \subseteq H^{m}(X) \]
be the images of $\mathcal{A}^{m}, \mathcal{E}^{m}$ respectively in $H^{m}(X)$.
$A^{m}$ is the image of $j^{*} : H^{m}(M) \to H^{m}(X)$, and
$E^{m}$ is the orthogonal complement of $A^{m}$.


\subsection{Hodge theory on EAC $G_{2}$-manifolds}
\label{g2topsubsec}

Let $M^{7}$ be an EAC \gtmfd{} with \gtstr{} $\varphi$ asymptotic
to \mbox{$\Omega + dt \wedge \omega$}.
$(\Omega,\omega)$ is a Calabi-Yau structure on the cross-section $X^{6}$
and induces decompositions of the harmonic forms on $X$ as described in
\S \ref{harmholsub}. Maps such as
$\Omega^{2}(X) \to \Omega^{4}(X), \; \sigma \mapsto \sigma \wedge \omega$
are $SU(3)$-equivariant, so by corollary \ref{hodgelapcor} they induce
maps between type components of the spaces of harmonic forms.
We now consider the relation between the type decompositions
and the decomposition (\ref{ae2eq}).

By remark \ref{cyldualrmk}, $*\varphi$ is asymptotic to
$\half \omega^{2} - dt \wedge \hat\Omega$, where $\hat\Omega$ is the unique
$3$-form on~$X$ (with the boundary orientation)
such that $\Omega + i \hat\Omega$ has type $(3,0)$ as discussed
in \S \ref{cydefsub}.


\begin{lem}
\label{aemapslem}
If $\tau \in \mathcal{E}^{m}$ then
$\tau \wedge \Omega \in \mathcal{E}^{m+3}$ and
$\tau \wedge \half \omega^{2} \in \mathcal{E}^{m+4}$.

If $\sigma \in \mathcal{A}^{m}$ then
$\sigma \wedge \Omega \in \mathcal{A}^{m+3}$,
$\sigma \wedge \half \omega^{2} \in \mathcal{A}^{m+4}$,
$\sigma \wedge \omega \in \mathcal{E}^{m+2}$ and
$\sigma \wedge \hat \Omega \in \mathcal{E}^{m+3}$.
\end{lem}

\begin{proof}
If $\chi \in \harm^{m+1}_{E}$ with $B_{e}\chi = \tau$ then
\begin{gather*}
\chi \wedge \varphi \in \harm^{m+4}_{0} \Rightarrow
dt \wedge \tau \wedge \Omega = B(\chi \wedge \varphi)
\in dt \wedge \mathcal{E}^{m+3} , \\
\chi \wedge *\varphi \in \harm^{m+5}_{0} \Rightarrow
dt \wedge \tau \wedge \half \omega^{2} = B(\chi \wedge *\varphi) \in
dt \wedge \mathcal{E}^{m+4} .
\end{gather*}
If $\chi \in \harm^{m}_{0}$ with $B_{a}\chi = \sigma$ then
\begin{gather*}
\chi \wedge \varphi \in \harm^{m+3}_{0} \Rightarrow
\sigma \wedge \Omega + dt \wedge \sigma \wedge \omega = B(\chi \wedge \varphi)
\in \mathcal{A}^{m+3} \oplus dt \wedge \mathcal{E}^{m+2} , \\
\chi \wedge *\varphi \in \harm^{m+4}_{0} \Rightarrow
\sigma \wedge \half \omega^{2} + dt \wedge \sigma \wedge \hat \Omega =
B(\chi \wedge *\varphi) \in
\mathcal{A}^{m+4} \oplus dt \wedge \mathcal{E}^{m+3} . \qedhere
\end{gather*}
\end{proof}

Hodge theory for compact manifolds allows us to identify the de Rham cohomology
of $X$ with the harmonic $m$-forms on $X$. The $L^{2}$-inner product on
$\harm^{m}_{X}$ therefore gives an inner product on $H^{m}(X)$, and
the Hodge star $* : \harm^{m}_{X} \to \harm^{6-m}_{X}$ gives isomorphisms
\[ * : H^{m}(X) \to H^{6-m}(X) . \]
This map is the composition of the metric isomorphism
$H^{m}(X) \cong (H^{m}(X))^{*}$ with Poincar\'e duality
$(H^{m}(X))^{*} \cong H^{6-m}(X)$.
Proposition \ref{aeprop} implies that there is an orthogonal direct sum
\begin{equation*}
\label{aeeq}
H^{m}(X) = A^{m} \oplus E^{m} ,
\end{equation*}
where $A^{m} = j^{*}(H^{m}(M))$ and $E^{m} = *A^{6-m}$.
Set $A^{k}_{d} = A^{k} \cap H^{k}_{d}(X)$ and
$E^{k}_{d} = E^{k} \cap H^{k}_{d}(X)$.

\begin{prop}
\label{aeisoprop}
Let $M^{7}$ an EAC \gtmfd{} with cross-section $X$. Then
\[ H^{2}_{6}(X) = A^{2}_{6} \oplus E^{2}_{6}, \;\;
H^{4}_{6}(X) = A^{4}_{6} \oplus E^{4}_{6} , \]
and the sums are orthogonal. Furthermore
\begin{enumerate}
\item \label{hodgeitem}
$H^{2}_{6}(X) \to H^{4}_{6}(X), [\alpha] \mapsto *[\alpha]$ maps
$A^{2}_{6}$ to $E^{4}_{6}$ and $E^{2}_{6}$ to $A^{4}_{6}$,

\item \label{omitem}
$H^{1}(X) \to H^{4}_{6}(X), [\alpha] \mapsto
[\alpha] \hcup [\Omega]$ maps $A^{1}$ to $A^{4}_{6}$ and
$E^{1}$ to $E^{4}_{6}$,

\item \label{om2item}
$H^{1}(X) \to H^{5}(X), [\alpha] \mapsto
[\alpha] \hcup [\half \omega^{2}]$
maps $A^{1}$ to $A^{5}$ and $E^{1}$ to $E^{5}$.
%\item $H^{1}(X) \to H^{4}_{6}(X), [\alpha] \mapsto
%[\alpha] \hcup [*\Omega]$ maps $A^{1} \to E^{4}_{6}, E^{1} \to A^{4}_{6}$
\end{enumerate}
\end{prop}

\begin{proof}
\ref{hodgeitem} is obvious, since $*$ maps $A^{m} \leftrightarrow E^{6-m}$.

$[\alpha] \mapsto [\alpha] \hcup [\Omega]$ is a bijection
$H^{1}(X) \to H^{4}_{6}(X)$. It maps $A^{1}$ into $A^{4}$ and
$E^{1}$ into $E^{4}$ by lemma \ref{aemapslem}, which implies \ref{omitem}.
It follows that $A^{1} \to A^{4}_{6}$ and $E^{1} \to E^{4}_{6}$
are both surjective and that
$H^{4}_{6}(X)$ splits as $A^{4}_{6} \oplus E^{4}_{6}$.
$H^{2}_{6}(X)$ splits too by~\ref{hodgeitem}.

\ref{om2item} follows from lemma \ref{aemapslem} in the same way.
\end{proof}

When $X$ is Kähler, the complex structure $J$ is parallel
and the Hodge Laplacian on forms commutes with the action of $J$.
Thus $\harm^{1}_{X}$ inherits a complex structure from $X$ in the Kähler case
and the inner product on $\harm^{1}_{X}$ is Hermitian.
We identify $H^{1}(X) \leftrightarrow \harm^{1}_{X}$ as usual to make
$H^{1}(X)$ a Hermitian vector space.

\begin{prop}
\label{realsubprop}
Let $M^{7}$ be an EAC \gtmfd{} with cross-section $X$. Then
\[j^{*} : H^{1}(M) \hookrightarrow H^{1}(X)\]
embeds $H^{1}(M)$ as a
Lagrangian subspace of $H^{1}(X)$ with the symplectic form
$<\cdot, J\cdot>$.
In particular $b^{1}(M) = \half b^{1}(X)$.
\end{prop}

\begin{proof}
By proposition \ref{bochnerprop}, $j^{*}$ maps $H^{1}(M)$ isomorphically to its
image $A^{1}$. The complex structure on $H^{1}(X)$ can be expressed as
\[ J[\alpha] = *([\alpha] \hcup [\half \omega^{2}]) . \]
It follows from proposition \ref{aeisoprop} that $J$ restricts to an
isomorphism $A^{1} \to E^{1}$. Thus
\[ H^{1}(X) = A^{1} \oplus JA^{1} . \qedhere \]
\end{proof}


\subsection{Spinors on EAC $Spin(7)$-manifolds}
\label{eacspspinorsub}

For compact \spmfd s the dimension of the space of positive
harmonic spinors
appears as a term in the dimension of the moduli space of torsion-free
\spstr s and also determines whether a simply-connected compact \spmfd{}
has holonomy exactly $Spin(7)$. For EAC \spmfd s the dimension of the
space of bounded harmonic positive spinors plays a similar role.

Let $M^{8}$ be an EAC \spmfd{} with cross-section $X^{7}$,
and $\spinorpm$ the real spinor bundles on $M$.
As in \S \ref{eacdiracsub} we let
$\harmspm_{0}$ denote the spaces of bounded and decaying
harmonic spinors on $M$ with respect to the Dirac Laplacian $\dirac^{2}$,
and let $\harms_{\infty}$ be the translation-invariant
spinors on the cylinder $X \times \bbr$.
Let $\spinorx$ be the bundle of spinors on the cross-section, and
$\harms_{X}$ the harmonic spinors on $X$.
Recall that the negative spinor bundle $\spinorm$ is isomorphic to $T^{*}M$
and that, because $M$ is scalar-flat, the elements of
$\harmspm_{0}$ and $\harms_{X}$ 
are parallel by the Lichnerowicz formula.

\begin{prop}
\label{dimharmspprop}
On an EAC \spmfd{} $M^{8}$
\[ \dim \harmsp_{0} = 1 + b^{1}(X) - b^{1}(M) . \]
\end{prop}

\begin{proof}
Because harmonic spinors are parallel, the boundary maps
$B : \harmspm_{0} \to \harms_{\infty}$ are injective and it follows from
proposition \ref{ltspinsplitprop} that
\[ \dim \harmsp_{0} = \dim \harms_{\infty} - \dim \harmsm_{0} . \]
Now $\harmsm_{0} \cong \harm^{1}_{0}$, while $\harms_{\infty} \cong
\harm^{0}_{X} \oplus \harm^{1}_{X}$. If $X$ is connected (i.e. $M$ has a
single end) then $\dim \harm^{0}_{X} = 1$.
Corollary \ref{degonehodgecor} implies that $\dim \harm^{1}_{0} = b^{1}(M)$,
so the formula holds.
If $X$ is not connected then $M$ is a product cylinder, and the result
is easy.
\end{proof} 

We can also deduce a splitting result similar to proposition \ref{aeisoprop}.
Fix a torsion-free EAC \spstr{} $\psi$ on $M$.
Denote bounded harmonic $m$-forms of type $d$ on $M$ by $\harm^{m}_{0,d}$.
The asymptotic limit of $\psi$ is $*_{\varphi}\varphi + dt \wedge \varphi$,
where $\varphi$ is a torsion-free \gtstr{} on the cross-section $X^{7}$
and $*_{\varphi}$ is the induced Hodge star (see proposition \ref{cylspprop}).
$\varphi$ induces type decompositions of $\Omega^{*}(M)$.
Let $\cala^{4}_{d} = \cala^{4} \cap \Omega^{4}_{d}(X)$ and
$\cale^{4}_{d} = \cale^{4} \cap \Omega^{4}_{d}(X)$.
Clearly $\cala^{4}_{1} \cong \bbr(*_{\varphi}\varphi)$,
and $\cale^{4}_{1} = 0$ unless $M$ is a product cylinder.
In any case $\harm^{4}_{X,1} = \cala^{4}_{1} \oplus \cale^{4}_{1}$,
but it is not clear \textit{a priori} that $\harm^{4}_{X,7}$ and
$\harm^{4}_{X,27}$ split in a similar way.

\begin{prop}
\label{spaesplitprop} 
Let $M$ be an EAC \spmfd. The boundary maps give isomorphisms
\begin{gather*}
B_{a} : \harm^{4}_{0,7} \isomap \cala^{4}_{7} , \\
B_{e} : \harm^{5}_{0,8} \isomap \cale^{4}_{1} \oplus \cale^{4}_{7} ,
\end{gather*}
and $\harm^{4}_{X}$ splits as
\[ \harm^{4}_{X} = \cala^{4}_{1} \oplus \cala^{4}_{7} \oplus \cala^{4}_{27}
\oplus \cale^{4}_{1} \oplus \cale^{4}_{7} \oplus \cale^{4}_{27} . \]
\end{prop}

\begin{proof}
The boundary maps are injective since both $\harm^{4}_{0,7}$ and
$\harm^{5}_{0,8}$ consist of parallel forms.
%Let $\cala^{4}_{1\oplus7} = \cala^{4} \cap \Omega^{4}_{1\oplus7}(X)$.
The dimensions of the images
$B_{e}(\harm^{5}_{0,8}) \subseteq \cale^{4}_{1\oplus7}$ and
$B_{a}(\harm^{4}_{0, 1\oplus7}) \subseteq \cala^{4}_{1\oplus7}$
are therefore the same as $B(\harmsp)$ and $B(\harmsm)$, respectively.
Dimension-counting using proposition \ref{ltspinsplitprop} tells us that
equality holds and that
$\harm^{4}_{X,1\oplus7} = \cala^{4}_{1\oplus7} \oplus \cale^{4}_{1\oplus7}$.

Since $\harm^{4}_{X,1}$ splits so does $\harm^{4}_{X,7}$ and hence also
$\harm^{4}_{X,27}$.
\end{proof}


\subsection{A topological criterion for $Hol = G_{2}$}
\label{topcondsub}

In this section we obtain a topological criterion for when the holonomy
group of an EAC \gtmfd{} $M^{7}$ is precisely
$G_{2}$ and not a proper subgroup.
As stated in corollary \ref{g2holcor} the holonomy group of a metric
defined by a torsion-free \gtstr{} is always a subgroup of $G_{2}$.
For compact \gtmfd s theorem \ref{cpttopcondthm} states that a necessary and
sufficient condition for full holonomy $G_{2}$ is that the manifold
has finite fundamental group.
We review the proof from Joyce \cite[p. 245]{joyce00}
and then generalise the result to the EAC case.

We first state two lemmas.
The first one is due to Cheeger and Gromoll, and is a corollary of their
line splitting theorem \cite{cheeger71}.
Note that all covering spaces
will be presumed to be equipped with the Riemannian metric pulled back
by the covering map.
In particular all covering maps will be local isometries, and all covering
transformations are isometries.

\begin{lem}[{\cite[Corollary $6.67$]{besse87}}]
\label{riccoverlem}
Let $M$ be a compact Ricci-flat Riemannian manifold. Then $M$ has a finite
cover isometric to a Riemannian product $T^{k} \times N$,
where $T^{k}$ is a flat torus (of dimension $k$ possibly $0$)
and $N$ is compact and simply-connected.
\end{lem}

\begin{lem}
\label{covredlem}
Let $G$ be a closed connected subgroup of $SO(n)$ and let $M^{n}$
be a connected Riemannian manifold with $Hol(M) \subseteq G$.
Then $Hol(M) = G$ if and only if $Hol(\tildm) = G$ for any
cover $\tildm$ of~$M$.
\end{lem}

\begin{proof}
Let $\unim$ be the universal cover of $M$. For any cover $\tildm$ of $M$
\[ Hol(\unim) \subseteq Hol(\tildm) \subseteq Hol(M) . \]
To prove the lemma it therefore suffices to show that if $Hol(M) = G$ then
$Hol(\unim) = G$. It is easy to see
that $Hol(\unim)$ is in fact the restricted holonomy group $Hol^{0}(M)$
of $M$, and by \cite[Corollary 10.48]{besse87} this is the identity component
of $Hol(M)$. Since $G$ connected it follows that
if $Hol(M) = G$ then $Hol^{0}(M) = G$.
\end{proof}

Now we prove the previously stated result that a compact \gtmfd{} has holonomy
exactly $G_{2}$ if and only if it has finite fundamental group.

\begin{proof}[Proof of theorem \ref{cpttopcondthm}.]
If $\pi_{1}(M)$ is infinite then let $\tildm = T^{k} \times N$ be a finite
cover of $M$ as given by lemma \ref{riccoverlem}. Clearly $\pi_{1}(M)$
infinite implies $k > 0$, so $Hol(\tildm) = Hol(N) \not = G_{2}$.
Hence $Hol(M)$ is a proper subgroup of $G_{2}$.

If $\pi_{1}(M)$ is finite consider the universal cover $\unim$ of $M$.
From theorem \ref{bergerthm} we find that up to conjugacy the only proper
subgroups of $G_{2}$ that can be
the holonomy group of a simply-connected Riemannian
manifold are $1, SU(2)$ and~$SU(3)$. Thus if $Hol(\unim)$ is not $G_{2}$ then
it fixes at least one vector in its action on $\bbr^{7}$. By proposition
\ref{simpleholprop}, there must exist a non-zero parallel $1$-form %$\phi$
on $\unim$ which, by corollary \ref{bochnercor}, is harmonic.
But since $\unim$ is compact there
is an isomorphism $\harm^{1} \to H^{1}(\unim)$ between harmonic forms and
de Rham cohomology. $\unim$ is simply-connected, so $b^{1}(\unim) = 0$ and
there can be no non-trivial harmonic $1$-forms on~$\unim$. Hence
$Hol(M) = Hol(\unim) = G_{2}$.
\end{proof}

When stating an EAC analogue of theorem \ref{cpttopcondthm} we need to take
into account that $M^{7}$ could be a product cylinder $X^{6} \times \bbr$.
If $X$ is an irreducible Calabi-Yau manifold then the fundamental group is
finite, but $Hol(M)$ is of course contained in $SU(3)$. The correct statement
in the EAC setting is:

\begin{thm}
\label{topcondthm}
Let $M^{7}$ be an EAC \gtmfd.
Then $Hol(M) = G_{2}$ if and only if the fundamental group $\pi_{1}(M)$ is
finite and neither $M$ nor any double cover of $M$ is homeomorphic
to a cylinder.
\end{thm}

To prove the theorem we use that by proposition \ref{bochnerprop} the space
of parallel $1$-forms on $M$ is exactly~$\harm^{1}_{0}$ and
that, by corollary \ref{degonehodgecor}, the natural map
$\harm^{1}_{0} \to H^{1}(M)$ from bounded harmonic forms to de Rham cohomology
is an isomorphism when $M$ has a single end.
%
In order to obtain the information we need about the structure of the
fundamental group of $M$ in the asymptotically cylindrical case, we quote
a result about groups of polynomial growth.

\begin{defn}
Let $G$ be a finitely generated group and $S$ a finite set of
generators of~$G$.
Let $N(k)$ be the number of elements of $G$ that can be written as a product
of less than $k$ elements of $S$. If $N(k)$ is bounded by a polynomial in $k$
then $G$ is said to have \emph{polynomial growth}.
\end{defn}

This definition is independent of the choice of finite generating set $S$.
The next result is the main theorem in Gromov \cite{gromov81} (see also
Kleiner \cite{kleiner07} for a shortened proof).

\begin{thm}
\label{gromovthm}
If $G$ is finitely generated and has polynomial growth then $G$ has a nilpotent
subgroup of finite index.
\end{thm}

\begin{cor}
\label{nilpotcor}
Let $M$ be an asymptotically cylindrical manifold with non-negative Ricci
curvature. Then the fundamental group $\pi_{1}(M)$ has a nilpotent subgroup
of finite index.
\end{cor}

\begin{proof}[Sketch proof.]
$M$ is homotopy equivalent to a compact manifold with boundary so $\pi_{1}(M)$
is finitely generated. Volume comparison arguments show that the volume of
balls in the universal cover of $M$ grows polynomially and this can be used
to deduce that $\pi_{1}(M)$ has polynomial growth
(see Milnor \cite{milnor68} for details).
Hence theorem \ref{gromovthm} applies.
\end{proof}

\begin{lem}
\label{topcaseslem}
Let $M$ be a Ricci-flat EAC manifold.
\begin{enumerate}
\item \label{cylcovercase}
If $M$ has a finite normal cover homeomorphic to a cylinder
then $M$ or a double cover of $M$ is homeomorphic to a cylinder.
\item \label{inftycase}
If $\pi_{1}(M)$ is infinite then $M$ has a finite cover $\tildm$ with
$b^{1}(\tildm) > 0$.
\end{enumerate}
\end{lem}

\begin{proof}
\begin{enumerate}
\item If $\tildm$ is a finite normal cover of $M$ homeomorphic to a cylinder
then it is isometric to a product cylinder $Y \times \bbr$.
%by theorem \ref{salurthm}.
$M$ is a quotient of $Y \times \bbr$ by a finite group $A$ of isometries. The
isometries are products of isometries of $Y$ and of $\bbr$ (since they
preserve the set of globally distance minimising geodesics
$\{\{y\} \times \bbr : y \in Y\}$). The elements of $A$ have
finite order, so they must act on the $\bbr$ factor as either the identity or as
reflections. Therefore the  subgroup $B \subseteq A$ which acts as the
identity on $\bbr$ is either all of~$A$, in which case $M$ is the cylinder
$(Y/B) \times \bbr$, or a normal subgroup of index~$2$, in which case
$(Y/B) \times \bbr$ is a cylindrical double cover of~$M$. 

\item
Let $G_{0} \subseteq \pi_{1}(M)$ be a nilpotent subgroup of finite index.
$G_{0}$ is soluble, so the derived series $G_{i+1} = [G_{i},G_{i}]$ reaches
$1$. Therefore there is a largest $i$ such that $G_{i} \subseteq \pi_{1}(M)$
has finite index.
Let $\tildm$ be the cover of $M$ corresponding to $G_{i} \subseteq \pi_{1}(M)$.
%Since $\tildm$ is a finite cover of $M$ it is an EAC manifold.
$G_{i}/G_{i+1}$ is an infinite Abelian group, so has non-zero rank. Hence
\[ b^{1}(\tildm) =
rk \left(\pi_{1}(\tildm)/[\pi_{1}(\tildm),\pi_{1}(\tildm)] \right)
= rk(G_{i}/G_{i+1}) > 0 . \qedhere \]
\end{enumerate}
\end{proof}

\noindent The lemma implies that if $M$ is an EAC \gtmfd{} then one of 4
possible cases holds:

\begin{enumerate}

\item \label{cylcase}
$\pi_{1}(M)$ is finite and $M$ is homeomorphic to a cylinder. Then
$M$ is isometric to $Y \times \bbr$ for some compact Calabi-Yau manifold
$Y^{6}$. The same arguments as in the proof of theorem \ref{cpttopcondthm}
show that the holonomy of $Y$ cannot be a proper subgroup of $SU(3)$.
Thus $Hol(M) = SU(3)$.

\item \label{dcylcase}
$\pi_{1}(M)$ is finite, $M$ has a single end and has a double cover
homeomorphic to a cylinder. This double cover has holonomy $SU(3)$
so $Hol(M) \not = G_{2}$.

\item \label{infcase}
$\pi_{1}(M)$ is infinite. Then $M$ has a finite cover $\tildm$ with
$b^{1}(M) > 0$. By theorem \ref{absderhamthm} together with proposition
\ref{bochnerprop} there is a parallel $1$-form on~$\tildm$,
so $Hol(\tildm) \subseteq SU(3)$ and $Hol(M) \not = G_{2}$.

\item \label{irredcase}
$\pi_{1}(M)$ is finite and neither $M$ nor any double cover of $M$ is
homeomorphic to a cylinder. Then the universal cover $\tildm$ of $M$ is an
EAC \gtmfd{} with a single end.
The only proper subgroups of $G_{2}$ that can be the holonomy group of
a complete simply-connected manifold are $1$, $SU(2)$ and $SU(3)$,
so if $Hol(\tildm)$ is not $G_{2}$ then there is a parallel vector field
on $\tildm$. But $b^{1}(\tildm) = 0$, so by corollary \ref{degonehodgecor} and
proposition \ref{bochnerprop} there are no parallel $1$-forms on~$\tildm$.
Hence $Hol(M) = G_{2}$. 
\end{enumerate}

\smallskip

$Hol(M)$ is exactly $G_{2}$ only in case \ref{irredcase},
so we have proved theorem \ref{topcondthm}.
We will find examples of EAC manifolds with holonomy exactly
$G_{2}$ in chapter \ref{pullchap} and
examples of the cases \ref{dcylcase} and \ref{infcase} are provided below.
%
We can also show that an EAC \gtmfd{} has holonomy exactly $G_{2}$ if
its cross-section (which is a Calabi-Yau $3$-fold) has holonomy exactly
$SU(3)$.

\begin{cor}
Let $M^{7}$ be an EAC \gtmfd{} with cross-section $X$ and suppose that $M$
is not finitely covered by a cylinder.
If $Hol(X) = SU(3)$ then $Hol(M) = G_{2}$.
\end{cor}

\begin{proof}
Suppose that $Hol(M)$ is a proper subgroup of $G_{2}$.
Then $\pi_{1}(M)$ is infinite, so $M$ has a finite cover $\tildm$ with
$b^{1}(\tildm) > 0$.
Let $\tilde X$ be the cross-section of $\tildm$. By proposition
\ref{realsubprop} $b^{1}(\tilde X) = 2b^{1}(\tildm) > 0$, so $Hol(\tilde X)$
is a proper subgroup of $SU(3)$. $\tilde X$ is a finite cover of $X$, so
it follows that $Hol(X)$ is a proper subgroup of $SU(3)$.
\end{proof}

Proposition \ref{realsubprop} also implies that if $b^{1}(X) > 0$ then
$b^{1}(M) > 0$, so $M$ is reducible. It is, however, not true in general
that $M$ is reducible if $X$ is reducible. In \S \ref{simplesub}\ref{ex2item}
we will see an example of an EAC \gtmfd{} with holonomy exactly $G_{2}$, whose
cross-section is a quotient of $T^{2} \times K3$ with $b^{1} = 0$.

\begin{ex}
There exist EAC manifolds $W^{6}$ with holonomy
precisely $SU(3)$ (see Kovalev \cite[Theorem $2.7$]{kovalev03}). Then we can
define a torsion-free \gtstr{} on the product $W \times S^{1}$ as in
proposition \ref{cylg2prop}. Of course $Hol(W \times S^{1})$ is not all of
$G_{2}$, but
just $SU(3)$. Furthermore $b^{1}(W \times S^{1}) > 0$,
so by theorem \ref{topcondthm} \emph{no} EAC
\gtstr{} on $W \times S^{1}$ can have holonomy exactly~$G_{2}$.
\end{ex}

\begin{ex}
\label{cyinvex}
Let $Y \subset \bbc P^{5}$ be the complex projective variety defined by the
equations $\sum X^{2}_{i} = 0, \;\sum X^{4}_{i} = 0$. $Y$ is a complete
intersection of hypersurfaces, so is a smooth complex 3-fold.
As described in \cite[p. 40]{joyce00} the adjunction formula can be used
to show that the first Chern class $c_{1}(Y)$ vanishes, and
the Lefschetz hyperplane theorem, stated in the form \cite[Theorem I]{bott59},
can be applied to show that $\pi_{1}(Y) = 1$.

Since the polynomials defining $Y$ are real,
the complex conjugation map on $\bbc P^{5}$  restricts to an
involution $a : Y \to Y$. $a$ is anti-holomorphic, and since the
defining polynomials have no roots over $\bbr$ the involution has no
fixed points.

Let $\omega_{FS}$ be the restriction of the Fubini-Study Kähler form to~$Y$.
The involution is anti-symplectic, i.e. $a^{*}\omega_{FS} = -\omega_{FS}$.
Since $c_{1}(Y) = 0$, Yau's solution to
the Calabi conjecture \cite{yau78} %(theorem \ref{cythm})
implies that there is a unique Kähler form $\omega$ in the cohomology class of
$\omega_{FS}$ such that the corresponding metric is Ricci-flat, making
$Y$ into a Calabi-Yau manifold. The cohomology class of $\omega_{FS}$
is preserved by $-a^{*}$ and $-a^{*}\omega$ is a Kähler form defining a
Ricci-flat metric, so the uniqueness part of Yau's theorem implies that
$-a^{*}\omega = \omega$ (cf. \cite[Proposition $15.2.2$]{joyce00}).

Pick a global holomorphic non-vanishing $3$-form $\phi$ on~$Y$.
$\overline{a^{*}\phi}$ is also holomorphic and therefore equal to
$\lambda^{2}\phi$ for some $\lambda \in \bbc$.
Replacing $\phi$ with $\lambda\phi$ we can assume
without loss of generality that $\lambda = 1$. Then $\Omega = \re \: \phi$ is
preserved by $a^{*}$.
We can rescale $\Omega$ to ensure that $(\Omega, \omega)$ is a
\cystr{} in the sense of definition \ref{cy3def}.

Now define a \gtstr{} on $Y \times \bbr$ by $\varphi =
\Omega + dt \wedge \omega$. By proposition \ref{cylg2prop},
$\varphi$ is torsion-free.
Let $M$ be the quotient of $Y \times \bbr$ by $a \times (-1)$.
$M$ has a single end and $\pi_{1}(M)$ has order 2.
$(a \times (-1))^{*}\varphi = a^{*}\Omega + (-dt)\wedge a^{*}\omega = \varphi$,
so $\varphi$ induces a well-defined torsion-free \gtstr{} on~$M$.
\end{ex}


\subsection{A topological criterion for $Hol = Spin(7)$}
\label{sptopcondsub}

As stated in corollary \ref{sp7holcor} the holonomy group of a metric
defined by a torsion-free \spstr{} is always a subgroup of $Spin(7)$.
In this section we obtain a topological criterion for when the holonomy
group of an asymptotically cylindrical \spmfd{} $M^{8}$ is precisely
$Spin(7)$ and not a proper subgroup.
For a compact \spmfd{} theorem \ref{spcpttopcondthm} states that
$Hol(M) = Spin(7)$ if and only if $\pi_{1}(M) = 1$ and $\ahat(M) = 1$.
We sketch the proof from Joyce \cite[\S 10]{joyce00} since the argument
for the EAC case is similar, even though the criterion obtained looks
different.

\begin{proof}[Proof of theorem \ref{spcpttopcondthm}]
Recall that any harmonic spinor on a compact \spmfd{} is parallel
and that the negative spinor bundle is isomorphic to $T^{*}M$.
As in the $G_{2}$ case, any compact manifold $M^{8}$ with holonomy exactly
$Spin(7)$ must have finite fundamental group. Then $b^{1}(M) = 0$,
so there are no non-zero harmonic negative spinors.
Because $Spin(7)$ fixes a unique line in the positive spin representation
$\spinrp{8}$ the dimension of the space of harmonic positive spinors
is $1$. Hence the index $\ahat(M)$ of the Dirac operator is $1$.
The same is true for any finite cover of $M$. Since $\ahat(M)$ is a
characteristic class it follows that $M$ can have no non-trivial finite cover,
so $M$ is simply-connected.

Conversely, if $M^{8}$ is a simply-connected \spmfd{} then Berger's list
(theorem \ref{bergerthm}) implies that the only possible holonomy groups of $M$
other than $Spin(7)$ are $SU(4)$, $Sp(2)$ and $SU(2) \times SU(2)$. These
fix a $2$-, $3$- and $4$-dimensional subspace of $\spinrp{8}$, respectively.
For a compact simply-connected \spmfd{} we can therefore use the
dimension $\ahat(M)$ of the space of parallel positive spinors to
distinguish between the four possible holonomy groups.
\end{proof}

In the EAC case it is still true that bounded harmonic spinors are parallel,
and the dimension of the space of parallel spinors is given in terms
of Betti numbers by proposition \ref{dimharmspprop}.
Unlike the compact case this is not a characteristic class,
so it is not immediately clear that EAC manifolds with holonomy $Spin(7)$
need to be simply-connected. It would be interesting to decide if that is the
case, but there are currently no known examples at all of EAC manifolds with
holonomy $Spin(7)$.

\begin{thm}
\label{sptopcondthm}
Let $M^{8}$ be an EAC \spmfd{}. Then $Hol(M)= Spin(7)$ if and only if
$\pi_{1}(M)$ is finite, neither $M$ nor any double cover of $M$ is
homeomorphic to a cylinder, and
the cross-section $\tilde X$ of the universal cover has $b^{1}(\tilde X) = 0$.
\end{thm}

\begin{proof}
If these conditions are satisfied then the universal cover $\tildm$
is a simply-connected manifold with cylindrical ends -- a single end,
in fact, by lemma \ref{topcaseslem}\ref{cylcovercase}.
As in the $G_{2}$ case we can deduce from corollary \ref{degonehodgecor} and
proposition \ref{bochnerprop} that there are no parallel $1$-forms on~$\tildm$.
As in the compact case the only possibile holonomy groups of $\tildm$
other than $Spin(7)$ are $SU(4)$, $Sp(2)$ and $SU(2) \times SU(2)$,
and we can use the dimension of the space of parallel positive spinors
to distinguish between them.
The condition $b^{1}(\tilde X) = 0$ implies that this is $1$ by proposition
\ref{dimharmspprop}. Therefore $\tildm$,
and hence also $M$, has holonomy exactly $Spin(7)$.

Conversely, if $\pi_{1}(M)$ is infinite then $M$ has a finite cover
$\tildm$ with $b^{1}(\tildm) > 0$ by lemma \ref{topcaseslem}\ref{inftycase}.
If $M$ has a cylindrical double cover then that must have a product metric.
If $\pi_{1}(M)$ is finite and the universal cover $\tildm$ has
cross-section $\tilde X$ with $b^{1}(\tilde X) > 0$ then the dimension
of the space of parallel spinors on $\tildm$ is at least $2$ by proposition
\ref{dimharmspprop}. In each of the three cases $M$ has a finite cover
with holonomy a proper subgroup of $Spin(7)$, so $M$ cannot have holonomy
exactly $Spin(7)$.
\end{proof}



\section{Deformations of EAC $G_{2}$-manifolds}
\label{cylgtmodsec}

In this section we construct the moduli space of exponentially asymptotically
cylindrical (EAC) torsion-free \gtstr s on a manifold with a cylindrical end.
We prove that the moduli space is smooth and find its dimension.
We also study the boundary map to the moduli space of \cystr s on the
cross-section.


\subsection{Results}

Let $M^{7}$ be a connected oriented manifold with cylindrical ends
and cross-section $X^{6}$.
%
For $\delta > 0$ let $\dcal{X}$ be the space of torsion-free EAC \gtstr s
with rate $\delta$ on $M$ (see definition \ref{cylg2def}).
$\dcal{X}$ is topologised as a subspace of the space of exponentially
asymptotically translation-invariant $3$-forms.

Let $\pcal{X} = \bigcup_{\delta>0} \dcal{X}$. If $\delta_{1} > \delta_{2} > 0$
then the inclusion
$\mathcal{X}_{\delta_{1}} \hookrightarrow \mathcal{X}_{\delta_{2}}$
is continuous, so we can give $\pcal{X}$ the direct limit topology,
i.e. $U \subseteq \pcal{X}$ is open if and only if $U \cap \dcal{X}$
is open in $\dcal{X}$ for all $\delta > 0$. Similarly let $\pcal{D}$ be the
group of %exponentially asymptotically cylindrical
EAC diffeomorphisms of~$M$ with any positive rate
(in the sense of definition \ref{cyldiffdef})
that are isotopic to the identity. $\pcal{D}$ acts on $\pcal{X}$ by pull-backs,
and the
\emph{moduli space of torsion-free %exponentially asymptotically cylindrical
EAC \gtstr s} on $M$ is the quotient $\pcal{M} = \pcal{X}/\pcal{D}$.

\begin{rmk}
\label{normalisermk}
The definition of an EAC \gtstr{} $\varphi$ that is used involves a
normalisation: if $t$ is the cylindrical coordinate on $M$ then
$\norm{\contrat} \to 1$ uniformly on $X$ as $t \to \infty$
(in the metric defined by $\varphi$)
so a scalar multiple $\lambda \varphi$ is not an EAC \gtstr{}.
This normalisation is the most convenient to work with, but a different
choice of normalisation (e.g. that $Vol(X) = 1$ in the induced metric
on the boundary) would of course give the same results.
Another interpretation is that $\bbrp$ acts on the moduli space of unnormalised
EAC \gtstr s by rescaling and that $\pcal{M}$ is the resulting quotient.
\end{rmk}

In the compact case theorem \ref{maincptthm} gives a description of the moduli
space of torsion-free \gtstr s using the natural projection map to the
de Rham cohomology.
In the EAC case, however, it is not enough to consider
\[ \pcal{M} \to H^{3}(M), \;\: \varphi\pcal{D} \mapsto [\varphi] . \]
We also need to consider the boundary values of $\varphi$ to get an adequate
description.
Any $\varphi \in \pcal{X}$ is asymptotic to some $\Omega + dt \wedge \omega$
with $(\Omega, \omega) \in \Omega^{3}(X) \times \Omega^{2}(X)$.
%(and by proposition \ref{cylg2prop} $(\Omega,\omega)$ defines a \cystr{}
%on $X$).
Let
\begin{equation}
\label{derhamprojeq}
\pi_{\mathcal{M}} : \pcal{M} \to H^{3}(M) \times H^{2}(X), \;\:
\varphi\pcal{D} \mapsto ([\varphi], [\omega]) .
\end{equation}
The main result of this section is

\begin{thm}
\label{maincylthm}
$\pcal{M}$ is a smooth manifold, and
$\pi_{\mathcal{M}} : \pcal{M} \to H^{3}(M) \times H^{2}(X)$ is an immersion.
\end{thm}

In order to prove theorem \ref{maincylthm} we make use of our understanding of
the deformations of the `boundary' of an EAC \gtmfd.
By proposition \ref{cylg2prop}, the cross-section $X$ is a compact Calabi-Yau
$3$-fold, and the moduli space $\caln$ of \cystr s on $X$ is a smooth
manifold by theorem \ref{cymodthm}.

In subsection \ref{propertiessub} we look at some local properties
of $\pcal{M}$. Its dimension is given by

\begin{prop}
\label{dimmodprop}
$\dim \pcal{M} = b^{4}(M) + \half b^{3}(X) - b^{1}(M) - 1$.
\end{prop}

We also study the properties of the \emph{boundary map} on $\pcal{M}$.
This is the natural map $B : \pcal{M} \to \mathcal{N}$
%where $\mathcal{N}$ is the moduli space of Calabi-Yau structures on $X$.
which sends a \gtstr{} on $M$ to the \cystr{} on $X$ defined by its asymptotic
limit. As before we
denote by $A^{m} \subseteq H^{m}(X)$ the image of the
pull-back map $j^{*} : H^{m}(M) \to H^{m}(X)$ in the long exact sequence
for relative cohomology (\ref{relexacteq}).
If $\varphi$ is asymptotic to $\Omega + dt \wedge \omega$ then
\begin{gather*}
[\Omega] = j^{*}([\varphi]) \in A^{3} , \\
\half [\omega^{2}] = j^{*}([*_{\varphi}\varphi]) \in A^{4} ,
\end{gather*}
so the image of $B : \pcal{M} \to \mathcal{N}$ is contained in
\begin{equation}
\label{nadefeq}
\mathcal{N}_{A} = \{ (\Omega, \omega)\xcal{D} \in \mathcal{N} :
[\Omega] \in A^{3}, [\omega^{2}] \in A^{4} \} .
\end{equation}
It turns out that, locally at least, these necessary conditions for
a point to be in the image are also sufficient.

\begin{thm}
\label{modsubmersethm}
The image of
\begin{equation}
\label{modsubmerseeq}
B : \pcal{M} \to \mathcal{N}_{A}
\end{equation}
is open in $\mathcal{N}_{A}$ and a submanifold of $\mathcal{N}$.
The map is a submersion onto its image.
\end{thm}

Since the methods used are entirely local they do not tell us anything
about the global properties of $\pcal{M}$ or the image of
(\ref{modsubmerseeq}).

We will show that the fibres of the submersion (\ref{modsubmerseeq})
are locally diffeomorphic to the compactly supported subspace
$H^{3}_{0}(M) \subseteq H^{3}(M)$. The fibre over
$(\Omega,\omega)$ corresponds to the moduli space of torsion-free \gtstr s
asymptotic to $\Omega + dt \wedge \omega$. Thus

\begin{cor}
\label{fixboundarycor}
The moduli space of torsion-free \gtstr s on $M^{7}$ exponentially asymptotic
to a fixed cylindrical \gtstr{} on $X^{6} \times \bbr$ is a manifold.
It is mapped locally diffeomorphically to an affine translate of
$H^{3}_{0}(M)$ by $\pi_{H} : \pcal{M} \to H^{3}(M)$.
\end{cor}

\noindent
In the proof of proposition \ref{dimmodprop} we find that
$\dim H^{3}_{0}(M) = b^{3}(M) - \half b^{3}(X)$, so
\[ \dim \mathcal{N}_{A} = b^{4}(M) - b^{3}(M) + b^{3}(X) - b^{1}(M) - 1 . \]
%
It follows from theorem \ref{maincylthm} that
\begin{equation}
\label{cyllageq}
\pcal{M} \to H^{3}(M) \oplus H^{4}(M), \;\:
\varphi\pcal{D} \to ([\varphi] , [*_{\varphi}\varphi])
\end{equation}
is an immersion. The image in $H^{3}(M) \oplus H^{4}(M)$ of a fibre of
$B : \pcal{M} \to \caln$ lies in an affine translate of
$H^{3}_{0}(M) \oplus H^{4}_{0}(M)$, which has a natural symplectic structure
since $H^{3}_{0}(M) \cong (H^{4}_{0}(M))^{*}$.

\begin{prop}
\label{cyllagembprop}
The restriction of \textup{(\ref{cyllageq})} to
the moduli space of torsion-free \gtstr s on $M^{7}$ exponentially asymptotic
to a fixed cylindrical \gtstr{} on $X^{6} \times \bbr$ is a Lagrangian
immersion to an affine translate of $H^{3}_{0}(M) \oplus H^{4}_{0}(M)$.
\end{prop}

Theorem \ref{salurthm} implies that if $M$ is a \gtmfd{} then either $M$ is
a cylinder $X \times \bbr$ (with a product metric) or $M$ has a single end.
If $M$ is a cylinder $X \times \bbr$ then the only possible torsion-free
\gtstr{} asymptotic to
a given cylindrical \gtstr{} $\varphi_{\infty}$ is $\varphi_{\infty}$ itself, so
the moduli space of asymptotically cylindrical torsion-free \gtstr s on $M$ is
equivalent to the moduli space of Calabi-Yau structures on $X$ (we can compute
that $H^{m}_{0}(X \times \bbr) = 0$ for all $m$, so this agrees with
corollary \ref{fixboundarycor}).
The moduli space will therefore only be interesting when $M$ has a single end,
though we will not need to assume this in the proof of
theorem \ref{maincylthm}.


\subsection{Proof outline}
\label{cylg2outlinesub}

We wish to prove that $\pcal{M}$ has the structure of a smooth manifold.
In order to do this we construct, as an intermediate step, moduli spaces
of torsion-free EAC \gtstr s with some fixed rate
$\delta > 0$. The arguments are similar to the proof of the compact version
(theorem \ref{maincptthm}), but each step needs to be adapted to the EAC
setting.

Each $\varphi \in \dcal{X}$ defines an EAC metric and hence a parameter
$\epsilon_{1}(\varphi)$ such that the associated Hodge Laplacian is Fredholm
on Hölder spaces with weights smaller than $\epsilon_{1}(\varphi)$
(see proposition \ref{lapindprop}).
When we study a neighbourhood of $\varphi$ we need to assume that
$\delta < \epsilon_{1}(\varphi)$, so we let
\begin{equation*}
\calxa = \{ \varphi \in \dcal{X} : \epsilon_{1}(\varphi) > \delta \} .
\end{equation*}
By lemma \ref{epsionectslem}, $\epsilon_{1}$ depends lower semi-continuously on
the asymptotic model so $\calxa$ is an open subset of $\dcal{X}$.
Let $\dcal{D}$ be the group of exponentially
cylindrical diffeomorphisms of $M$ with rate $\delta$ and
\mbox{$\dcal{M} = \calxa/\dcal{D}$}.

In generalising the proof from the compact case we make use of the Hodge
theory for EAC manifolds developed in \S \ref{hodgesub}.
In order to apply the implicit function theorem to show that the pre-moduli
space is a smooth manifold we need an EAC version of the chain rule proposition
\ref{fibremapprop}.

\begin{prop}
\label{cylfibremapprop}
Let $M$ be an EAC manifold with rate $\delta$, $E$ and $F$ vector bundles
associated to $TM$ and $\Psi : E \to F$ a smooth fibre-preserving map 
that is exponentially asymptotically translation-invariant with rate $\delta$.
Then $\Psi$ induces a smooth
map of sections $\holdad{k}(E) \to \holdad{k}(F)$.
\end{prop}

%\begin{proof}
%xxx
%\end{proof}

We now consider how to adapt the steps of \S \ref{cptsec} in order to
construct pre-moduli spaces of torsion-free EAC \gtstr s near
$\varphi \in \calxa$.

Given $\varphi \in \calxa$, let %$\varphi_{\infty} =
$\Omega + dt \wedge \omega = B(\varphi)$ denote the asymptotic limit of
$\varphi$.
If we identify $\Omega + dt \wedge \omega$ with the pair $(\Omega,\omega)$
then, by proposition \ref{cylg2prop},
$\Omega + dt \wedge \omega$ defines a \cystr{} on $X$.
%To simplify the problem of describing a neighbourhood of $\varphi\dcal{D}$
%in $\dcal{M}$ we use our understanding of the deformation theory of \cystr s.
In order to simplify the problem of finding a slice for the
$\mathcal{D}$-action at $\varphi$ we use the deformation theory for compact
Calabi-Yau $3$-folds developed in \S \ref{cymodsec}. In particular,
proposition \ref{cypremodprop} ensures that there is a pre-moduli space
$\mathcal{Q}$ of Calabi-Yau structures near $\Omega + dt \wedge \omega$.
Let
\begin{equation*}
\qcal{X} = \{ \psi \in \calxa : B(\psi) \in \mathcal{Q} \}
\end{equation*}
and let $\qcal{D} \subseteq \dcal{D}$ be the subgroup of diffeomorphisms
asymptotic to automorphisms of the cylindrical \cystr{}
$\Omega + dt \wedge \omega$. By proposition \ref{autinvprop}, $\qcal{D}$ acts
on $\qcal{X}$, and we will see that $\qcal{X}/\qcal{D}$ maps homeomorphically
to an open subset of $\dcal{M}$.

We use slice arguments to study a neighbourhood of $\varphi\qcal{D}$ in
$\qcal{X}/\qcal{D}$. In order to be able to apply analysis results we need
to use Banach spaces of forms, so we work with weighted Hölder $\holdad{k}$
spaces for some fixed $k \geq 1$ and $\alpha \in (0,1)$.
Note that the boundary values of elements of $\qcal{X}$ must lie in
\begin{equation}
\label{qaeq}
\mathcal{Q}_{A} = \{\Omega' + dt \wedge \omega' \in \mathcal{Q}
: [\Omega'] \in A^{3}, [\omega'^{2}] \in A^{4} \} ,
\end{equation}
where $A^{m}$ is the image of $j^{*} : H^{m}(M) \to H^{m}(X)$ (cf. discussion
before theorem \ref{modsubmersethm}). We use a cut-off function $\rho$
for the cylinder on $M$ to
consider $\rho\mathcal{Q}_{A}$ as a subspace of smooth asymptotically
translation-invariant $3$-forms supported on the cylinder of $M$ and set
\[ \ztreq \subseteq \holdad{k}(\Lambda^{3}) + \rho \mathcal{Q}_{A} \]
to be the subspace of closed forms. Then $\qcal{X}$ embeds
continuously into $\ztreq$.

The main technical steps in the construction of the pre-moduli space near
$\varphi$ are
\begin{enumerate}
\item to show that $\mathcal{Q}_{A}$ is a submanifold of $\mathcal{Q}$
(proposition \ref{qasubmfdprop}), so that $\ztreq$ is a manifold,
\item to find a complement $\slt$ in $T_{\varphi}\ztreq$ for the tangent
space to the $\qcal{D}$-orbit at $\varphi$
(proposition \ref{sltprop})
and to pick a submanifold $\mathcal{S} \subseteq \ztreq$ with
$T_{\varphi}\mathcal{S} = \slt$,
\item
to show that the space of torsion-free
\gtstr s $\dcal{R} \subseteq \mathcal{S}$ is a submanifold
(proposition \ref{premodprop}),
\item to show that the elements of $\dcal{R}$ are smooth and EAC
with rate $\delta$
%, so can be considered as a pre-moduli space $\dcal{R}$
(proposition \ref{cylregprop}).
\end{enumerate}

To complete the proof of theorem \ref{maincylthm} it then only remains to
explain how to adapt the slice arguments from \S \ref{modconstrsec} to the
EAC case.
This is done in \S \ref{modconstrsub}.
Repeated use is made of the
regularity result for isometries of EAC manifolds proposition
\ref{eacisomprop}.
We show that $\dcal{R}$ is homeomorphic to a neighbourhood in
$\qcal{X}/\qcal{D}$ (and therefore in~$\dcal{M}$).
Hence $\dcal{M}$ is a manifold for any $\delta > 0$.
We then show that $\dcal{M}$ is homeomorphic to an open
subset of $\pcal{M}$ for any $\delta > 0$ and deduce that $\pcal{M}$ is
a manifold.

\begin{rmk}
The last step means that if $\varphi \in \pcal{X}$ is EAC with rate
$\delta_{0}(\varphi)$ then for any
$0 < \delta < \min \{\delta_{0}(\varphi), \epsilon_{1}(\varphi)\}$
the pre-moduli
space $\dcal{R}$ gives a chart near $\varphi$ not only in~$\dcal{M}$, but also
in $\mathcal{M}_{\delta'}$ for any $\delta' > \delta$ and hence in $\pcal{M}$.
In other words, $\dcal{R}$ is essentially independent of $\delta$ if $\delta$
is chosen sufficiently small.
\end{rmk}


\subsection{The boundary values}

As explained above we are restricting our attention to determining the
space of torsion-free \gtstr s in $\ztreq$ whose boundary values
lie in a certain space $\mathcal{Q}_{A}$. To make this approach work we first
of all need to show that $\mathcal{Q}_{A}$ is a manifold.

Let $X^{6}$ be the cross-section of an EAC \gtmfd{} $M^{7}$, and
$(\Omega, \omega)$ a Calabi-Yau structure on $X$ defined by the limit
of a torsion-free EAC \gtstr{} on $M$.
Let $\mathcal{Q}$ be the pre-moduli space of Calabi-Yau structures near
$(\Omega, \omega)$ and equivalently to (\ref{qaeq}) define
\begin{equation*}
\mathcal{Q}_{A} = \{(\Omega', \omega') \in \mathcal{Q}
: [\Omega'] \in A^{3}, [\omega'^{2}] \in A^{4} \} .
\end{equation*}
Since $\mathcal{Q}$ is diffeomorphic to a neighbourhood in the moduli space
$\mathcal{N}$ of \cystr s on~$X$, $\mathcal{Q}_{A}$ is homeomorphic to a
neighbourhood in the subspace $\mathcal{N}_{A} \subseteq \caln$
defined by~(\ref{nadefeq}).

Recall that by proposition \ref{cypremodprop} the tangent space at
$(\Omega, \omega)$ to the pre-moduli space $\mathcal{Q}$ is the space
$\harm_{SU}$ of harmonic tangents to the space of $SU(3)$-structures.
%
As before let $E^{m} \subseteq H^{m}(X)$ be the orthogonal complement of
$A^{m}$ and let $\mathcal{A}^{m}, \mathcal{E}^{m} \subseteq \harm^{m}_{X}$
denote the respective spaces of harmonic representatives.
%
By lemma \ref{aemapslem}, $\tau \mapsto \omega \wedge \tau$ maps
$\mathcal{A}^{2} \to \mathcal{E}^{4}$ and $\mathcal{E}^{2} \to \mathcal{A}^{4}$.
Hence the linearisation of the condition $[\omega'^{2}] \in A^{4}$ is
$[\tau] \in E^{2}$, and we would expect the tangent space to $\mathcal{Q}_{A}$
at $(\Omega,\omega)$ to be
\begin{equation*}
\harm_{SU,A} =
\{ (\sigma, \tau) \in \harm_{SU}
: \sigma \in \mathcal{A}^{3}, \tau \in \mathcal{E}^{2} \} .
\end{equation*}
This is indeed the case.

\begin{prop}
\label{qasubmfdprop}
Let $(\Omega,\omega)$ be the Calabi-Yau structure induced on the cross-section
$X^{6}$ of an EAC \gtmfd{} $M^{7}$ and $\mathcal{Q}$ the
pre-moduli space of \cystr s near $(\Omega,\omega)$.
Then $\mathcal{Q}_{A} \subseteq \mathcal{Q}$ is a submanifold and
\[ T_{(\Omega, \omega)}\mathcal{Q}_{A} = \harm_{SU,A} . \]
\end{prop}

\begin{proof}
The map $\mathcal{Q} \to H^{3}(X)$ is a submersion, so
\[ \mathcal{Q}' =
\{(\Omega',\omega') \in \mathcal{Q} : [\Omega'] \in A^{3} \} \]
is a submanifold of $\mathcal{Q}$.
By proposition \ref{aeisoprop}
\[ H^{4}(X) = A^{4}_{1} \oplus A^{4}_{6} \oplus A^{4}_{8}
\oplus E^{4}_{6} \oplus E^{4}_{8} , \]
where as before $A^{4}_{d} = H^{4}_{d}(X) \cap A^{4}$ and
$E^{4}_{d} = H^{4}_{d}(X) \cap E^{4}$.
%
Let
\[ P_{E,8} : H^{4}(X) \to E^{4}_{8} \]
be the orthogonal projection. For $(\Omega', \omega') \in \mathcal{Q}'$,
let ${E^{m}}'$ be the orthogonal complement of $A^{m}$ with respect to the
metric defined by $(\Omega', \omega')$, and let
$P_{A'} : H^{m}(X) \to A^{m}$ and $P_{E'} : H^{m}(X) \to {E^{m}}'$ be the
respective projections.
Let
\[ F : \mathcal{Q'} \to E^{4}_{8}, \;\:
(\Omega',\omega') \mapsto P_{E,8}P_{E'}[\omega'\wedge\omega'] . \]
We prove that $\mathcal{Q}_{A}$ is a submanifold of $\mathcal{Q}'$ by
showing that it is the zero set of~$F$ and that $F$ has surjective derivative
at $(\Omega,\omega)$.

Suppose $F(\Omega',\omega') = 0$ and let $a = P_{E'}[\omega'\wedge\omega']$.
Write $a = b + c$, with $b \in A^{4}$,
$c \in E^{4}$. $P_{E,8}a = 0 \Rightarrow \pi_{8}c = 0$,
so $c \in E^{4}_{6}$.
Since $E^{1} \to E^{4}_{6}, \; v \mapsto [\Omega] \hcup v$ is an isomorphism
$c = [\Omega] \hcup v$ for some $v \in E^{1}$. In the inner product
$\inner{\;,\;}'$ on $H^{*}(X)$ defined by $(\Omega',\omega')$,
\begin{equation*}
\inner{a,a}' \;=\; \inner{a,[\Omega] \hcup v}' \;=\;
\inner{a, [\Omega] \hcup v - [\Omega'] \hcup P_{E'}v}' \\ \leq
\norm{a}'(\norm{[\Omega-\Omega']}'\norm{v}' + \norm{P_{A'}v}') .
\end{equation*}
The RHS can be estimated by $\norm{[\Omega-\Omega']}(\norm{a}')^{2}$ for
$(\Omega',\omega')$ close to $(\Omega,\omega)$.
Hence for $(\Omega',\omega')$ sufficiently close to $(\Omega,\omega)$
\[ F(\Omega',\omega') = 0 \Rightarrow P_{E'}[\omega'^{2}] = 0
\Rightarrow [\omega'^{2}] \in A^{4} . \]
So $\mathcal{Q}_{A} \subseteq \mathcal{Q}'$ is the zero set of $F$.
It remains to show that $F$ has surjective derivative.
%
If $(\sigma, \tau) \in (\mathcal{A}^{3} \times \harm^{2}_{X}) \cap \harm_{SU} =
T_{(\Omega,\omega)}\mathcal{Q}'$ then, since $[\omega^{2}] \in A^{4}$,
\[DF_{(\Omega,\omega)}(\sigma, \tau) = P_{E,8}P_{E}(2[\omega \wedge \tau])
= 2P_{E,8}[\omega \wedge \tau] . \]
Since $\mathcal{A}^{2}_{8} \to \mathcal{E}^{4}_{8}, \;
\tau \mapsto \omega \wedge \tau$ is a bijection,
the derivative maps the space $0 \times \mathcal{A}^{2}_{8}$
%$ \subseteq T_{(\Omega,\omega)}\mathcal{Q}'$
onto $E^{4}_{8}$.

By the implicit function theorem $\mathcal{Q}_{A}$ is a manifold
and the tangent space at
$(\Omega,\omega)$ is
\[ \ker DF_{(\Omega,\omega)} = \harm_{SU,A} . \qedhere \]
\end{proof}

\begin{cor}
\label{hsuacor}
The map $\harm_{SU,A} \to \mathcal{A}^{3}, \; (\sigma,\tau) \mapsto \sigma$
is surjective with kernel $0 \times \mathcal{E}^{2}_{8}$.
\end{cor}

\begin{proof}
The last part of the proof of the proposition actually shows that 
$\mathcal{Q}_{A} \to A^{3}$ is a submersion, so
$\harm_{SU,A} \to \mathcal{A}^{3}$ is surjective. This could also be deduced
from lemma \ref{aemapslem}. By definition of $\harm_{SU}$, the kernel
consists of those $(0, \tau) \in 0 \times \mathcal{E}^{2}$ satisfying the
conditions (\ref{leqs}), which reduce to $\pi_{1}\tau = \pi_{6}\tau = 0$.
\end{proof}

Proposition \ref{qasubmfdprop} implies directly that a neighbourhood
%in $\mathcal{N}_{A}$
of the image of $B : \pcal{M} \to \mathcal{N}_{A}$ is a manifold.
The rest of theorem \ref{modsubmersethm} follows too, once we have proved the
main result that $\pcal{M}$ is a manifold.
We will return to this in \S \ref{propertiessub}.

\begin{rmk}
If $b^{1}(X) = 0$ then the proof of proposition \ref{qasubmfdprop} simplifies
and it is possible to show that $\caln_{A}$ is itself a submanifold of
$\caln$. However, we do not want to exclude the case $b^{1}(X) \not = 0$,
since many of the interesting examples for the gluing construction
discussed in \S \ref{gluechap} have reducible cross-section.
\end{rmk}


\subsection{The slice}

Fix $k \geq 1$, $\delta > 0$, $\alpha \in (0,1)$ and $\varphi \in \calxa$.
We will find a direct complement $\slt$ in $T_{\varphi}\ztreq$ to the
$\qcal{D}$-orbit. Then we will define a submanifold
$\mathcal{S} \subseteq \ztreq$ whose tangent space at $\varphi$ is $\slt$.
$\mathcal{S}$ will be used as a slice in $\ztreq$ for the $\qcal{D}$-action
at $\varphi$.

The fixed \gtstr{} $\varphi$ is used to define an EAC metric and a Hodge star.
It also defines type decompositions of the exterior bundles
(\ref{gtformdecompeq}) and spaces of
harmonic forms, as described in \S \ref{harmholsub}.
Recall that the map
$TM \to \Lambda^{2}_{7}, \; v \mapsto v \lrcorner \varphi$ is a bundle
isomorphism and that Lie derivatives
$\mathcal{L}_{V}\varphi = d(V \lrcorner \varphi)$ are precisely exterior
derivatives of $2$-forms of type~$7$.

Restricting our attention to \gtstr s in $\ztreq$ is convenient
because the asymptotic values of elements of $T_{\varphi}\ztreq$ are harmonic.
Recall the notation for harmonic forms from the summary in \S \ref{hodgesub},
%the summary on page \pageref{hodgesummary},
in particular that $\harm^{m}_{0}$
and $\harm^{m}_{\infty}$ denote the spaces of bounded harmonic forms on
$M$ and harmonic translation-invariant forms on $X \times \bbr$ respectively.

For convenience we identify translation-invariant $3$-forms on $X \times \bbr$
with pairs of $3$- and $2$-forms on $X$ by
$\sigma + dt \wedge \tau \leftrightarrow (\sigma, \tau)$. This identifies the
tangent spaces $\harm_{SU}$ and $\harm_{SU,A}$ of $\mathcal{Q}$ and
$\mathcal{Q}_{A}$ with subspaces $\harm^{3}_{SU}$ and
$\harm^{3}_{SU,A} \subseteq \harm^{3}_{\infty}$.
%of the space of harmonic translation-invariant forms on $X \times \bbr$.
Let
\begin{equation*}
\ztre_{cyl} \subseteq \holdad{k}(\Lambda^{3}) \oplus \rho \harm^{3}_{SU,A}
\end{equation*}
be the subspace of closed forms.
Clearly $T_{\varphi}\ztreq \subseteq \ztre_{cyl}$; we show below that
equality holds.
The tangent space to the pre-moduli space of torsion-free \gtstr s at $\varphi$
will turn out to be the subspace
\begin{equation*}
\label{hcyleq}
\harm^{3}_{cyl} \subseteq \ztre_{cyl}
\end{equation*}
of harmonic forms. This is exactly the subspace of elements of 
$\harm^{3}_{0}$ which are tangent to cylindrical deformations of the \gtstr{},
i.e. whose boundary values lie in $\harm^{3}_{SU}$. The boundary map
$B : \harm^{3}_{0} \to \harm^{3}_{\infty}$ maps $\harm^{3}_{cyl}$ precisely
onto $\harm^{3}_{SU,A}$. Together with the Hodge decomposition
theorem \ref{hodgedecompthm} it follows that
\begin{equation}
\label{zcylspliteq}
\ztre_{cyl} = \harm^{3}_{cyl} \oplus \holdad{k}[d\Lambda^{2}] .
\end{equation}

\begin{rmk}
$d\holdad{k+1}(\Lambda^{m-1})$ is the space of exterior derivatives of decaying
forms, while we use $\holdad{k}[d\Lambda^{m}]$ to denote the space of exact
decaying forms.
$d\holdad{k+1}(\Lambda^{m-1}) \subseteq \holdad{k}[d\Lambda^{m}]$ is a closed
subspace of finite codimension.
\end{rmk}

\begin{lem}
\label{zqmfdlem}
$\ztre_{\mathcal{Q}}$ is a manifold and $T_{\varphi}\ztreq = \ztre_{cyl}$.
\end{lem}

\begin{proof}
If $\psi$ is a $3$-form asymptotic to an element
$(\Omega',\omega') \in \mathcal{Q}_{A}$
then the condition $[\Omega'] \in A^{3}$ implies that
$d\psi \in d\holdad{k}(\Lambda^{3})$. Therefore
\[ d : \holdad{k}(\Lambda^{3}) + \rho \mathcal{Q}_{A} \to
d\holdad{k}(\Lambda^{3}) \]
is a submersion and the result follows from the implicit function theorem.
\end{proof}

Let $\dkaq$ be the group of diffeomorphisms of $M$ which are isotopic to the
identity and $\holdad{k+1}$-asymptotic to a cylindrical automorphism of the
cylindrical \gtstr{} $\Omega + dt \wedge \omega$. The elements of a
neighbourhood of the identity in $\dkaq$ can be written as
$\exp (V + \rho V_{\infty})$, where $V$ is a $\holdad{k+1}$ vector field on $M$
and $V_{\infty}$ is a translation-invariant vector field on $X \times \bbr$
with $\mathcal{L}_{V_{\infty}}(\Omega + dt \wedge \omega) = 0$, i.e.
$V_{\infty} \in (\harm^{1}_{\infty})^{\sharp}$.
Therefore if we let
\begin{equation*}
\label{deq}
D = \rho (\harm^{1}_{\infty})^{\sharp} \lrcorner \varphi
\subseteq \Omega^{2}_{7}(M)
\end{equation*}
then the tangent space to the $\dkaq$-orbit at $\varphi$ is
\[ \{ \mathcal{L}_{V + \rho V_{\infty}}\varphi : 
V \in \holdad{k+1}(TM), V_{\infty} \in (\harm^{1}_{\infty})^{\sharp} \} =
d(\holdad{k+1}(\Lambda^{2}_{7}) \oplus D) . \]
As a direct complement in $\ztre_{cyl}$ we can take
$K = \harm^{3}_{cyl} \oplus W$, where $W$ is the space of decaying exact forms
of type $27$. Like in the compact case this is the kernel of the
formal adjoint of $d : \Omega^{2}_{7}(M) \to \Omega^{3}(M)$
(see remark \ref{bryantrmk}).

\begin{defn}
Let $W = \holdad{k}[d\Lambda^{2}] \cap \Omega^{27}(M)$.
\end{defn}

\begin{prop}
\label{sltprop}
\begin{equation*}
\ztre_{cyl} = d(\holdad{k+1}(\Lambda^{2}_{7}) \oplus D) \oplus
\harm^{3}_{cyl} \oplus W .
\end{equation*}
\end{prop}

\begin{proof}
By (\ref{zcylspliteq}) it suffices to show
\begin{equation}
\label{wspliteq}
\holdad{k}[d\Lambda^{2}] = d(\holdad{k+1}(\Lambda^{2}_{7}) \oplus D) \oplus W .
\end{equation}
As in the proof of proposition \ref{ortsumprop} we can
identify the spinor bundle on the \gtmfd{} $M$ both with
$\Lambda^{0} \oplus \Lambda^{2}_{7}$
and with $\Lambda^{3}_{1\oplus7}$ so that the Dirac operator on exponentially
asymptotically translation-invariant spinors (\ref{eatdiraceq}) is
identified with
\begin{equation*}
\holdad{k+1}(\Lambda^{0} \oplus \Lambda^{2}_{7}) \to
\holdad{k}(\Lambda^{3}_{1 \oplus 7}), \;\:
(f, \eta) \mapsto \pi_{1\oplus7}d\eta + *(df \wedge \varphi) .
\end{equation*}
As explained in \S \ref{eacdiracsub}, the image of this map is the \ltwoorth{}
complement of the decaying harmonic spinors. On a scalar-flat manifold
there are no harmonic decaying spinors, so the map is surjective.
Thus, if $\beta \in \holdad{k}[d\Lambda^{2}]$ then
\[ \pi_{1\oplus7}\beta = \pi_{1\oplus7}d\eta + *(df \wedge \varphi) \]
for some $\eta \in \holdad{k+1}(\Lambda^{2}_{7}) \oplus D$,
$f \in \holdad{k+1}(\Lambda^{0}) \oplus \rho\harm^{0}_{\infty}$.
Integrating by parts like in (\ref{dfzeq}) shows $df = 0$.
Hence the decaying exact form
$\beta - d\eta$ has type $27$, i.e. $\beta - d\eta \in W$.
\end{proof}

We want to take as our slice for the $\dcal{D}$-action at $\varphi$ a
submanifold $\mathcal{S} \subseteq \ztreq$ with
$T_{\varphi}\mathcal{S} = \slt$.
To this end we pick a map
\begin{equation}
\label{expdefeq}
\exp : U \to \ztreq
\end{equation}
on a neighbourhood $U$ of $0$ in $\ztre_{cyl} = T_{\varphi}\ztreq$,
such that $D\exp_{0} = id$. We also insist that $\exp$ is a translation on the
decaying forms and that it maps smooth forms to smooth forms.
We can do this since by (\ref{zcylspliteq}) the decaying forms have a
finite-dimensional complement of smooth forms in $\ztre_{cyl}$.
We then choose
\begin{equation}
\label{slicedefeq}
\mathcal{S} = \exp (\slt \cap U) .
\end{equation}

\ignore{
\begin{rmk}
%\label{cylsliceinvrmk}
Note that the stabiliser $\cali_{\varphi} \subset \pcal{D}$ is
compact by lemma \ref{eaccptstablem} (except in the uninteresting case when
the EAC \gtmfd{} $M$ is a product cylinder), and that we may take $\exp$ to be
invariant under the action of the stabiliser $\cali_{\varphi}$.
It would be feasible to prove an analogue of theorem \ref{slice2thm},
claiming that $\exp$ maps the smooth exponentially translation-invariant forms
in $U$ homeomorphically to the smooth exponentially translation-invariant forms
in $\ztreq$, %This is sufficient to allow us
although that is not the approach we take.
\end{rmk}
}

\subsection{Smoothness of the pre-moduli space}

Let $\dcal{R} \subseteq \mathcal{S}$ be the subset of $\holdad{k}$
torsion-free \gtstr s.
$\dcal{R}$ is the pre-moduli space of torsion-free \gtstr s near $\varphi$.
In order to show that $\dcal{R}$ is a submanifold we will
exhibit it as the zero set of a function $F$ with surjective derivative and
apply the implicit function theorem.

Recall that by theorem \ref{graythm}\ref{torfreeitem}, a closed \gtstr{} $\psi$
is torsion-free if and only if $d\Theta(\psi) = 0$, where $\Theta$ is the
non-linear map $\psi \mapsto *_{\psi}\psi$. Thus
\[ \dcal{R} = \{ \psi \in \mathcal{S} : d\Theta(\psi) = 0 \} . \]
If $\psi \in \ztreq$ then $\psi$ is asymptotic to a torsion-free
cylindrical \gtstr{} $\Omega' + dt \wedge \omega'$,
so $d\Theta(\psi)$ decays. Moreover, elements of $\ztreq$
are asymptotic to elements of $\mathcal{Q}_{A} \subseteq \mathcal{Q}$
by definition. Therefore $\Theta(\psi)$ is asymptotic
to $\half \omega'^{2} - dt \wedge \hat\Omega'$, with $[\omega'^{2}] \in A^{4}$
(cf. (\ref{qaeq}) and remark \ref{cyldualrmk}).
This implies that $d\Theta(\psi) \in d\holdad{k+1}(\Lambda^{4})$.

$\Theta : \Lambda^{3}_{+}T^{*}M \to \Lambda^{4}T^{*}M$ is point-wise smooth,
so by the chain rule
\begin{equation*}
\ztreq \to d\holdad{k+1}(\Lambda^{4}), \;\: \psi \to d\Theta(\psi)
\end{equation*}
is a smooth function.
We need to adjust this map to obtain a function with surjective derivative.
%
If $\beta$ is a $3$-form such that
$d^{*}\beta \in d^{*}\holdad{k+1}(\Lambda^{3})$ then by the Hodge decomposition
theorem \ref{hodgedecompthm} there is a unique
$P_{E}(\beta) \in \holdad{k+1}[d\Lambda^{2}]$ such that
$d^{*}\beta = d^{*}P_{E}(\beta)$.
We can think of $P_{E}(\beta)$ as the exact part of $\beta$.
We then let $P_{W}(\beta)$ be the image of $P_{E}(\beta)$ under the projection
$\holdad{k+1}[d\Lambda^{2}] \to W$ in the direct sum decomposition
(\ref{wspliteq}).

\begin{defn}
For $\psi$ close to $\varphi$ in $\ztreq$ let
\begin{equation}
F(\psi) = P_{W}(*\Theta(\psi)) .
\end{equation}
\end{defn}

Clearly $d\Theta(\psi) = 0 \Rightarrow F(\psi) = 0$. We need to show that
that the converse also holds, so that we do not `lose any information' by
considering zeros of $F$ instead of $\psi \mapsto d\Theta(\psi)$.

\begin{prop}
\label{zerosetprop}
For $\psi \in \ztreq$ sufficiently close to $\varphi$, $\psi$ is
torsion-free if and only if $F(\psi) = 0$.
\end{prop}

\begin{proof}
We adapt the proof of proposition \ref{appropriateprop}.
For $\psi \in \ztreq$ the
tangent space to the $\cald^{1}_{\calq}$-orbit at $\psi$ can be written as
\[ T_{\psi} = d(\holdad{1}(\Lambda^{2}_{7,\psi}) \oplus
\rho (\harm^{1}_{\infty})^{\sharp}\lrcorner \psi ), \]
where $\Lambda^{2}_{7,\psi}$ is the bundle of type $7$ $2$-forms defined
by the \gtstr{} $\psi$.
The linear map
\begin{equation*}
W \oplus \holdad{1}(TM) \oplus \harm^{1}_{\infty} \to
\holdad{0}[d\Lambda^{2}],
\;\: (w, V, \beta) \mapsto w + d((V +\rho\beta^{\sharp}) \lrcorner \psi)
\end{equation*}
is surjective at $\psi = \varphi$ by (\ref{wspliteq}).
Since it depends continuously on $\psi$,
\[ \holdad{0}[d\Lambda^{2}] = W + T_{\psi} \]
%d(\holdad{k+1}(\Lambda^{2}_{7,\psi}) \oplus b_{\psi(H^{1}(X)})) \]
for any $\psi$ sufficiently close to $\varphi$.

We deduce from theorem \ref{graythm}\ref{pi7item} that
$*d\Theta(\psi) = d^{*}P_{E}(*\Theta(\psi))$ is point-wise orthogonal
to $\Lambda^{2}_{7,\psi}$ when $d\psi = 0$.
Therefore $P_{E}(*\Theta(\psi))$ is \ltwoorth{} to $T_{\psi}$
(integrating by parts).
%$d((\holdad{k+1}(TM) \oplus \rho (\harm^{1})^{\sharp}) \lrcorner \psi)$.
$F(\psi) = 0$ means that $P_{E}(*\Theta(\psi))$ is \ltwoorth{} to $W$ too,
so $P_{E}(*\Theta(\psi)) = 0$.
\end{proof}

By proposition \ref{cylfibremapprop}, $F : \ztreq \to W$ is a smooth
function. Next we show that it %$F : \mathcal{S} \to W$
satisfies the hypotheses of the implicit function theorem.

% ignore the prop stating the formula for the derivative
% moved to background 

\begin{prop}
\label{cylfderprop}
$DF_{\varphi} : \ztre_{cyl} \to W$ is $0$ on
$d(\holdad{k+1}(\Lambda^{2}_{7}) \oplus D) \oplus \harm^{3}_{cyl}$
and $-id$ on~$W$.
\end{prop}

\begin{proof}
Same as the compact version \ref{fderprop}.
\end{proof}

We have taken the pre-moduli space $\dcal{R}$ near $\varphi$ to consist of the
torsion-free EAC \gtstr s in the slice $\mathcal{S}$. However, we can only
prove that it has the properties we want close to $\varphi$. We will therefore
repeatedly replace $\mathcal{S}$ by a neighbourhood of $\varphi$ in
$\mathcal{S}$ in order to ensure that $\dcal{R} \subseteq{S}$ has the
desired properties.
%
The first step is to ensure that $\dcal{R}$ is a manifold.
Proposition \ref{zerosetprop} shows that if $\mathcal{S}$ is sufficiently small
then $\dcal{R}$ is the zero set of $F$ in $\mathcal{S}$.
Applying the implicit function theorem to $F : \mathcal{S} \to W$ we deduce
%Since $T_{\varphi}\mathcal{S} = K = \harm^{3}_{cyl} \oplus W$ proposition
%\ref{cylfderprop} means that the implicit function theorem can be applied
%to $F : \mathcal{S} \to W$. We deduce

\begin{prop}
\label{premodprop}
If the slice $\mathcal{S}$ near $\varphi$ is shrunk sufficiently then
the space $\dcal{R}$ of torsion-free EAC \gtstr s in $\cals$ is a manifold.
Its tangent space at $\varphi$ is $\harm^{3}_{cyl}$.
\end{prop}

The implicit function theorem implies also that a small neighbourhood of
$\varphi$ in $\calx_{k,\calq}$, the space of torsion-free \gtstr s
in $\ztreq$, is a manifold.


\subsection{Regularity}
\label{cylregsub}

To finish the proof of theorem \ref{maincylthm} we need to show
that if the slice $\mathcal{S}$ is taken sufficiently small then the
torsion-free \gtstr s in $\mathcal{S}$ are smooth and
asymptotically cylindrical.
We first prove a regularity
result about solutions of elliptic operators which are
`$\holdad{l}$-asymptotically cylindrical', and then generalise the
regularity argument for the compact case from \S \ref{cptregsub}.
We start by quoting an interior estimate.

\begin{thm}
\label{intestthm}
Let $B_{R}, B_{2R} \subset \bbr^{q}$ be the balls centred on the origin of
radius $R$ and $2R$ respectively. Let $A$ be an elliptic order $r$ operator
with $C^{l,\alpha}$ coefficients acting on the $\bbr^{p}$-valued functions
on $B_{2R}$.

If $u \in C^{r,\alpha}(B_{2R},\bbr^{p})$ and
$Au \in C^{l,\alpha}(B_{2R}, \bbr^{p})$ then
$u|_{B_{R}} \in C^{l+r,\alpha}(B_{R},\bbr^{p})$.
Furthermore
\begin{equation}
\norm{u|_{B_{R}}}_{C^{l+r,\alpha}} \leq K\left(
\norm{Au}_{C^{l,\alpha}} + \norm{u}_{C^{0}} \right) ,
\end{equation}
where the constant $K$ depends on $q, k, \alpha$, the $C^{l,\alpha}$-norm
of the coefficients of $A$ and a lower bound for the operator norm of
the symbol of $A$ evaluated at unit vectors.
\end{thm}

\begin{proof}
This is essentially a paraphrase of Theorems $6.2.5$ and $6.2.6$ in
\cite{morrey66}.
\end{proof}

\begin{defn}
Say that the differential operator $A$ on $M$ is \emph{$\holdad{l}$-asymptotic}
to the translation-invariant operator $A_{\infty}$ on the cylinder
$X \times \bbr$ if the difference between the coefficients (as in definition
\ref{asyopdef}) of $A$ and $A_{\infty}$ is $\holdad{l}$.
\end{defn}

It is well-known how to go from an interior estimate like theorem
\ref{intestthm} to global estimates for appropriate norms on an asymptotically
cylindrical manifold. We wish to use $\holdad{l}$-norms. The theorem
below is similar to for example \cite[Theorem $6.3$]{mazya78}.

\begin{thm}
\label{cylregthm}
Let $M^{n}$ be an asymptotically cylindrical manifold, $E$ a vector bundle
on $M$ associated to $TM$
and $A$ a linear elliptic order $r$ differential operator on the sections
of $E$ that is $\holdad{l}$-asymptotic to a translation-invariant operator
with $C^{l,\alpha}$ coefficients.

If $s \in \holdad{r}(E)$ and $As \in \holdad{l}(E)$ then
$s \in \holdad{l+r}(E)$.
Furthermore
\begin{equation}
\norm{s}_{\holdad{l+r}} < K\left( \norm{As}_{\holdad{l}} +
\norm{s}_{C^{0}_{\delta}} \right)
\end{equation}
for some $K > 0$ independent of $s$.
\end{thm}

\begin{proof}
Working just on the compact part $M_{0}$ of $M$ this is easy. We can take
coordinate charts that identify neighbourhoods $U \subset M_{0}$ with
$B_{2R} \subset \bbr^{n}$ and apply theorem \ref{intestthm} to deduce that
on the inverse image $U'$ of $B_{R}$
\begin{equation}
\norm{s|_{U'}}_{C^{l+r,\alpha}} < K\left( \norm{As|_{U}}_{C^{l,\alpha}}
+ \norm{s|_{U}}_{C^{0}} \right)
\end{equation}
for some $K > 0$ independent of $s$. $K$ is allowed to depend on $U$, but
since we can cover $M_{0}$ by finitely many $U$ we can take $K$ large enough
to ensure that
\begin{equation}
\label{cptesteq}
\norm{s|_{M_{0}}}_{C^{l+r,\alpha}} < K\left( \norm{As|_{M_{0}}}_{C^{l,\alpha}}
+ \norm{s|_{M_{0}}}_{C^{0}} \right) .
\end{equation}
Now we consider the non-compact part $M_{\infty} = X \times \bbrp$.
Let $V$ be a neighbourhood in $X$, $f : V \to B_{2R} \subset \bbr^{n-1}$
a diffeomorphism and $V'$ the inverse image of $B_{R}$ under $f$.
For $t \in \bbrp$ let $I_{t}$ and $I'_{t}$ be the intervals
$(t-2R, t+2R), (t-R, t+R)$ respectively. By theorem \ref{intestthm}
\begin{equation}
\label{striponeeq}
\norm{s|_{V' \times I'_{t}}}_{C^{l+r,\alpha}} < K\left(
\norm{As|_{V \times I_{t}}}_{C^{l,\alpha}} +
\norm{s|_{V \times I_{t}}}_{C^{0}} \right) .
\end{equation}
Here the constant $K$ depends on $q, k, \alpha$, a bound on $V \times I_{t}$
for the $C^{l,\alpha}$-norm of the coefficients of~$A$,
a lower bound on $V \times I_{t}$ for the
operator norm of the symbol of $A$ at unit vectors and finally also on
the diffeomorphism $f$ (e.g. on how $f$ distorts the metric).
Since $A$ is $\holdad{l}$-asymptotically translation-invariant we can find
global bounds for the $C^{l,\alpha}$-norms of the coefficients of $A$ and
for the operator norm of the symbol. Thus we can take $K$ in
(\ref{striponeeq}) independent of $t$. Furthermore we can scale
(\ref{striponeeq}) by $e^{\delta t}$ in order to replace the norms by weighted
norms. Hence we deduce
\begin{equation*}
\norm{s|_{V' \times \bbrp}}_{\holdad{l+r}} < K\left(
\norm{As|_{V \times \bbrp}}_{\holdad{l}} +
\norm{s|_{V \times \bbrp}}_{C^{0}_{\delta}} \right) .
\end{equation*}
Since we can cover $M_{\infty}$ by finitely many strips $V' \times \bbrp$
it follows that we can take $K$ large enough that
\begin{equation*}
\norm{s|_{M_{\infty}}}_{\holdad{l+r}} < K\left(
\norm{As|_{M_{\infty}}}_{\holdad{l}} +
\norm{s|_{M_{\infty}}}_{C^{0}_{\delta}} \right) .
\end{equation*}
Together with (\ref{cptesteq}) this proves the theorem.
\end{proof}

\begin{rmk}
With some additional algebra theorem \ref{cylregthm} can be a generalised
to show regularity of `$\holdad{r}$-asymptotically translation-invariant'
solutions.
\end{rmk}
%mod3P.tex

Now consider again a \gtmfd{} $M$ with a torsion-free \gtstr{}~$\varphi$,
and the pre-moduli space $\dcal{R}$ of torsion-free \gtstr s
in the slice $\mathcal{S} = \exp(\slt \cap U) \subseteq \ztreq$ for the
$\qcal{D}$-action at $\varphi$.
We use theorem \ref{cylregthm} in a boot-strapping argument to show
that the elements of $\dcal{R}$ are EAC. A priori they are
$\holdad{k}$-asymptotic to elements of $\mathcal{Q}_{A}$.
%, for some $k \geq 1$.
As in proposition \ref{premodprop} we can only work close to $\varphi$, and
must replace $\mathcal{S}$ by a neighbourhood of $\varphi$ in $\mathcal{S}$.

%\hfuzz=5pt

\begin{prop}
\label{cylregprop}
If the slice $\mathcal{S}$ near $\varphi$ is shrunk sufficiently then the
pre-moduli space $\dcal{R} \subseteq \mathcal{S}$ consists of %EAC \gtstr s.
smooth exponentially asymptotically translation-invariant forms.
\end{prop}

%\hfuzz=0pt

\begin{proof}
We want to show that if $\psi \in \mathcal{S}$ is sufficiently close to
$\varphi$ and $d\Theta(\psi) = 0$ then $\psi$ is smooth and
exponentially asymptotically translation-invariant.
Set $\psi = \varphi + \beta$. Then
\[ D(d\!*\!d\Theta)_{\varphi} =
-dd^{*} \circ (\fourthird\pi_{1} + \pi_{7} - \pi_{27}) , \]
so we can write
\[ d\!*\!d\Theta(\varphi + \beta) =
-dd^{*}(\fourthird\pi_{1}\beta + \pi_{7}\beta - \pi_{27} \beta) +
P(\beta, \nabla \beta, \nabla^{2}\beta) + R(\beta, \nabla \beta) , \]
where $P$ consists of the quadratic terms of $d\!*\!d\Theta(\varphi+\beta)$
that involve $\nabla^{2}\beta$, and $R$ consists of the remaining quadratic
terms. $P$ and $R$ depend only point-wise on their arguments and $P$
is linear in $\nabla^{2}\beta$.

By the definition of the map $\exp$ (\ref{expdefeq}) we can write
$\beta = \kappa + \gamma$, with $\kappa \in W$ and
$\gamma$ smooth and exponentially asymptotic to some element of
$\mathcal{Q}_{A}$.
As $\kappa$ is closed of type $27$
\[ -dd^{*}(\fourthird\pi_{1}\kappa + \pi_{7}\kappa - \pi_{27}\kappa) =
\triangle \kappa . \]

Considering $\beta$ and $\nabla \beta$ as fixed we can define a
second-order linear differential operator
$A : \zeta \mapsto P(\beta, \nabla\beta, \nabla^{2}\zeta)$. Then
the condition $d\!*\!d\Theta(\psi) = 0$ is equivalent to 
\begin{equation}
\label{lineareq2}
(\triangle + A)\kappa = -R +
dd^{*}(\fourthird\pi_{1}\gamma + \pi_{7}\gamma - \pi_{27}\gamma) - A\gamma .
\end{equation}

If $\beta = 0$ then $A = 0$, so $\triangle + A$ is elliptic. Ellipticity
is an open condition, so $\triangle + A$ is in fact elliptic for any
$\beta$ sufficiently small in the uniform norm.

Now suppose $\kappa$ is $\holdad{l+1}$ and is a solution of
(\ref{lineareq2}). $R$ and the coefficients of $A$ depend smoothly on $\kappa$
and $\nabla \kappa$. Therefore $\triangle + A$ and the RHS of (\ref{lineareq2})
are $\holdad{l}$-asymptotically translation-invariant.
%since $\gamma$ is smooth and exponentially translation-invariant
Since the RHS of (\ref{lineareq2}) is decaying \textit{a priori} it is
$\holdad{l}$.
If $\beta$ is sufficiently small that $\triangle + A$ is elliptic then by
theorem \ref{cylregthm} $\kappa$ is $\holdad{l+2}$.
Since $\kappa$ is $\holdad{1}$ it is $\holdad{l}$ for all
$l$ by induction.

Hence $\psi = \varphi + \kappa + \gamma$ is smooth and exponentially
asymptotically translation-invariant.
\end{proof}



\subsection{Constructing the moduli space}
\label{modconstrsub}

For each $\varphi \in \calxa$ we have constructed a pre-moduli space
$\dcal{R}$. $\dcal{R}$ is a manifold, its elements are smooth and EAC
and its tangent space at $\varphi$ is $\harm^{3}_{cyl}$.
To complete the proof of the main theorem \ref{maincylthm}
we now provide slice arguments to show that we can take the pre-moduli
spaces $\dcal{R}$ as coordinate charts to define a smooth structure on
$\dcal{M}$.
We use the same ideas as in the compact case and study the charts for
$\pcal{M}$ in terms of the projection to the de Rham cohomology
which appears in the statement of the main theorem \ref{maincylthm}.
\begin{equation*}
\pi_{\mathcal{M}} : \pcal{X} \to H^{3}(M) \times H^{2}(X), \;\:
\varphi \mapsto ([\varphi], [B_{e}(\varphi)]) .
\end{equation*}
%where we use $B_{a}(\beta) + dt \wedge B_{e}(\beta)$ to denote the
%asymptotic limit of an asymptotically translation-invariant form $\beta$.
%%% $B_{a}$ and $B_{e}$ are both linear maps.

We first check that $\pi_{\mathcal{M}}$ is an embedding on $\dcal{R}$.
If we allow ourselves to shrink $\dcal{R}$ this
amounts to showing that the derivative of $\pi_{\mathcal{M}}$ at $\varphi$ is
injective. The derivative is
\begin{equation*}
(\pi_{H}, \pi_{H,e}) : \harm^{3}_{cyl} \to H^{3}(M) \times H^{2}(X), \;\:
\beta \mapsto ([\beta], [B_{e}(\beta)])
\end{equation*}
and the kernel consists of harmonic, exact, decaying forms, so
%by integration by parts
it is trivial.

Recall from subsection \ref{cylg2outlinesub} that we chose a pre-moduli space
$\mathcal{Q}$ near the Calabi-Yau structure $(\Omega,\omega)$ on $X$ defined
by the asymptotic limit of $\varphi$, and that $\qcal{X} \subseteq \calxa$
is the subset of torsion-free \gtstr s whose asymptotic limits lie in
$\mathcal{Q}$. $\qcal{D} \subseteq \dcal{D}$ is the subgroup of smooth EAC
diffeomorphisms of $M$ whose asymptotic limits lie in the automorphism group
of $(\Omega,\omega)$. $\qcal{D}$ acts on $\qcal{X}$ by proposition
\ref{autinvprop} and as an intermediate step for our slice result we prove
that $\dcal{R}$ is a coordinate chart for $\qcal{X}/\qcal{D}$.

\begin{prop}
\label{qsliceprop}
The natural map $\dcal{R} \to \qcal{X}/\qcal{D}$ is a homeomorphism onto
a neighbourhood of $\varphi\qcal{D}$.
\end{prop}

\begin{proof}
Recall that a small neighbourhood of $\varphi$ in $\calx_{k,\calq}$,
the space of torsion-free \gtstr s in $\ztreq$, is a manifold.
Its tangent space at $\varphi$ is
$d(\holdad{k+1}(\Lambda^{2}_{7}) \oplus D) \oplus \harm^{3}_{cyl}$
by proposition \ref{cylfderprop}. The first term is the tangent space to the
$\dkaq$-orbit of $\varphi$, so the derivative at $(\varphi, id)$ of
\begin{equation*}
\dcal{R} \times \dkaq \to \ztreq, \;\: (\beta, \phi) \mapsto \phi^{*}\beta
\end{equation*}
is surjective. By the submersion theorem it is an open map on a neighbourhood
of $(\varphi,id)$.
Using the regularity of isometries of EAC manifolds from proposition
\ref{eacisomprop} it follows (as in corollary \ref{smoothslicecor}) that
%Thus if $U \subseteq \dcal{R}$ is a small neighbourhood of $\varphi$ then
%$U\dkaq$ is open in the closure of $\qcal{X}$ in $\ztreq$.
%If $\psi \in \dcal{R}$, $\phi \in \dkaq$ and $\phi^{*}\psi$ is smooth
%and EAC then $\phi$ is also smooth and EAC by proposition \ref{eacisomprop}.
%Therefore $U\dkaq \cap \qcal{X} = U\qcal{D}$, and $U\qcal{D}$ is open
%in~$\qcal{X}$.
%
%Hence
$\dcal{R} \to \qcal{X}/\qcal{D}$ is an open map. It is also injective,
since $\pi_{\mathcal{M}}$ is $\dcal{D}$-invariant and
injective on $\dcal{R}$.
\end{proof}

For our argument to work we may need to shrink $\mathcal{Q}$ by replacing it
with a neighbourhood of $(\Omega,\omega)$ in $\mathcal{Q}$.

\begin{lem}
\label{qmodlem}
If the pre-moduli $\mathcal{Q}$ of Calabi-Yau structures is shrunk
sufficiently then
$\qcal{X}/\qcal{D}$ is homeomorphic to a neighbourhood of $\varphi\dcal{D}$
in $\calxa/\dcal{D}$.
\end{lem}

\begin{proof}
The natural map $f : \qcal{X}/\qcal{D} \to \calxa/\dcal{D}$ is injective
by proposition \ref{autinvprop}.

Let $\mathcal{Y}$ be the space of \cystr s on $X$.
One of the properties of the pre-moduli space $\calq$ is that there is
%The construction of the
%moduli space of \cystr s on $X$ uses slice results similar to proposition
%\ref{qsliceprop}. This allows us to define on
a neighbourhood $U$ of $(\Omega,\omega)$ in $\mathcal{Y}$
and a continuous map
\begin{equation*}
P : U \to C^{\infty}(TX) \times \mathcal{Q}, \;\:
(\beta, \gamma) \mapsto (V, \Omega', \omega')
\end{equation*}
such that $(\beta,\gamma) = (\exp V)^{*}(\Omega',\omega')$ for any
$(\beta,\gamma) \in U$ (the map is given by restricting a local right inverse
of the submersion (\ref{gkeq}) to a small $U$).

Let $\mathcal{X}_{U} = \{ \psi \in \calxa : B(\psi) \in U \}$.
If $\psi \in \mathcal{X}_{U}$ let $V = P(B(\psi))$,
$\phi = \exp \rho V \in \dcal{D}$ and $g(\psi) = \phi^{*}\psi$.
Then $B(g(\psi)) \in \mathcal{Q}$, so $\psi \in \qcal{X}$.
Obviously $f(g(\psi)\qcal{D})) = \psi\dcal{D}$. Since $f$ is injective
$g$ induces a well-defined map $\mathcal{X}_{U}\dcal{D} \to \qcal{X}/\qcal{D}$.
$g$ is an inverse for $f$ on a neighbourhood of $\varphi\dcal{D}$ in
$\calxa/\dcal{D}$, so the result follows.
\end{proof}

\begin{thm}
\label{dmodthm}
$\dcal{M}$ has a unique smooth structure such that
\[ \pi_{\mathcal{M}} : \dcal{M} \to H^{3}(M) \times H^{2}(X) \]
is an immersion.
\end{thm}

\begin{proof}
Same as theorem \ref{modmfdthm}.
%Proposition \ref{qsliceprop} and lemma \ref{qmodlem} show that
%for any $\varphi \in \calxa$ the pre-moduli space $\dcal{R}$
%is homeomorphic to a neighbourhood $U$ of $\varphi\dcal{D}$ in~$\dcal{M}$,
%and
%\[ \pi_{\mathcal{M}} : U \to H^{3}(M) \times H^{2}(X) \]
%is a homeomorphism onto an embedded manifold. To prove that we can use the maps
%$\dcal{R} \to U$ as coordinate charts for $\dcal{M}$ we need to check that
%the transition functions are smooth. But on an overlap $U_{1} \cap U_{2}$
%both charts define the unique smooth structure for which
%$\pi_{\mathcal{M}} : U_{1} \cap U_{2} \to H^{3}(M) \times H^{2}(X)$
%is an embedding, so we are done.
\end{proof}

If $\delta_{1} > \delta_{2} > 0$ then
$\mathcal{M}_{\delta_{1}} \to \mathcal{M}_{\delta_{2}}$ is injective by
proposition \ref{eacisomprop}, and $\mathcal{M}_{\delta_{1}}$ must be an
open submanifold of $\mathcal{M}_{\delta_{2}}$ since $\pi_{\mathcal{M}}$
is an immersion on both spaces.
Similarly $\dcal{M} \to \pcal{M}$ is injective for any $\delta > 0$, so
\[ \pcal{M} = \bigcup_{\delta > 0} \dcal{M} . \]
To finish the proof of the main theorem \ref{maincylthm} it remains only
to observe

\begin{lem}
For any $\delta > 0$ the natural map $\dcal{M} \to \pcal{M}$ is
a homeomorphism to an open subset.
\end{lem}

\begin{proof}
We need to show that $\dcal{M} \to \pcal{M}$ is open, i.e. that
if $U \subseteq \calxa$ with $U\dcal{D}$ open in $\dcal{X}$ then
$U\pcal{D}$ is open in $\pcal{X}$. By the definition of the topology on
$\pcal{X}$ this means that $U\pcal{D} \cap \mathcal{X}_{\delta'}$ is open
in $\mathcal{X}_{\delta'}$ for any $\delta' > \delta$.
But proposition \ref{eacisomprop} implies that
$U\pcal{D} \cap \mathcal{X}_{\delta'} = U\mathcal{D}_{\delta'}$, which
is open in $\mathcal{X}_{\delta'}$ since $\dcal{M} \to \mathcal{M}_{\delta'}$
is a local diffeomorphism.
\end{proof}

This concludes the proof of theorem \ref{maincylthm}.


\subsection{Properties of the moduli space}
\label{propertiessub}

We look at some local properties of the moduli space $\pcal{M}$
on an EAC \gtmfd{} $M$,
which follow from the existence of a pre-moduli space $\mathcal{R}$
with tangent space $\harm^{3}_{cyl}$.

First, the boundary map $B$ maps $\harm^{3}_{cyl}$ onto $\harm^{3}_{SU,A}$,
so proposition \ref{qasubmfdprop} implies that
$B : \mathcal{R} \to \mathcal{Q}_{A}$ is a submersion.
As $\mathcal{Q}_{A}$ is homeomorphic to an open set in $\mathcal{N}_{A}$
it follows that $B : \pcal{M} \to \mathcal{N}_{A}$ is a submersion onto its
image and we have proved theorem \ref{modsubmersethm}.

We can now deduce corollary \ref{fixboundarycor}. It suffices to
show that the fibres of $B : \pcal{M} \to \mathcal{N}_{A}$ are locally
diffeomorphic to the compactly supported subspace
$H^{3}_{0}(M) \subseteq H^{3}(M)$. 

\begin{lem}
Let $\varphi$ be an EAC torsion-free \gtstr{} on $M^{7}$, $\mathcal{R}$ the
pre-moduli space of EAC torsion-free \gtstr s near $\varphi$ and $\mathcal{Q}$
the pre-moduli space of \cystr s near~$B(\varphi)$.% The map
\begin{equation*}
\label{fibrecharteq}
\pi_{H} : \mathcal{R} \to H^{3}(M) %, \;\: \psi \mapsto [\psi]
\end{equation*}
maps a neighbourhood of the fibre of $B :\mathcal{R} \to \mathcal{Q}_{A}$
containing $\varphi$ diffeomorphically to an open subset of
the affine space $[\varphi] + H^{3}_{0}(M)$.
\end{lem}

\begin{proof}
If $\psi$ is in the same fibre as $\varphi$ then $\psi - \varphi$ is
exponentially decaying, so lemma \ref{cylfylem} implies that
$[\psi - \varphi] \in H^{3}_{0}(M)$.
The tangent space to the fibre at $\varphi$ is the kernel of the derivative
of the submersion $B$, i.e. the subspace $\harm^{3}_{+}$ of decaying forms 
in $\harm^{3}_{cyl} = T_{\varphi}\mathcal{R}$. By theorem
\ref{derhamthm} $\harm^{3}_{+} \cong H^{3}_{0}(M)$, and the result follows.
\end{proof}

We can also use the pre-moduli space charts to check that
$\defstrp \to H^{3}(M) \oplus H^{4}(M)$ restricts to Lagrangian immersions
of the fibres of the boundary map. To prove proposition \ref{cyllagembprop} it
suffices to prove that for each $\varphi$ the derivative is a Lagrangian
embedding on the tangent space to the pre-moduli space.

\begin{lem}
Let $\varphi$ be an EAC torsion-free \gtstr{} on $M^{7}$ and $\mathcal{R}$ the
pre-moduli space %of EAC torsion-free \gtstr s
near $\varphi$.
Then the derivative of $\calr \to H^{3}(M) \oplus H^{4}(M)$
embeds the tangent space at $\varphi$ to the fibre of $B : \calr \to \calqa$
as a Lagrangian subspace of $H^{3}_{0}(M) \oplus H^{4}_{0}(M)$.
\end{lem}

\begin{proof}
As above the tangent space to the fibre of $\varphi$ is
$T_{\varphi}\calr = \harm^{3}_{+}$. Since harmonic $3$-forms of type $1$ or $7$
are parallel the elements of $\harm^{3}_{+}$ have type $27$.
By proposition \ref{derthetaprop}, the derivative on the fibre is
\[ \harm^{3}_{+} \to H^{3}_{0}(M) \oplus H^{4}_{0}(M), \;\:
\chi \mapsto ([\chi], -[*\chi]) . \]
If $\chi_{1}, \chi_{2} \in \harm^{3}_{+}$ then the symplectic pairing of their
images in $H^{3}_{0}(M) \oplus H^{4}_{0}(M)$ is just
\begin{equation*}
\linner{\chi_{1}, - \chi_{2}} - \linner{- \chi_{1}, \chi_{2}} = 0 .
\end{equation*}
Hence the derivative on the fibre is a Lagrangian inclusion.
\end{proof}

Finally, to confirm the formula for the dimension in proposition
\ref{dimmodprop} we just have to compute the dimension of $\harm^{3}_{cyl}$.
Recall from subsection \ref{hodgesub} that $A^{m}$ is the image of
$j^{*} : H^{m}(M) \to H^{m}(X)$,
that $H^{m}(X) = A^{m} \oplus E^{m}$
is an orthogonal direct sum and
that the harmonic representatives of the summands are denoted by
$\mathcal{A}^{m}$ and $\mathcal{E}^{m}$ respectively.

\hfuzz=4pt
\begin{lem}
\label{hmidlaglem}
Let $M^{4k+3}$ be an oriented EAC %exponentially asymptotically cylindrical
manifold with cross-section~$X$. Then
$A^{2k+1} \subseteq H^{2k+1}(X)$ has dimension $\half b^{2k+1}(X)$.
\end{lem}
\hfuzz=0.1pt

\begin{proof}
$H^{2k+1}(X)$ is a symplectic vector space under the Poincar\'e pairing.
In particular $b^{2k+1}(X)$ is even.
%
$* : H^{2k+1}(X) \to H^{2k+1}(X)$ maps $A^{2k+1}$ isomorphically to its
orthogonal complement $E^{2k+1}$. The Poincar\'e pairing on $H^{2k+1}(X)$
can be expressed as $<\cdot, *\cdot>$, so $A^{2k+1} \subseteq H^{2k+1}(X)$
is a Lagrangian subspace.
\end{proof}

In particular for any EAC \gmfd{} $M$ with cross-section $X$ the long exact
sequence (\ref{relexacteq}) for relative cohomology gives
\begin{equation}
\label{b3cptdimeq}
\dim H^{3}_{0}(M) = b^{3}(M) - \half b^{3}(X) .
\end{equation}

\begin{lem}
\label{dimmodlem}
$\dim \harm^{3}_{cyl} = b^{4}(M) + \half b^{3}(X) - b^{1}(M) - 1$.
\end{lem}

\begin{proof}
As before we let $E^{2}_{d} = E^{2} \cap H^{2}_{d}(X)$.
As a consequence of corollary \ref{hsuacor} and theorem \ref{absderhamthm}
we find that
$\pi_{H} : \harm^{3}_{cyl} \to H^{3}(M)$ is surjective and the kernel
is mapped bijectively to $E^{2}_{8}$ by
$\pi_{H,e} : \harm^{3}_{cyl} \to H^{2}(X)$.
Hence $\dim \harm^{3}_{cyl} = b^{3}(M) + \dim E^{2}_{8}$.

The dimension of $E^{2}$ can be computed from the long exact sequence
(\ref{relexacteq}) for relative cohomology together with
(\ref{b3cptdimeq}).
\begin{multline*}
\dim E^{2} = \dim \ker (e : H^{3}_{cpt}(M) \to H^{3}(M)) \\ = 
b^{4}(M) - \dim H^{3}_{0}(M) = b^{4}(M) - b^{3}(M) +\half b^{3}(X) .
\end{multline*}
By propositions \ref{aeisoprop} and \ref{bochnerprop}
\[ \dim E^{2}_{6} = \dim A^{1} = b^{1}(M),
\;\: \dim E^{2}_{1} = \dim A^{0} = 1 . \]
Hence
\begin{equation*}
\dim E^{2}_{8} =  b^{4}(M) - b^{3}(M) +\half b^{3}(X) - b^{1}(M) - 1 . \qedhere
\end{equation*}
\end{proof}


\section{Deformations of EAC $Spin(7)$-manifolds}
\label{cylspmodsec}

In this section we prove that the moduli space of torsion-free exponentially
asymptotically cylindrical \spstr s on a manifold
with cylindrical ends is smooth and study some of its local properties.
The method is very similar to that for the $G_{2}$ case.


\subsection{Results}

Let $M^{8}$ be a connected oriented manifold with cylindrical ends
and cross-section $X^{7}$.
%
For $\delta > 0$ let $\dcal{X}$ be the space of torsion-free EAC
\spstr s with rate $\delta$ on $M$
(see definition \ref{cylsp7def}).
$\dcal{X}$ has the topology of a subspace of the space of exponentially
asymptotically translation-invariant $4$-forms.

Let $\pcal{X} = \bigcup_{\delta>0} \dcal{X}$. If $\delta_{1} > \delta_{2} > 0$
then the inclusion
$\mathcal{X}_{\delta_{1}} \hookrightarrow \mathcal{X}_{\delta_{2}}$
is continuous, so we can give $\pcal{X}$ the direct limit topology.
%i.e. $U \subseteq \pcal{X}$ is open if and only if $U \cap \dcal{X}$
%is open in $\dcal{X}$ for all $\delta > 0$. 
Let $\pcal{D}$ be the group of %exponentially asymptotically cylindrical
EAC diffeomorphisms of~$M$ with any positive rate
(in the sense of definition \ref{cyldiffdef})
that are isotopic to the identity. $\pcal{D}$~acts on $\pcal{X}$ by pull-backs,
and the
\emph{moduli space of torsion-free %exponentially asymptotically cylindrical
EAC \spstr s} on $M$ is the quotient $\pcal{M} = \pcal{X}/\pcal{D}$.

\begin{rmk}
The definition of an EAC \spstr{} involves a normalisation of the
asymptotic limit (cf. remark \ref{normalisermk}).
\end{rmk}

The main result of this section is

\begin{thm}
\label{mainspcylthm}
$\pcal{M}$ is a smooth manifold and
$\pi_{H} : \pcal{M} \to H^{4}(M)$ is an immersion.
\end{thm}

Since $M$ is non-compact, the notion of self-duality does not make sense for
arbitrary classes in~$H^{4}(M)$, but only for compactly supported ones.
Let $H^{4}_{+}(M) \subseteq H^{4}_{0}(M)$ be the self-dual subspace and denote
its dimension by $b^{4}_{+}(M)$.  The dimension of the moduli space is given by

\begin{prop}
\label{spdimmodprop}
$\dim \pcal{M} = b^{4}(M) - b^{4}_{+}(M) - b^{1}(M) + 1 + b^{1}(X)$.
\end{prop}

\begin{rmk}
The term $b^{4}(M) - b^{4}_{+}(M)$ is the dimension of the space of bounded
anti-self-dual forms, while $-b^{1}(M) + 1 + b^{1}(X)$ is the dimension of the
space of bounded parallel positive spinors on the \spmfd{} $M$.
This is analogous to the terms in the dimension formula for the moduli space
on a compact \spmfd{} (see remark \ref{sptermrmk}).
\end{rmk}

The asymptotic limit of a torsion-free EAC \spstr{} is
of the form $*_{\varphi}\varphi + dt \wedge \varphi$,
where $\varphi$ is a torsion-free \gtstr{} on the cross-section $X^{7}$
and $*_{\varphi}$ the induced Hodge star on $X$
(see proposition \ref{cylspprop}).
This gives a natural boundary map $B : \pcal{M} \to \mathcal{N}$
where $\mathcal{N}$ is the moduli space of torsion-free \gtstr s on $X$.
As before, let $A^{m} \subseteq H^{m}(X)$ denote the image of the
pull-back map $j^{*} : H^{m}(M) \to H^{m}(X)$. % in the long exact sequence
%for relative cohomology (\ref{relexacteq}).
If a torsion-free \spstr{} $\psi$ is asymptotic to
$*_{\varphi}\varphi + dt \wedge \varphi$ then
\begin{equation*}
[*_{\varphi}\varphi] = j^{*}([\psi]) \in A^{4} ,
\end{equation*}
so the image of $B : \pcal{M} \to \mathcal{N}$ is contained in a
subset determined by the topology of the pair $(M,X)$
\begin{equation}
\label{spnadefeq}
\mathcal{N}_{A} = \{ \varphi\xcal{D} \in \mathcal{N} :
[*\varphi] \in A^{4} \} .
\end{equation}
$\caln \to H^{4}(X)$ is a local diffeomorphism by theorem \ref{maincptthm},
so $\caln_{A}$ is a submanifold of $\caln$.

\begin{thm}
\label{spsubmersethm}
\begin{equation}
\label{spsubmerseeq}
B : \pcal{M} \to \mathcal{N}_{A}
\end{equation}
is a submersion.
\end{thm}

Since the methods used are entirely local they do not tell us anything
about the global properties of $\pcal{M}$ or the image of
(\ref{spsubmerseeq}).

We will show that the fibres of the submersion (\ref{spsubmerseeq})
are locally diffeomorphic to the compactly supported anti-self-dual cohomology
$H^{4}_{-}(M) \subseteq H^{4}_{0}(M)$. The fibre over
$\varphi$ corresponds to the moduli space of torsion-free \spstr s
asymptotic to $*\varphi + dt \wedge \varphi$. Thus

\begin{cor}
\label{spfixboundarycor}
The moduli space of torsion-free \spstr s on $M$ exponentially asymptotic to a
fixed cylindrical \spstr{} on $X \times \bbr$ is a manifold locally
diffeomorphic to $H^{4}_{-}(M)$.
\end{cor}


\subsection{Proof outline}
\label{cylspoutlinesub}

We use a set-up entirely analogous to the $G_{2}$ case.
Each $\psi \in \dcal{X}$ defines an EAC metric, and hence a parameter
$\epsilon_{1}(\psi)$ (cf.~proposition \ref{lapindprop}). Let
\begin{equation*}
\calxa = \{ \psi \in \dcal{X} : \epsilon_{1}(\psi) > \delta \} .
\end{equation*}
$\epsilon_{1}$ depends lower semi-continuously on the
asymptotic model by lemma~\ref{epsionectslem}, so
$\calxa$ is an open subset of $\dcal{X}$.
Let $\dcal{D}$ be the group of exponentially
cylindrical diffeomorphisms of $M$ with rate $\delta$, and
\mbox{$\dcal{M} = \calxa/\dcal{D}$}.

We now consider how to adapt the steps of section \ref{cptsec} to
construct pre-moduli spaces of torsion-free EAC \spstr s near
$\psi \in \calxa$.

Given $\psi \in \calxa$, the asymptotic limit has the form
$B(\psi) = *\varphi + dt \wedge \varphi$ where $\varphi$ is a torsion-free
\gtstr{} on $X$.
%To simplify the problem of describing a neighbourhood of $\psi\dcal{D}$
%in $\dcal{M}$ we use our understanding of the deformation theory of \cystr s.
In \S \ref{cptsec} we saw that there is a pre-moduli space
of torsion-free \gtstr s near $\varphi$ which we now denote by $\calq$.
We can identify $\calq$ with a space of torsion-free cylindrical \spstr s
on $X \times \bbr$, and let
\begin{equation*}
\qcal{X} = \{ \psi \in \calxa : B(\psi) \in \mathcal{Q} \} .
\end{equation*}
%and $\qcal{D} \subseteq \dcal{D}$ the subgroup of diffeomorphisms
%asymptotic to automorphisms of the cylindrical \spstr{}
%$*\varphi + dt \wedge \varphi$.
%By proposition \ref{autinvprop} $\qcal{D}$ acts on $\qcal{X}$.
%
%We use slice arguments to study a neighbourhood of $\psi\qcal{D}$ in
%$\qcal{X}/\qcal{D}$. In order to be able to apply analysis results we need
%to use Banach spaces of forms, so we work with weighted Hölder $\holdad{k}$
%spaces, for some fixed $k \geq 1, \alpha \in (0,1)$.
Note that the boundary values of elements of $\qcal{X}$ must lie in
\begin{equation}
\label{spqaeq}
\mathcal{Q}_{A} = \{*\varphi' + dt \wedge \varphi' \in \mathcal{Q}
: [*\varphi'] \in A^{4} \}
\end{equation}
where $A^{m}$ is the image of $j^{*} : H^{m}(M) \to H^{m}(X)$ (cf. discussion
before theorem \ref{spsubmersethm}). Let $\rho$ be a cut-off function
for the cylinder on $M$. $\rho\mathcal{Q}_{A}$ can be identified with
a subspace of smooth asymptotically
translation-invariant $4$-forms supported on the cylinder of $M$. Let
\[ \calcfq \subseteq \holdad{k}(\Lambda^{4}) + \rho \mathcal{Q}_{A} \]
be the subspace of forms which take values in $\spspace$, in other
words the space of \spstr s which are $\holdad{k}$-asymptotic to elements
of $\calqa$. Then $\qcal{X} \hookrightarrow \calcfq$
continuously.

Since $\calq \to H^{4}(X)$ is a local diffeomorphism it is,
unlike in the $G_{2}$ case, immediately clear that
$\calqa \subseteq \calq$ is a submanifold. Its tangent space can be considered
as a subspace of the translation-invariant harmonic $4$-forms on the cylinder,
\[ \harmgta = T_{\varphi}\calqa \subset \harm^{4}_{\infty} . \]
We will see that it is precisely the image under the boundary map $B$ of
the space $\harm^{4}_{cyl}$ of harmonic tangents at $\psi$ to the space of
$\holdad{k}$-asymptotically translation-invariant \spstr s.

In the next subsections we describe how to set up a slice and find
a smooth pre-moduli space $\dcal{R}$ near $\psi$ with
$T_{\psi}\dcal{R} = \harm^{4}_{cyl}$, generalising from the compact case.
The regularity and slice arguments that complete the proof of
theorem \ref{mainspcylthm} are very similar to the $G_{2}$ case and therefore
omitted. Just as in the $G_{2}$ case, one finds that
the elements of $\dcal{R}$ are smooth, that $\dcal{R} \to \dcal{M}$
can be used as a coordinate chart, and that
$\pcal{M} = \bigcup_{\delta > 0} \dcal{M}$ is therefore a smooth manifold.

Once the coordinate charts are set up it is easy to deduce proposition
\ref{spdimmodprop}, theorem \ref{spsubmersethm} and corollary
\ref{spfixboundarycor}. This is done in \S \ref{sppropertiessub}.


\subsection{The slice}

Fix $k \geq 1$, $\delta > 0$, $\alpha \in (0,1)$ and $\psi \in \calxa$.
We find a submanifold $\mathcal{S} \subseteq \calcfq$
whose tangent space at $\psi$ is a direct complement to the
diffeomorphism orbit.

The fixed \spstr{} $\psi$ is used to define an EAC metric and a Hodge star.
It also defines type decompositions of the exterior bundles
(\ref{sp7formdecompeq}) and of the
spaces of harmonic forms, as described in subsection \ref{harmholsub}.
Note that the map
$TM \to \Lambda^{3}_{8}, \; v \mapsto v \lrcorner \psi$ is a bundle
isomorphism. Therefore Lie derivatives
$\mathcal{L}_{V}\psi = d(V \lrcorner \psi)$ are precisely exterior
derivatives of $3$-forms of type $8$.

Recall the notation for harmonic forms from %subsection \ref{hodgesub},
the summary in \S \ref{hodgesummary},
in particular that $\harm^{m}_{0}$
and $\harm^{m}_{\infty}$ denote the spaces of bounded harmonic forms on
$M$ and harmonic translation-invariant forms on $X \times \bbr$ respectively.
%
\gtstr s on $X$ are identified with cylindrical \spstr s on $X \times \bbr$
by $\varphi' \leftrightarrow *_{\varphi'}\varphi' + dt \wedge \varphi'$.

The torsion-free \gtstr{} $\varphi$ on the cross-section $X^{7}$ induced by
$\psi$ defines type decompositions of $\Omega^{*}(X)$.
In view of proposition \ref{derthetaprop}, the tangent space to $\calqa$
corresponds to
\begin{equation*}
\harmgta = \{ \chi + dt \wedge *(\threequart \pi_{1} + \pi_{7} - \pi_{27})\chi
\: : \: \chi \in \cala^{4} \} \subseteq \harm^{4}_{\infty} .
\end{equation*}
Recall that the tangent space to the space of \spstr s consists of
sections of $\Lambda^{4}_{1\oplus7\oplus35}$.
$T_{\psi}\calcfq$ is therefore the space of $4$-forms of type $1$, $7$ and $35$
which are $\holdad{k}$-asymptotic to elements of $\harmgta$. Let
$\harm^{4}_{cyl} \subset T_{\psi}\calcfq$ be the subspace of harmonic forms.

\begin{lem}
\label{gtasurjlem}
$B : \harm^{4}_{cyl} \to \harmgta$ is surjective.
\end{lem}

\begin{proof}
Proposition \ref{spaesplitprop} implies that $\cala^{4}$ splits into type
components, so $\harmgta$ splits as
\begin{equation*}
%\label{gtaspliteq}
\harmgta = \{ \chi + dt \wedge * \threequart \chi : \chi \in \cala^{4}_{1} \}
\oplus \{ \chi + dt \wedge * \chi : \chi \in \cala^{4}_{7} \}
\oplus \{ \chi - dt \wedge * \chi : \chi \in \cala^{4}_{27} \} .
\end{equation*}
It implies also that $\harm^{4}_{0,7} \subset \harm^{4}_{cyl}$ is mapped
onto the middle term by the boundary map $B$.

For $\chi \in \cala^{4}_{27}$, the EAC Hodge decomposition
theorem \ref{absderhamthm}
implies that there is some $\phi \in \harm^{4}_{abs}$ such that
$B(\phi) = \chi$. Then $\phi-*\phi$ lies in the space $\harm^{4}_{0,35}$
of bounded harmonic anti-self-dual forms and
$B(\phi - *\phi) = \chi - dt \wedge *\chi$,
so $\phi - *\phi \in \harm^{4}_{cyl}$.
%
Finally, $*\varphi + dt \wedge \threequart \varphi$ can be written
as $\frac{7}{8}B(\psi) + \eightth (*\varphi - dt \wedge \varphi)$,
and the second term lies in the image of $\harm^{4}_{0,35}$.
\end{proof}

Let $\dkaq$ be the group of diffeomorphisms of $M$ which are isotopic to the
identity and $\holdad{k+1}$-asymptotic to a cylindrical automorphism of the
cylindrical \spstr{} $*_{\varphi}\varphi + dt \wedge \varphi$.
$\dkaq$ acts on $\qcal{X}$ by proposition \ref{autinvprop}.
The elements of a
neighbourhood of the identity in $\dkaq$ can be written as
$\exp (V + \rho V_{\infty})$, where $V$ is a $\holdad{k+1}$ vector field on $M$
and $V_{\infty}$ is a translation-invariant vector field on $X \times \bbr$
with $\mathcal{L}_{V_{\infty}}(*_{\varphi}\varphi + dt \wedge \varphi) = 0$,
i.e. $V_{\infty} \in (\harm^{1}_{\infty})^{\sharp}$.
Therefore if we set
\begin{equation*}
%\label{deq}
D = \rho (\harm^{1}_{\infty})^{\sharp} \lrcorner \psi
\subseteq \Omega^{3}_{8}(M)
\end{equation*}
then the tangent space to the $\dkaq$-orbit at $\psi$ is
\[ \{ \mathcal{L}_{V + \rho V_{\infty}}\psi : 
V \in \holdad{k+1}(TM), V_{\infty} \in (\harm^{1}_{\infty})^{\sharp} \} =
d(\holdad{k+1}(\Lambda^{3}_{8}) \oplus D) . \]

\begin{defn}
Let $W \subseteq \holdad{k}(\Lambda^{4}_{35})$ be the \ltwoorth{} subspace
to the decaying harmonic forms $\harm^{4}_{+}$.
\end{defn}

\begin{prop}
%\label{sltprop}
\begin{equation*}
T_{\psi}\calcfq = d(\holdad{k+1}(\Lambda^{3}_{8}) \oplus D) \oplus
\harm^{4}_{cyl} \oplus W .
\end{equation*}
\end{prop}

\begin{proof}
The Hodge decomposition theorem \ref{hodgedecompthm} implies that a non-zero
decaying exact form cannot be anti-self-dual, so
$W \cap d(\holdad{k+1}(\Lambda^{3}_{8}) \oplus D) = 0$ and the sum is direct.
It follows from lemma \ref{gtasurjlem} that any element of
$T_{\psi}\calcfq$ can be written as $\phi + \beta$, with
$\phi \in \harm^{4}_{cyl}$ and
$\beta \in \holdad{k}(\Lambda^{4}_{1\oplus7\oplus35})$ \ltwoorth{} to the
decaying harmonic forms. It therefore suffices to show that
$\pi_{1\oplus7}\beta$ lies in the image of
\[ \pi_{1\oplus7}d : \holdad{k+1}(\Lambda^{3}_{8}) \oplus D \to
\holdad{k}(\Lambda^{4}_{1\oplus7}) . \]
Just as in the compact case proposition \ref{sportsumprop} this is
easily proved by interpreting the map as the Dirac operator
(cf. (\ref{eatdiraceq}) is Fredholm).
\end{proof}

We want to take as our slice for the $\dcal{D}$-action at $\psi$ a
submanifold $\mathcal{S} \subseteq \calcfq$ with
$T_{\psi}\mathcal{S} = \harm^{4}_{cyl} \oplus W$.
To this end we pick a map
\begin{equation}
%\label{expdefeq}
\exp : U \to \calcfq
\end{equation}
on a neighbourhood $U$ of $0$ in $T_{\psi}\calcfq$,
such that $D\exp_{0} = id$. We also insist that $\exp$ maps
decaying forms to decaying forms and smooth forms to smooth forms.
We can do this since %by (\ref{zcylspliteq})
the decaying forms have a finite-dimensional complement of smooth forms in
$T_{\psi}\calcfq$. We then choose
\begin{equation}
%\label{slicedefeq}
\mathcal{S} = \exp ((\harm^{4}_{cyl} \oplus W) \cap U) .
\end{equation}


\subsection{Smoothness of the pre-moduli space}

Let $\dcal{R} \subseteq \mathcal{S}$ be the subset of $\holdad{k}$
torsion-free \spstr s.
$\dcal{R}$ is the pre-moduli space of torsion-free \spstr s near $\psi$.

If $\psi' \in \calcfq$ then $\psi'$ is asymptotic to a torsion-free
cylindrical \spstr, so $d\psi'$ decays. Moreover, elements
of $\calcfq$ are asymptotic to elements of
$\mathcal{Q}_{A} \subseteq \mathcal{Q}$ by definition.
Therefore $\psi'$ is asymptotic to $*_{\varphi'}\varphi' + dt \wedge \varphi'$
with $[*_{\varphi'}\varphi'] \in A^{4}$,
so that $d\psi' \in d\holdad{k+1}(\Lambda^{4})$.

Recall that by theorem \ref{salamonthm}, a closed \spstr{} $\psi'$
is torsion-free if and only if $d\psi' = 0$. Thus $\dcal{R}$ is the
zero set of
\[ d : \mathcal{S} \to d\holdad{k+1}(\Lambda^{4}) . \]
To show that $\dcal{R}$ is a manifold we check that the derivative
at $\psi$ is surjective.

\begin{prop}
The map $d : T_{\psi}\calcfq \to d\holdad{k+1}(\Lambda^{4})$ is $0$ on
$d(\holdad{k+1}(\Lambda^{2}_{7}) \oplus D) \oplus \harm^{4}_{cyl}$
and bijective on~$W$.
\end{prop}

\begin{proof}
Any closed anti-self-dual form is harmonic, so $W$ contains no closed forms.
Conversely, any element of $d\holdad{k+1}(\Lambda^{4})$ can be written as
$d\beta$ with $\beta$ coexact by the Hodge decomposition theorem
\ref{hodgedecompthm}. Then $\beta - *\beta \in W$ and
$d(\beta - *\beta) = d\beta$.
\end{proof}

Thus we can apply the implicit function theorem to deduce that
a small neighbourhood of $\varphi$ in $\calx_{k,\calq}$,
the space of torsion-free \spstr s in $\calcfq$, is a manifold.
Also, $\dcal{R}$ is a submanifold of $\cals$.

\begin{prop}
%\label{premodprop}
If the slice $\mathcal{S}$ near $\psi$ is shrunk sufficiently then
the space $\dcal{R}$ of torsion-free EAC \spstr s in $\cals$ is a manifold.
Its tangent space at $\psi$ is $\harm^{4}_{cyl}$.
\end{prop}

The projection to de Rham cohomology $\pi_{H} : \harm^{4}_{cyl} \to H^{4}(M)$
is injective. This is because the boundary conditions on
$\phi \in \harm^{4}_{cyl}$ imply that if $j^{*}[\phi] = 0$ then $\phi$
is decaying, and therefore cannot be exact by integration by parts.
Hence
\[ \pi_{H} : \dcal{R} \to H^{4}(M) \]
is an embedding if $\dcal{R}$ is taken small enough.
We can then use regularity and slice arguments just like for the $G_{2}$ case
(\S \ref{cylregsub}--\ref{modconstrsub}) to show

\begin{prop}
If the slice $\cals$ is taken small enough then the elements of
$\dcal{R} \subset \cals$ are smooth and EAC.
\end{prop}

\begin{prop}
$\dcal{R} \to \dcal{M}$ is a homeomorphism onto an open subset.
\end{prop}

\noindent
Using pre-moduli spaces as charts we deduce

\begin{thm}
Let $M^{8}$ be an EAC \spmfd.
Then $\dcal{M}$ has a unique smooth structure such that
\[ \pi_{H} : \dcal{M} \to H^{4}(M) \]
is an immersion.
\end{thm}

Finally $\dcal{M} \to \pcal{M}$ is a homeomorphism onto an open subset
for any $\delta > 0$, so the union $\pcal{M} = \bigcup_{\delta > 0} \dcal{M}$
is a smooth manifold.
This completes the proof of the main theorem \ref{mainspcylthm}.


\subsection{Properties of the moduli space}
\label{sppropertiessub}

Now that we have found pre-moduli spaces that can be used as charts for
$\pcal{M}$ we can deduce some simple results about the local properties
of the moduli space. For a start, lemma \ref{gtasurjlem} means that
$B : T_{\psi}\calr \to T_{\varphi}\calqa$ is surjective, 
so $B : \calr \to \calqa$
is a submersion and theorem \ref{spsubmersethm} follows.

To find the dimension of the moduli space we just need to evaluate
the dimension of $T_{\psi}\calr = \harm^{4}_{cyl}$. From the proof
of lemma \ref{gtasurjlem} we see that
\begin{equation*}
\dim \harm^{4}_{cyl} = \dim \harm^{4}_{0,1\oplus7} + \dim \harm^{4}_{0,35} .
\end{equation*}
The first term corresponds to the bounded harmonic positive spinors,
with dimension given by proposition \ref{dimharmspprop}.
The second term corresponds to the bounded anti-self-dual harmonic forms.
This splits into two parts: the decaying anti-self-dual forms
$\harm^{4}_{+,35}$ and $\{ \phi - * \phi : \phi \in \harm^{4}_{E} \}$.
Its image in $H^{4}(M)$ is therefore a direct complement of the self-dual
compactly supported cohomology $H^{4}_{+}(M)$, so
\[ \dim \harm^{4}_{0,35} = b^{4}(M) - b^{4}_{+}(M) . \]
This proves proposition \ref{spdimmodprop}.

Finally, $\harm^{4}_{+,35}$ is the tangent space to the fibre of $\psi$
in the submersion $\calr \to \calqa$. The decaying anti-self-dual forms
represent precisely the compactly supported anti-self-dual cohomology
$H^{4}_{-}(M)$, so the projection from $\calr$ to the affine subspace
$[\psi] + H^{4}_{-}(M) \subset H^{4}(M)$ is a local diffeomorphism.
This proves corollary \ref{spfixboundarycor}.

%To prove proposition \ref{spdimmodprop} we just need to write
%$\dim \harm^{4}_{cyl}$ in terms of Betti numbers using Hodge theory.
%The surjective map $B : \harm^{4}_{cyl} \to \harm^{4}_{G_{2}}$ corresponds
%to the derivative of $B : \pcal{M} \to \calqa$, which proves theorem
%\ref{spsubmersethm}.


\chapter{$G$-metrics and Ricci-flat deformations}
\label{gricchap}

Let $G$ be one of the exceptional holonomy groups $Spin(7)$ or $G_{2}$.
We have previously discussed how a torsion-free \gstr{}
on a manifold $M$ (of dimension $8$ or $7$ respectively) 
determines a \emph{\gmetric},
i.e. a Riemannian metric of holonomy $\subseteq G$, and studied the deformation
theory of EAC \gstr s.
We now consider instead the deformations of \gmetric s
on compact and EAC manifolds.

\gmetric s on a manifold $M$ are Ricci-flat for $G = SU(n)$, $Sp(n)$,
$Spin(7)$ or $G_{2}$.
Wang \cite{wang91} proves a local converse for the case when $M$ is compact:
any small Ricci-flat deformation of a \gmetric{} still has holonomy contained
in $G$.
In other words the moduli space $\defmetg$ of \gmetric s is an open subset of
the moduli space $\defmetz$ of Ricci-flat metrics.
Wang proves the result case by case, but asks if there is a general proof.
We observe that the problem can be reduced in a uniform way to showing
unobstructedness for deformations of torsion-free \gstr s.
This has in turn been given a uniform treatment by Goto \cite{goto04}.

We prove furthermore that $\defmetg$ is a smooth manifold.
For $G = Spin(7)$ or $G_{2}$ we show also that the moduli space $\defstrg$
of torsion-free \gstr s is a locally trivial fibre bundle over $\defmetg$
and generalise the results to the case when $M$ is an EAC \gmfd.


\section{Results}

Let $G$ be one of the Ricci-flat holonomy groups $SU(n)$, $Sp(n)$,
$Spin(7)$ or $G_{2}$. In the cases $Spin(7)$, $G_{2}$ or $SU(3)$ we explained
in \S \ref{groupsec} how a \gmetric{} on a manifold $M$
of the appropriate dimension can be defined in terms of a \gstr, i.e.
a section of a subbundle $\gspace \subset \Lambda^{*}T^{*}M$, which is
torsion-free and in particular closed.
The tangent space to $\Gamma(\gspace)$ at a \gstr{} $\chi$
consists of the sections of the bundle of point-wise tangents to $\gspace$
at $\chi$, which is a vector subbundle $E_{\chi} \subseteq \Lambda^{*}T^{*}M$
associated to the \gstr.
Since $E_{\chi}$ is a bundle of forms the Hodge Laplacian acts
on~$\Gamma(E_{\chi})$. When $\chi$ is torsion-free this is the same as
the Lichnerowicz Laplacian from \S \ref{harmholsub}.
The same approach can be taken for
$SU(n)$ and $Sp(n)$. We do not need the details, but they can be found in
\cite{goto04}.

The group $\mathcal{D}$ of diffeo\-morphisms of $M$ isotopic to the
identity acts on the space of torsion-free \gstr s by pull-backs
and the quotient is the moduli space $\defstrg$ of torsion-free \gstr s.
Goto \cite{goto04} proves that the deformations of torsion-free \gstr s
are unobstructed in the following sense:

\begin{prop}
\label{gpremodprop}
Let $G = SU(n)$, $Sp(n)$, $Spin(7)$ or $G_{2}$, $M$ a compact \gmfd,
and $\chi$ a torsion-free \gstr{} on $M$.
Then there is a submanifold $\qcalg$ of the space of $C^{1}$ \gstr s such that
\begin{enumerate}
\item \label{smoothitem}
the elements of $\qcalg$ are smooth torsion-free \gstr s,
\item the tangent space to $\qcalg$ at $\chi$ is the space of harmonic
sections of $E_{\chi}$,
\item the natural map $\qcalg \to \defstrg$ is a homeomorphism onto a
neighbourhood of $\chi\mathcal{D}$ in $\defstrg$.
\end{enumerate}
\end{prop}

We proved this proposition for the cases $Spin(7)$, $G_{2}$ and $SU(3)$ in
\S \ref{cptchap}.
The spaces $\qcalg$ are pre-moduli spaces of torsion-free \gstr s and 
can be used as coordinate charts for~$\defstrg$.

$\mathcal{D}$ also acts on the space of Riemannian metrics, and we let
$\defmetg$ and $\defmetz$ denote the moduli spaces of \gmetric s and Ricci-flat
metrics respectively.
In \S \ref{ricdefgsec} we review the deformation theory of Ricci-flat
metrics and prove

\begin{thm}
\label{cptopenthm}
Let $G = SU(n)$, $Sp(n)$, $Spin(7)$ or $G_{2}$, and let $M$ be a compact \gmfd.
Then $\defmetg$ is open in $\defmetz$.
Moreover, $\defmetg$ is a smooth manifold and the natural map
\begin{equation*}
%\label{mainmapeq}
m : \defstrg \to \defmetg
\end{equation*}
that sends a torsion-free \gstr{} to the metric it defines is a submersion.
\end{thm}

\begin{rmk}
It is easy to see that $\defmetg$ is also closed in $\defmetz$, so it is a
union of connected
components.
It is an open problem whether there exist \emph{any} compact Ricci-flat
manifolds without a holonomy reduction.
\end{rmk}

\begin{rmk}
The quotient of the space of \gmetric s by the group of \emph{all}
diffeomorphisms of $M$ (not just the ones isotopic to the identity)
is a quotient of~$\defmetg$ with discrete fibres and in general an orbifold
(cf. remark \ref{orbirmk}).
\end{rmk}

The case $G = G_{2}$ of theorem \ref{cptopenthm} was proved by
M.Y. Wang \cite[Theorem $3.1$B]{wang91}.
For $G = Sp(n)$ or $Spin(7)$, Wang showed that $\defmetg \subseteq \defmetz$ is
open.
The case $G = SU(n)$ is a special case of a more general result by Koiso on
Einstein deformations of Kähler-Einstein metrics.

Let $X^{2n}$ be a compact Kähler-Einstein manifold. Koiso
\cite[Theorem $0.7$]{koiso83} shows that if the Einstein constant $e$
(equivalently the first Chern class $c_{1}(X)$) is non-positive and the complex
deformations of $X$ are unobstructed, then any small Einstein deformation of
the metric is Kähler with respect to some perturbed complex structure.
In other words the map from the moduli space of Kähler-Einstein structures
to the moduli space of Einstein metrics is open
(see e.g. \cite[\S 12J]{besse87} for a discussion).
The proof shows that near any Kähler-Einstein metric there is a smooth
pre-moduli space of Einstein metrics, so that the moduli space of
Kähler-Einstein metrics is an orbifold.
As explained in \S \ref{tiansub},
Tian \cite{tian87} and Todorov \cite{todorov89} show that on a compact
Calabi-Yau %(i.e. Ricci-flat Kähler)
manifold the obstructions to the complex deformations vanish.
Hence most of theorem \ref{cptopenthm} for $G = SU(n)$
follows from Koiso's theorem.

\begin{rmk}
Dai, X. Wang and Wei \cite{dai05} use the fact that $\defmetg$ is open
in $\defmet$ to deduce that any scalar-flat deformation of a Ricci-flat
\gmetric{} on a compact manifold remains a \gmetric.
\end{rmk}

The proof of theorem \ref{cptopenthm} given in \S \ref{ricdefgsec} is
a simplification
of Wang's argument for the case $G = G_{2}$. We observe that the
point-wise surjectivity of the derivative of $m$ follows from 
the equivariance properties of the Laplacians on manifolds with reduced
holonomy discussed in subsection \ref{harmholsub}.
This makes it easy to see that the proof applies also for the other Ricci-flat
holonomy groups, provided that the deformations of torsion-free \gstr s
are unobstructed.
%Secondly, we summarise the parts of the deformation
%theory for Ricci-flat metrics that are required, rather than quoting the
%results of Koiso. This makes it easier to generalise the result to non-compact
%settings.

The proof works also on non-compact manifolds if the deformation theory for
the \gstr s is in place.
For EAC \gtmfd s and \spmfd s we proved in \S \ref{eacgchap}
%sections \ref{cylgtmodsec} and \ref{cylspmodsec}
that there is a smooth moduli space $\defstrp$ of torsion-free
EAC \gstr s. Let $\defmetp$ denote the moduli space of EAC Ricci-flat metrics.

\begin{thm}
\label{eacopenthm}
Let $G = Spin(7)$ or $G_{2}$, and let $M$ be an EAC \gmfd.
Then $\defmetg$ is open in $\defmetp$.
Moreover, $\defmetg$ is a smooth manifold and the natural map
\begin{equation*}
m : \defstrp \to \defmetg
\end{equation*}
is open.
\end{thm}

In \cite{kovalev05} Kovalev proves the analogous result for EAC Calabi-Yau
manifolds, by an extension of Koiso's arguments for the compact Kähler-Einstein
case.
The discussion in \S \ref{eacopensub} of deformations of EAC Ricci-flat
metrics is similar to that in \cite{kovalev05}.

For $G = Spin(7)$ and $G_{2}$
we also study the structure of the map $m$ in greater detail.
As described in chapter \ref{eacgchap} the topology of a compact \gmfd{} $M$
determines whether each \gmetric{} on
$M$ has holonomy \emph{exactly}~$G$.
In \S \ref{bundlesec} we explain that the fibres of
$m$ are determined up to diffeomorphism by the topology in a similar way,
and show

\begin{thm}
\label{cptfibredefthm}
Let $G = Spin(7)$, $G_{2}$ or $SU(3)$, and let $M$ be a compact \gmfd.
Then
\begin{equation*}
m : \defstrg \to \defmetg
\end{equation*}
is a locally trivial fibre bundle. The typical fibre is a disjoint union of
real projective spaces.

When $M$ has holonomy exactly $G$ the typical fibre of $m$ is a point
if $G = Spin(7)$ or~$G_{2}$, and $S^{1}$ if $G = SU(3)$.
\end{thm}

The proof uses the fact that point-wise the space of \gstr s defining the same
inner product is
$SO(8)/Spin(7) \cong SO(7)/G_{2} \cong SO(6)/SU(3) \cong \bbr P^{7}$.
To be precise, the point-wise fibre can in each case be identified with
the space of lines in a real rank $8$ spin representation.
In principle, the result is an application of the observation of
Wang \cite{wang89} that a torsion-free \gstr{} can be recovered explicitly
from a parallel spinor (or an appropriate set of spinors). 
In the case $G = SU(n)$ for $n \geq 4$ the fibres of $m$ would be identified
in terms of the space of parallel \emph{pure} spinors, which is less
straight-forward to determine topologically.
Nevertheless theorem \ref{cptfibredefthm} is, like theorem \ref{cptopenthm},
the analogue of a result for Calabi-Yau manifolds:
\cite[Theorem $12.103$]{besse87} states that the moduli space of Calabi-Yau
structures on a compact manifold is a locally trivial fibration with compact
fibres over the moduli space of Calabi-Yau metrics (but does not describe
the fibres further).

The fact that $\defmetg$ is smooth suggests that it may be as natural to
consider in general as the moduli space of \gstr s $\defstrg$
(of course it makes no difference for irreducible \gtmfd s and \spmfd s).
For example Karigiannis and Leung \cite{karigiannis07} and
Grigorian and Yau \cite{grigorian08} both study the geometry of the
moduli space $\defstr$ of torsion-free \gtstr s on a compact \gtmfd.
They find that the curvature computations are more involved for \gtmfd s
$M^{7}$ with $b^{1}(M) > 0$. This is because curvature has different
expressions on the three components $H^{3}_{1}(M)$, $H^{3}_{7}(M)$ and
$H^{3}_{27}(M)$ of the tangent space to~$\defstr$
(which is isomorphic to $H^{3}(M)$). If $b^{1}(M) = 0$ then the $H^{3}_{7}(M)$
component vanishes, and the computations simplify.
On the other hand the tangent space to $\defmetgt$ is isomorphic to
$H^{3}_{1\oplus27}(M)$, whether $b^{1}(M) = 0$ or not.


%The global properties of the moduli space $\defstrg$ are not very well
%understood for irreducible \gmfd s, and theorem \ref{cptfibredefthm}
%does not provide any new information in this case (the definition of an
%\cystr{} includes a choice of a normalised holomorphic $(3,0)$-form,
%so the $S^{1}$ fibre in the case $G = SU(3)$ just corresponds to multiplying
%the form by a unit norm complex scalar).
%For locally reducible \gmfd s global questions
%about the fibration $m$, e.g. whether distinct connected components of the
%fibre can lie in the same component in~$\defstrg$, seem to be as difficult as
%questions about the global structure of $\defstrg$ for irreducible \gmfd s.

The main requirement for proving theorem \ref{cptfibredefthm} is
theorem \ref{cptopenthm}, so it can be
generalised to non-compact manifolds under similar assumptions.
In \S \ref{eacfibresub} we prove

\begin{thm}
\label{eacfibredefthm}
Let $G = Spin(7)$ or $G_{2}$, and let $M$ be an EAC \gmfd.
Then
\begin{equation*}
m : \defstrp \to \defmetg
\end{equation*}
is a locally trivial fibre bundle. The typical fibre is a disjoint union of
real projective spaces.

If $M$ has holonomy exactly $G$ then $m$ is a diffeomorphism.
\end{thm}



\section{Ricci-flat deformations of $G$-metrics}
\label{ricdefgsec}


\subsection{Killing vector fields}
\label{killsub}

Before we discuss the deformations of Ricci-flat metrics we make some remarks
about \emph{Killing vector fields}. These are the infinitesimal isometries of a
Riemannian manifold $(M,g)$,
i.e. vector fields $V$ such that the Lie derivative
$\mathcal{L}_{V}g$ vanishes.

\begin{defn}
Given a metric $g$ on $M$ let
$\delta^{*} : \Omega^{1}(M) \to \Gamma(S^{2}(T^{*}M))$ be the symmetric part of
the Levi-Civita connection
$\nabla : \Omega^{1}(M) \to \Gamma(T^{*}M \otimes T^{*}M)$.
\end{defn}

\noindent
The formal adjoint $\delta$ of $\delta^{*}$ is the restriction of
$\nabla^{*} : \Gamma(T^{*}M \otimes T^{*}M) \to \Omega^{1}(M)$ to
the symmetric part $\Gamma(S^{2}(T^{*}M))$.

\begin{prop}[{\cite[Lemma $1.60$]{besse87}}]
\label{delstarprop}
Let $g$ be a Riemannian metric on a manifold $M$ and $V$ a vector field.
Then $\mathcal{L}_{V}g = 2\delta^{*}V^{\flat}$, where $V^{\flat}$ denotes the
$1$-form $g(V, \cdot)$.
\end{prop}

The second Bianchi identity implies that
\begin{equation}
\label{bianchieq}
(2\delta + d\:\tr)Ric = 0
\end{equation}
for any Riemannian metric. The operator $2\delta + d\:\tr$ is sometimes called
the Bianchi operator, and it satisfies also the following useful identity.

\begin{lem}[{\cite[(14)]{kovalev05}}]
\label{delstarlem}
If $(M,g)$ is a Ricci-flat manifold then
\[ (2\delta + d\:\tr)\delta^{*} = \triangle . \]
\end{lem}

\begin{proof}
The anti-symmetric part of $\nabla$ on $\Omega^{1}(M)$ is $\half d$, so
$\delta^{*} = \nabla - \half d$.
Also $\tr \; \delta^{*} = d^{*}$ on~$\Omega^{1}(M)$.
Using proposition \ref{weitzenprop} we obtain
\[ (2\delta + d\:\tr)\delta^{*} =
2\nabla^{*}\nabla - \nabla^{*}d +d\:\tr\:\delta^{*} =
2\nabla^{*}\nabla - d^{*}d - dd^{*} = \triangle . \qedhere\]

\end{proof}

\begin{prop}
\label{killingprop}
Let $(M,g)$ be a Ricci-flat manifold. If $V$ is a Killing field then
the $1$-form $V^{\flat}$ is harmonic.
If $M$ is compact then the converse also holds.
\end{prop}

\begin{proof}
$\delta^{*}V^{\flat} = 0 \Rightarrow \triangle V^{\flat} = 0$ by
lemma \ref{delstarlem}.
%
Trivially $\nabla^{*}V^{\flat} = 0 \Rightarrow \delta^{*}V^{\flat} = 0$,
and if $M$ is compact then
$\triangle V^{\flat} = 0 \Rightarrow \nabla^{*}V^{\flat} = 0$
by corollary \ref{bochnercor}.
\end{proof}

This implies that, for any of the Ricci-flat holonomy groups $G$,
the space of infinitesimal automorphisms of a compact
\gmfd{} is $(\harm^{1})^{\sharp}$.


\subsection{Deformations of Ricci-flat metrics}
\label{ricdefsub}

We summarise some deformation theory for Ricci-flat metrics. This is mostly
taken from the deformation theory for Einstein metrics as explained in
\cite[\S 12C]{besse87}, specialised to the Ricci-flat case.
It turns out that in general we cannot prove
that the moduli space of Ricci-flat metrics is a manifold.

Let $M^{n}$ be a compact manifold. The diffeomorphism group $\mathcal{D}$ acts
on the space of Ricci-flat metrics on $M$ by pull-backs.
We define the moduli space $\defmetz$ of Ricci-flat metrics to be the quotient
of the space of Ricci-flat metrics by $\mathcal{D}$. (We do not divide by the
rescaling action of $\bbrp$ too, as is done in \cite{besse87}.)

Take $k \geq 2$, and let $g$ be a Ricci-flat Riemannian metric on $M$.
In order to study a neighbourhood of $g\mathcal{D}$ in $\defmetz$
we use slice arguments explained in \S \ref{modconstrsec}.
We include the space of smooth Riemannian metrics in the Hölder space
$\holda{k}(S^{2}T^{*}M)$, and let $\kpcal{D}$ be the $\holda{k+1}$ completion
of $\mathcal{D}$ ($\kpcal{D}$ is generated by $\exp$ of $\holda{k+1}$
vector fields).

By proposition \ref{delstarprop} the tangent space to the $\kpcal{D}$-orbit
at $g$ is $\delta^{*}_{g}\holda{k+1}(\Lambda^{1})$. 
Let $\slt$ be the kernel of $2\delta_{g} + d\:\tr_{g}$ in
$\holda{k}(S^{2}T^{*}M)$.
Because $g$ is Ricci-flat harmonic $1$-forms are parallel, and therefore
\ltwoorth{} to the image of $2\delta_{g} + d\:\tr_{g}$. It follows from
lemma \ref{delstarlem} and the
Fredholm alternative for $\triangle_{g}$ on $\Omega^{1}(M)$ that
there is a direct sum decomposition
\begin{equation*}
\holda{k}(S^{2}T^{*}M) = \delta^{*}_{g}\holda{k+1}(\Lambda^{1}) \oplus \slt .
\end{equation*}
We use a neighbourhood $\mathcal{S}$ of $g$ in $\slt$ as a slice for the
$\mathcal{D}$-action.
%We can therefore use a local right inverse for the submersion (\ref{gkeq}) in
%the basic slice result theorem \ref{slicethm} to define a smooth local
%projection map
%\[ P : \mathcal{U} \to \mathcal{S} \]
%from a neighbourhood $\mathcal{U}$ in $\holda{k}(S^{2}T^{*}M)$ to a
%neighbourhood
%$\mathcal{S}$ of $g$ in $\slt$, with the property that for
%any $g' \in \mathcal{U}$ there is a $\phi \in \kpcal{D}$ such that
%$P(g') = \phi^{*}g'$. $\mathcal{S}$ is a slice for the $\mathcal{D}$-action.


\begin{rmk}
This is not exactly the same choice of slice as in \cite{besse87}.
It has been used before by Biquard \cite{biquard00}
%and is also used by
%Kovalev \cite{kovalev05} to deal with Ricci-flat deformations of EAC metrics.
and Kovalev \cite{kovalev05}.
\end{rmk}

Let $\qcalz$ be the space of Ricci-flat (not \textit{a priori} smooth)
metrics in $\mathcal{S}$ -- this
is the \emph{pre-moduli space of Ricci-flat metrics} near $g$.
The linearisation of the Ricci curvature functional at a Ricci-flat metric
is given by (cf. \cite[(12.28')]{besse87})
\begin{equation}
\label{infriceq}
(DRic)_{g}h = \triangle_{L} h + \delta^{*}_{g}(2\delta_{g} + d\:\tr_{g})h ,
\end{equation} 
where $\triangle_{L}$ denotes the Lichnerowicz Laplacian on $S^{2}T^{*}M$
in the sense of definition \ref{lichdef}.
In particular, on the tangent space $\slt$ to the slice
the linearisation reduces to $\triangle_{L}$.
It is elliptic so its kernel has finite dimension. Moreover, the kernel
is contained in $K$: differentiating the Bianchi identity (\ref{bianchieq})
at the Ricci-flat metric $g$ gives
\[ (2\delta_{g} + d\:\tr_{g})(DRic)_{g} = 0 , \]
and hence
\[ \triangle_{L}h = 0 \; \Rightarrow \;
\triangle(2\delta_{g} + d\:\tr_{g})h = 0
\; \Rightarrow \; (2\delta_{g} + d\:\tr_{g})h = 0 . \]

\begin{defn}
The space of \emph{infinitesimal Ricci-flat deformations} of $g$ is the
kernel $\varepsilon(g)$ of $\triangle_{L}$ in
$\Gamma(S^{2}(T^{*}M))$.
\end{defn}

If $h \in \Gamma(S^{2}T^{*}M)$ is tangent to a curve of Ricci-flat metrics
in the slice $\mathcal{S}$ then of course $h \in \varepsilon(g)$.
The converse is not true; in general there may be elements in $\varepsilon(g)$
which are not tangent to any curve of Ricci-flat metrics.
Thus $\qcalz$ need not be a manifold with tangent space $\varepsilon(g)$.

The image of $DRic_{g}$ is the \ltwoorth{} complement $K'$ to $\varepsilon(g)$
in $K$. Let $P_{g}$ be the \ltwoorth{} projection to $K'$.
The Ricci curvature functional is real analytic. We can apply the implicit
function theorem to the composition
\begin{equation}
\label{ricprojeq}
F : \cals \to K' : \;\: h \mapsto  P_{g}Ric(h)
\end{equation}
to deduce that there is a real analytic submanifold $Z \subseteq \mathcal{S}$
whose tangent space at $g$ is precisely $\varepsilon(g)$ and which contains
$\qcalz$ as a real analytic subset. The
analyticity implies that \emph{if} every element of $\varepsilon(g)$ is tangent
to a curve of Ricci-flat metrics then in fact $\qcalz$
contains a neighbourhood of $g$ in $Z$. Thus the pre-moduli space
$\qcalz$ is a manifold in this case (cf. \cite[Corollary $3.5$]{koiso83}).

The elements of $Z$ are smooth by elliptic regularity, and
when $\qcalz = Z$ one can deduce from corollary \ref{smoothslicecor} that
$\qcalz \to \defmetz$ is open. In general the argument needs to be modified
a bit. One needs to extend (\ref{ricprojeq}) to a function on a neighbourhood
$U$ of $g$ in $\holda{k}(S^{2}T^{*}M)$ such that $F^{-1}(0)$ is a manifold
containing the Ricci-flat metrics \emph{and} %ensure that
$Z\kpcal{D} \cap U \subseteq F^{-1}(0)$. One way to do this is to first define
a smooth map $f : U \to g\kpcal{D}$ with the property that
$f(\phi^{*}g') = \phi^{*}g$ when $\phi \in \kpcal{D}$ and $g' \in Z$.
If $f(h) = \phi^{*}g$ then $P_{f(h)}$ is a projection to $\phi^{*}K'$,
and we can take
\begin{equation}
F : U \to K', \;\: h \mapsto P_{g}P_{f(h)}Ric(h) .
\end{equation}
Then
\[ Z \times \kpcal{D} \to F^{-1}(0) \]
is an open map near $(g, id)$ by the submersion theorem, and we can proceed
as in the proof of theorem \ref{slicethm} and corollary \ref{smoothslicecor}
to deduce that $\qcalz \to \defmetz$ is open.
%We can show that $\qcalz$ consists of smooth elements by a regularity
%argument like proposition \ref{cptregprop} and then deduce
%from corollary \ref{smoothslicecor} that $\qcalz \to \defmetz$ is open.
Theorem \ref{sliceinvthm} implies that in fact this is injective up to the
action of the stabiliser $\cali_{g}$. %since the slice is $\cali_{g}$-invariant
Moreover, proposition \ref{killingprop} implies that all stabilisers of
elements in $\qcalz$ have the same dimension, so at least their identity
components must be equal. As $\cali_{g}$ is compact its action on $\qcalz$
factors through a finite group (cf. \cite[$12.25$]{besse87}).
Hence

\begin{thm}
\label{koisothm}
Let $M$ be a compact manifold and $g$ a Ricci-flat metric on $M$.
Let $\qcalz$ be the pre-moduli space of Ricci-flat metrics near $g$
and $\cali_{g}$ the stabiliser of $g$ in $\cald$.
Then $\qcalz/\cali_{g}$ is homeomorphic to
a neighbourhood of $g\cald$ in $\defmetz$.
In particular, if every element of $\varepsilon(g)$ is integrable then
$\defmetz$ is an orbifold near $g\cald$.
\end{thm}

%(although this is not
%strictly required for the proof of theorem \ref{cptopenthm}).
%since assertion \ref{smoothitem} of proposition
%\ref{gpremodprop} above provides all the necessary regularity. xxx

\begin{rmk}
\label{orbirmk}
Clearly the argument would give the same result even if we were to consider the
moduli space of Ricci-flat metrics given by dividing by the action of
the full diffeomorphism group of $M$.
\end{rmk}

\ignore{
\begin{rmk}
In \cite[Lemma $2.6$]{koiso83} Koiso uses instead of $\mathcal{S}$ a slice
constructed by Ebin \cite{ebin70}, and shows that any
Einstein metric in this slice is smooth.
% maybe cite koiso80 instead?
\end{rmk}
}


\subsection{Proof of theorem \ref{cptopenthm}}
\label{proof1sub}

Let $M$ be a compact \gmfd, $\Gamma(\gspace)$ the space of \gstr s on $M$ and
\begin{equation}
\label{mbundleeq}
m : \Gamma(\gspace) \to \Gamma(S^{2}T^{*}M), \;\: \chi \mapsto g_{\chi}
\end{equation}
the natural map that sends a \gstr{} to the metric it defines.
In order to prove theorem \ref{cptopenthm} we show first that for any
torsion-free
\gstr{} $\chi$ the derivative of $m$ maps the tangent space to the
pre-moduli space $\qcalg$ at $\chi$ onto the space $\varepsilon(g_{\chi})$
of infinitesimal Ricci-flat deformations.

The tangent space to $\Gamma(\gspace)$ at $\chi$ is the space of differential
forms $\Gamma(E_{\chi})$, where $E_{\chi} \subseteq \Lambda^{*}T^{*}M$
is a vector subbundle associated to the \gstr{} defined by $\chi$.
Fibre-wise $\gspace$ is a $GL(\bbr^{n})$-orbit and $E_{\chi}$ is the tangent
space $\glalg_{n}\chi$ to the orbit. Because $m$ is $GL(\bbr^{n})$-equivariant
its derivative takes $a\chi \mapsto ag_{\chi}$ for any $a \in \glalg_{n}$,
which maps onto the fibre of $S^{2}T^{*}M$.
Hence the derivative
\begin{equation}
\label{mdereq}
Dm_{\chi} : \Gamma(E_{\chi}) \to \Gamma(S^{2}T^{*}M)
\end{equation}
is surjective. Furthermore, the derivative
is $G$-equivariant with respect to the \gstr{} defined by $\chi$.
Since $\triangle_{L}$ is the Lichnerowicz Laplacian
on $S^{2}T^{*}M$,
lemma \ref{lichlem} implies that the diagram below commutes.

\begin{diagram}[PostScript=Rikicki,balance]
\Gamma(E_{\chi})   &    \rTo^{Dm_{\chi}}  &  \Gamma(S^{2}T^{*}M) \\
\dTo^\triangle\;\; &  & \dTo_{\triangle_{L}} \\
\Gamma(E_{\chi})   &    \rTo^{Dm_{\chi}}  &  \Gamma(S^{2}T^{*}M) 
\end{diagram}

\noindent Hence

\begin{lem}
\label{mdersurjlem}
If $\chi$ is a torsion-free \gstr{} then $Dm_{\chi}$ maps the harmonic sections
of $E_{\chi}$ onto the space $\varepsilon(g_{\chi})$ of infinitesimal
Ricci-flat deformations.
\end{lem}

So let $\chi$ be any torsion-free \gstr{} on $M$ and $\qcalg$ the pre-moduli
space of torsion-free \gstr s near $\chi$. As described in subsection
\ref{ricdefsub}, there is a slice at $g_{\chi}$ for the $\mathcal{D}$-action on
the metrics and a local projection map $P$ to the slice. The Ricci-flat
metrics in the slice are contained in a submanifold $Z$, and the tangent space
to $Z$ at $\chi$ is $\varepsilon(g_{\chi})$.
\begin{equation}
\label{pomeq}
P \circ m : \qcalg \to Z
\end{equation}
is a well-defined smooth map and lemma \ref{mdersurjlem}
means that its derivative at $\chi$ is surjective.
Therefore every element of $\varepsilon(g_{\chi})$ is tangent to
a path of Ricci-flat metrics, so $\qcalz$ is a manifold.
By the submersion theorem, the image $\defmetg$ in $\defmetz$ contains
a neighbourhood of~$g\cald$.

The pre-images of $g_{\chi}$ under $m$ are defined by differential forms
which are harmonic with respect to $g_{\chi}$. By Hodge theory they represent
distinct cohomology classes. Because elements of $\cali_{g_{\chi}}$ are isotopic
to the identity they must therefore fix the fibre,
so $\cali_{g_{\chi}} = \cali_{\chi}$. 
Now, if $g' \in \qcalz$ then
$g' = \phi^{*}m(\chi')$ for some $\chi' \in \qcalg$ and $\phi \in \kpcal{D}$
because (\ref{pomeq}) is a submersion. As $\cali_{\chi}$
acts trivially on $\qcalg$ by proposition \ref{autinvprop} it follows
that the conjugate $\cali_{g_{\chi}}^{\phi}$ fixes $g'$.
But then
$\cali_{g_{\chi}}^{\phi} \subseteq \cali_{g'} \subseteq \cali_{g_{\chi}}$
by theorem \ref{sliceinvthm}, so in fact
$\cali_{g_{\chi}}^{\phi} = \cali_{g_{\chi}}$.
Thus $\cali_{g_{\chi}}$ fixes~$g'$.

Now theorem \ref{koisothm} implies that $\qcalz$ is homeomorphic to a
neighbourhood of $\defmetz$. Thus $\defmetg$ is a manifold near $g\cald$
and the proof of theorem \ref{cptopenthm} is complete.


\subsection{The asymptotically cylindrical case}
\label{eacopensub}

The proof of theorem \ref{cptopenthm} only used the compactness assumption
to access certain deformation results for \gstr s and Ricci-flat metrics.
For the cases $G = G_{2}$ and $Spin(7)$ we found pre-moduli spaces of EAC
\gstr s in \S \ref{eacgchap}, with properties analogous to
proposition \ref{gpremodprop}.

\begin{prop}
\label{cylgpremodprop}
Let $G = Spin(7)$ or $G_{2}$, $M$ an EAC \gmfd{}
and $\chi$ a torsion-free EAC \gstr{} on $M$.
Then there is a submanifold $\qcalg$ of the space of $C^{1}$ \gstr s such that
\begin{enumerate}
\item
the elements of $\qcalg$ are smooth EAC torsion-free \gstr s,
\item the tangent space to $\qcalg$ at $\chi$ is the space of bounded harmonic
sections of $E_{\chi}$,
\item the natural map $\qcalg \to \defstrp$ is a homeomorphism onto a
neighbourhood of $\chi\pcal{D}$ in~$\defstrp$.
\end{enumerate}
\end{prop}

In order to prove the EAC version theorem \ref{eacopenthm} we
only need to explain how to set up the deformation theory for EAC Ricci-flat
metrics.
We define the slices with same equations as in the compact case in
\S \ref{ricdefsub} and use the same reasoning as for deformations of EAC
\gtmfd s in \S \ref{modconstrsub} to make the slice arguments work on
EAC manifolds.
The resulting approach is similar to that of Kovalev \cite{kovalev05},
who considers Ricci-flat deformations of EAC Calabi-Yau manifolds.

Let $M^{n}$ be a manifold with cylindrical ends and cross-section $X^{n-1}$.
Let $\defmetp$ be the quotient of the space of EAC Ricci-flat metrics (with
any exponential rate) by the group $\pcal{D}$ of EAC diffeomorphisms of $M$
isotopic to the identity.
We pick an EAC Ricci-flat metric $g$ on $M$ and study a neighbourhood
of $g\pcal{D}$ in $\defmetp$.
By definition, the asymptotic limit of $g$ is a cylindrical metric
$dt^{2} + g_{X}$ on $X \times \bbr$,
where $g_{X}$ is a Ricci-flat metric on $X$.

We work with weighted Hölder spaces of sections. Let $k \geq 2$,
$\alpha \in (0,1)$, and $\delta > 0$ be less than the exponential rate of $g$.
The metric $g$ defines a Hodge Laplacian on $1$-forms and a
Lichnerowicz Laplacian on symmetric bilinear forms, which are both
asymptotically translation-invariant operators.
We require that $\delta$ is small enough that the Laplacians are Fredholm
on $\holdad{k}$ spaces, as we may according to theorem \ref{indthm}.

We proved in \S \ref{ricdefsub} that there is a real analytic submanifold
$Z \subset \holda{k}(S^{2}T^{*}X)$ which contains representatives of all
diffeomorphism classes of Ricci-flat metrics on $X$ close to $g_{X}$.
Its tangent space $T_{g_{X}}Z = \varepsilon(g_{X})$ is the space of
Lichnerowicz harmonic sections of $S^{2}T^{*}X$.

Let $\scrmkz$ denote the space of $\holda{k}$ metrics on $M$ which are
$\holdad{k}$-asymptotic to cylindrical metrics $dt + g_{X}^{2}$ such that
$g_{X} \in Z$.
If $\rho$ is a cut-off function for the cylinder then $\rho Z$ can be identified
with a space of bilinear forms on $M$, and $\scrmkz$ is an open subset
\begin{equation*}
\scrmkz \subset \holdad{k}(S^{*}T^{*}M) + \rho Z .
\end{equation*}
Similarly let $\caldkpz$ be the set of EAC diffeomorphisms with rate $\delta$
which are asymptotic to elements of the isometry group $\cali_{g_{X}}$ of
$g_{X}$.
Then $\scrmkz$ contains representatives of all diffeomorphism classes of
Ricci-flat metrics near $g$ and, because $Z$ is $\cali_{g_{X}}$-invariant,
theorem \ref{sliceinvthm} implies that any isometry between elements of
$\scrmkz$ must lie in $\caldkpz$ (a similar argument for simplifying the
problem by a slice at the boundary was used for the $G_{2}$ case in
lemma \ref{qmodlem}).

We therefore identify a slice in $\scrmkz$ for the action of $\caldkpz$ at $g$.
The tangent space to $\scrmkz$ at $g$ is
\begin{equation*}
T_{g}\scrmkz = \holdad{k}(S^{*}T^{*}M) \oplus \rho \varepsilon(g_{X}) .
\end{equation*}
The tangent space at the identity of $\caldkpz$ corresponds to vector fields
which are $\holdad{k}$-asymptotic to translation-invariant Killing vector
fields on the cylinder, i.e. to elements of~$(\harm^{1}_{\infty})^{\sharp}$,
where $\harm^{1}_{\infty}$ denotes the translation-invariant harmonic
$1$-forms on the cylinder $X \times \bbr$.
By proposition \ref{delstarprop} the tangent space to the $\caldkpz$-orbit
at $g$ is
\begin{equation*}
\delta^{*}_{g}(\holdad{k}(\Lambda^{1}) \oplus \rho \harm^{1}_{\infty}) .
\end{equation*}
%
Let $\slt$ be the kernel of $2\delta_{g} + d\:\tr_{g}$ in $T_{g}\scrmkz$.

\begin{lem}
Let $M$ be a Ricci-flat EAC manifold with a single end.
Then
\begin{equation}
T_{g}\scrmkz = \slt \oplus
\delta^{*}_{g}(\holdad{k}(\Lambda^{1}) \oplus \rho \harm^{1}_{\infty}) .
\end{equation}
\end{lem}

\begin{proof}
$(2\delta_{g} + d\:\tr_{g})\delta^{*} = \triangle_{g}$ according to
lemma \ref{delstarlem}, so it suffices to show that the image of
$2\delta_{g} + d\:\tr_{g} : T_{g}\scrmkz \to \holdad{k-1}(\Lambda^{1})$
is contained in the image of
\begin{equation*}
\triangle : \holdad{k+1}(\Lambda^{1}) \oplus \rho \harm^{1}_{\infty}
\to \holdad{k-1}(\Lambda^{1}) .
\end{equation*}
It follows from proposition \ref{lapindprop} that this has index $0$,
so its image is the \ltwoorth{} complement to its kernel $\harm^{1}_{0}$,
the space of bounded harmonic $1$-forms.

Now, if $h \in T_{g}\scrmkz$ and $\beta \in \harm^{1}_{0}$ then
the difference between $\inner{\delta_{g}h, \beta}$ and
$\inner{h, \delta^{*}_{g}\beta} \: = 0$ is the divergence of the contraction
of $h$ with $\beta$. The boundary condition on $h$ ensures that the
asymptotic limit of the contraction has no $dt$-component, so the integral
of the divergence is $0$. Hence
\[ \linner{\delta_{g}h, \beta} = 0 . \]
The hypothesis that $M$ has a single end ensures that the asymptotic limit
of $\beta$ has no $dt$-component (corollary \ref{degonehodgecor}),
so integration by parts also applies to show that
$\linner{d\:\tr_{g}h, \beta} = 0$. Thus the image of $2\delta_{g} + d\:\tr_{g}$
is \ltwoorth{} to $\harm^{1}_{0}$.
\end{proof}

Now we can use a real analytic $\cali_{g}$-invariant submanifold
$\cals \subset \scrmkz$ with $T_{g}\cals = \slt$ as a slice for the
$\caldkpz$-action.
Let $\qcalz \subset \cals$ be the subset of Ricci-flat metrics.
As in the compact case $\qcalz$ is an analytic subset of an analytic
submanifold $Z' \subset \cals$, defined as the zero set of the composition
of the Ricci functional $\cals \to \holdad{k-2}(S^{2}T^{*}M)$ with the
projection onto the image of its derivative at $g$.
On $K$ the derivative of the Ricci functional is the Lichnerowicz Laplacian,
so $T_{g}Z'$ is the space of harmonic sections of $S^{2}T^{*}M$,
exponentially asymptotic to sections of $S^{2}T^{*}X$ (i.e. the asymptotic
limit has no $dt$-components). We call this the space of infinitesimal
Ricci-flat EAC deformations $\varepsilon(g)$.

In general we can use by regularity and slice arguments like in
\S \ref{cylregsub} -- \ref{modconstrsub} that $Z$ consists of smooth
EAC metrics and extend the proof of theorem \ref{koisothm}, thus proving
that $\qcalz/\cali_{g}$ is homeomorphic to a neighbourhood
of $g$ in $\defmetp$.

We have now set up the deformation theory for EAC Ricci-flat metrics
that is required, together with the unobstructedness of deformations of
torsion-free EAC \gstr s for $G = G_{2}$ and $Spin(7)$ proved in
\S \ref{eacgchap}, in order to prove theorem \ref{eacopenthm} by the same
argument as for the compact case.


\section{$\defstrg$ as a fibre bundle}
\label{bundlesec}

Let $G = Spin(7)$, $G_{2}$ or $SU(3)$, and $M$ a compact \gmfd.
We have established that the natural map
\begin{equation}
\label{mgmapeq}
m : \defstrg \to \defmetg
\end{equation}
is a submersion. We now prove that it is a smooth fibre bundle.
We use the correspondence between torsion-free \gstr s and parallel spinors
to show that the fibres are determined by the topology of $M$,
and then deduce theorem \ref{cptfibredefthm}.
We will prove the case $G = Spin(7)$ in detail, and remark
on the minor adaptions required for $G = G_{2}$ or $SU(3)$.


\subsection{The case $G = Spin(7)$}
\label{fibresub}

Let $M$ be a compact \spmfd.
We begin by observing that the fibres of $m$ in
$\Gamma(S^{2}T^{*}M)$ and in the moduli space $\defstr$ are equivalent.

\begin{lem}
\label{distinctlem}
Let $g$ be a \spmetric{} on $M$. Then the fibre of
$m : \defstrsp \to \defmetsp$ over $g\cald$ is homeomorphic to the fibre of
$m : \Gamma(\spspace) \to \Gamma(S^{2}T^{*}M)$ over $g$.
\end{lem}

\begin{proof}
If $\psi$ is a \spstr{} such that $m(\psi) = \phi^{*}g$ for some
$\phi \in \cald$ then obviously $m((\phi^{-1})^{*}\psi) = g$. Thus the natural
map from fibre over $g$ to the fibre over $g\cald$ is surjective.

Suppose that two torsion-free \spstr s which define the
same metric are $\mathcal{D}$-equivalent.
Then they represent the same class in $H^{4}(M)$. Since they are both harmonic
(with respect to the same metric) they must be equal. Therefore the map is
injective too.
\end{proof}

Now consider the point-wise model for \spstr s which was described in
\S \ref{sp7sub}.
$\lmsp(\bbr^{8})^{*} \subseteq \Lambda^{4}(\bbr^{8})^{*}$ is
the $GL(\bbr^{8})$-orbit of the standard \spstr{} $\psi_{0}$
and each element defines an inner product on $\bbr^{8}$.
The subset that defines the same inner product as~$\psi_{0}$
(i.e. the standard inner product on $\bbr^{8}$) is the orbit $SO(8)\psi_{0}$.
Recall that % (\ref{spspineq}), 
the positive real spin representation $\spinrp{8}$ of $Spin(8)$
splits into irreducible components
\begin{equation*}
\spinrp{8} = \bbr \oplus \bbr^{7}
\end{equation*}
under the action of $Spin(7)$. The $\bbr$ term defines an element
$\spl \in \bbp \spinrp{8}$.
Since $SO(8)$ acts transitively on $\bbp \spinrp{8}$ with stabiliser $Spin(7)$
at $\spl$ it follows that there is a natural bijection
$SO(8)\psi_{0} \cong \bbp \spinrp{8}$.

$Spin(7)$ acts irreducibly on $\bbr^{7}$, so $\spl$ is its unique fixed point
in $\bbp \spinrp{8}$. Thus $\psi_{0}$ is the unique \spstr{} defining the
standard inner product that is fixed by $Spin(7)$.
Hence if $g$ is a \spmetric{} with $Hol(g) = Spin(7)$ then the fibre over $g$
is a point, and $m : \defstr \to \defmetsp$ is a diffeomorphism.

Let $I$ be the set of spin structures on $M^{8}$.
Given a \spmetric{} $g$ on $M$, for each $i \in I$ let $F_{i}$ be the space of
parallel sections of the positive spinor bundle defined by $i$.
Any torsion-free \spstr{} inducing $g$ determines a spin structure $i$
and a parallel spinor in $F_{i}$. The point-wise considerations show that
the fibre of $m$ over $g$ is homeomorphic to the disjoint union of projective
spaces
\[ \bigsqcup_{i \in I} \bbp F_{i} . \]
For each $i$ and $g$ either $F_{i} = 0$ or its dimension is given
by the topological formula (\ref{cptspindimeq}).
The fact that $m : \defstr \to \defmetsp$ is open implies that, for each $i$,
triviality of $F_{i}$ depends only on the connected component
of $\defmetsp$ containing $g\cald$.

If $g'$ is another \spmetric{} let $F_{i}'$ denote the corresponding spaces of
parallel spinors. If $g'$ is close to $g$ then we may define a linear map
$F_{i} \to F_{i}'$, and because they have equal dimension it is easy to see
that it is bijective for $g'$ sufficiently close to $g$.
If we let $\qcalsp$ be the pre-moduli space of \spmetric s near $g$ then this
defines a local diffeomorphism
\[ \qcalsp \times \bigsqcup_{i \in I} \bbp F_{i} \to \defstr . \]
Transition functions between such maps are isomorphisms of projective varieties
on the fibres. The maps can therefore be used as local trivialisations for
$\defstr$ as a smooth fibre bundle over each connected component of $\defmetsp$.

It remains to check that the fibres are the same over different connected
components. A pair of spin structures $i$, $j$ determines an element
of $H^{1}(M, \bbz_{2})$ (the unique class whose pull-back to the total
space of the frame bundle of $M$ is the difference of the Stiefel-Whitney
classes of $i$ and~$j$), which in turn determines a double cover
$\tildm$ of $M$ (cf. \cite[\S II.1]{lawson89}).
$i$ and $j$ lift to the same spin structure on $\tildm$. If $g$ is a \spmetric{}
on $M$ and $F_{i}$ and $F_{j}$ are both non-empty then the dimension of
the space on parallel spinors on $\tildm$ is twice that of~$F_{i}$.
By the topological formula (\ref{cptspindimeq}) for the dimension of the
space of parallel spinors on a \spmfd{} this is a topological property of
$\tildm$. Hence the number of non-empty connected components $\bbp F_{i}$
of the fibre of $m$ above a \spmetric{} $g$ is determined by the topology
of $M$, independently of the choice of $g$.
This concludes the proof of theorem \ref{cptfibredefthm} for the case
$G = Spin(7)$.

\begin{rmk}
Another way to think of this is that if we fix a torsion-free \spstr{} $\psi$
on $M$ then the fibre of $m$ containing $\psi$ can be identified with the
parallel sections of the projective spinor bundle $\bbp \spinorp$ determined
by $\psi$. Any such section corresponds to a parallel spinor either on $M$
or on a double cover of $M$.
\end{rmk}


\subsection{The cases $G = G_{2}$ and $SU(3)$}

All that needs to be changed to extend the proof of theorem
\ref{cptfibredefthm} above to the case when $M^{7}$ is a compact \gmfd{}
is the point-wise model used in the identification of the fibres.

Recall that the spin representation $\spinr{7}$ splits into irreducible
components
\begin{equation}
\label{gtdecompeq2}
\spinr{7} \cong \bbr \oplus \bbr^{7}
\end{equation}
under the action of $G_{2}$. The $\bbr$ term defines an element
$\spl \in \bbp \spinr{7}$. $SO(7)$ acts transitively on~$\bbp \spinr{7}$
and the stabiliser of $u$ is~$G_{2}$,
so there is a natural identification $SO(7)\varphi_{0} \cong \bbp \spinr{7}$
of the $SO(7)$-orbit of the standard \gtstr{} with $\bbp \spinr{7}$.
For any \gtmetric{} $g$ on $M$ the fibre of $m$ over $g$ is identified
with $\bigsqcup_{i \in I}\bbp F_{i}$, where $F_{i}$ is the space of parallel
spinors defined by $g$ and a spin structure $i$.
The rest of the proof of theorem \ref{cptfibredefthm} for $G = G_{2}$ is
identical to the case $G = Spin(7)$.

\begin{rmk}
\label{twistrmk}
Let $\varphi$ be a torsion-free \gtstr{} on a compact manifold $M^{7}$.
The isomorphism (\ref{gtdecompeq2}) implies that the spinor bundle
on $M$ defined by $\varphi$ is $S \cong \bbr \oplus TM$.
We can therefore identify the fibre of $m$ over $\varphi$ with parallel
sections of $\bbp(\bbr \oplus TM)$.
This identification can be interpreted as given by `twisting' $\varphi$.

If $\psi$ is a \gtstr{} on a vector space $V$ and $v \in V$ is a unit vector
then the $2$-form $v \lrcorner \psi$ can be identified with an element
$A_{v, \psi} \in \soalg(V)$.
If $v$ is a unit vector in $\bbr^{7}$ and $\theta \in \bbr$ then the
identification of
$\bbp \spinr{7}$ with $SO(7)\varphi_{0}$ maps the line containing
$\cos \theta + v \sin \theta$
to $\varphi_{\theta} = \exp(2\theta A_{v, \varphi_{0}})\varphi_{0}$.
Note that $\varphi_{\theta}$ satisfies a flow equation considered
by Karigiannis \cite{karigiannis05},
\begin{equation*}
\textstyle{\frac{d}{d\theta}}
\varphi_{\theta} = A_{v, \varphi_{\theta}}\varphi_{\theta} .
\end{equation*}
The torsion-free \gtstr{} $\varphi$ can in this sense be `twisted' by
parallel vector fields on $M$ to produce other torsion-free \gtstr s
defining the same metric and all elements in the connected component of
$\varphi$ in its fibre in $m : \defstrgt \to \defmetgt$ arise this way.
\end{rmk}

Finally we prove theorem \ref{cptfibredefthm} for $G = SU(3)$. Let $M^{6}$ be a
compact Calabi-Yau $3$-fold.
To identify the fibres of $m : \defstrsu \to \defmetsu$ we consider the
spin representation $\spinr{6}$ of $Spin(6)$. This is the unique
irreducible real rank $8$ representation of $Spin(6)$
(it can be regarded as a rank $4$
complex representation in two non-equivalent ways).
As an $SU(3)$-representation
\begin{equation}
\spinr{6} \cong \bbc \oplus \bbc^{3},
\end{equation}
where $SU(3)$ acts naturally on $\bbc^{3}$. Let $\bbp \spinr{6}$ be the
space of (real) lines in $\spinr{6}$. $SO(6)$ acts transitively
on $\bbp \spinr{6}$ and the stabiliser of an element of
$\bbp \bbc \subset \bbp \spinr{6}$ is $SU(3)$.
If we fix once and for all an element $\spl \in \bbp \bbc$ then this determines
an identification
$SO(6)(\Omega_{0},\omega_{0}) \cong \bbp \spinr{6}$
of the $SO(6)$-orbit of the standard Calabi-Yau structure with
$\bbp \spinr{6}$.
For any Calabi-Yau metric $g$ on $M$ the fibre of $m$ over $g$ is identified
with $\bigsqcup_{i \in I}\bbp F_{i}$, where $F_{i}$ is the space of parallel
spinors defined by $g$ and a spin structure $i$.

$SU(3)$ acts irreducibly on $\bbc^{3}$, so its fixed points in $\bbp \spinr{6}$
are exactly $\bbp \bbc \cong S^{1}$. Thus when $M$ is an irreducible
Calabi-Yau manifold the fibres of $m : \defstr \to \defmetsu$ are $S^{1}$.
This corresponds to the
$S^{1}$-action on $\defstrsu$ given by multiplication of the holomorphic
$(3,0)$-form $\Omega + i \hat\Omega$ by a unit complex scalar.
The rest of the proof is identical to the case $G = Spin(7)$.

%\begin{rmk}
%$G_{2}$ can be considered as the automorphism group of the normed algebra
%of octonions, and the isomorphism (\ref{gtdecompeq2}) means that $\spinr{7}$
%can be given an algebra structure that is invariant under the $G_{2}$ action.
%The collection of subspaces $K_{i}$ from page x is closed
%under this product, so both their number and dimension must be powers of $2$.
%Thus for the case $G = G_{2}$ the number of components of the typical fibre
%of $m : \defstrg \to \defmetg$ is $1$, $2$, $4$ or $8$, while the dimension of
%the fibres is $0$, $1$, $3$ or~$7$.
%
%Since $SU(3) \subset G_{2}$ the same holds for $G = SU(3)$ (except that the
%fibres must have dimension at least $1$).
%\end{rmk}


\begin{ex}
\label{cyrotex}
Let $X^{6}$ a simply connected manifold with a Calabi-Yau structure
$(\Omega,\omega)$. $\varphi = \Omega + dt \wedge \omega$ defines
a torsion-free \gtstr{} on $M = S^{1} \times X$.
The typical fibre of $\defstrgt \to \defmetgt$ is~$S^{1}$.
The fibre containing $\varphi$ corresponds exactly to the $S^{1}$ fibre
of \cystr s defining the same metric as $(\Omega,\omega)$.
\end{ex}

\begin{ex}
In example \ref{cyinvex} we constructed an irreducible compact Calabi-Yau
manifold $X^{6}$ with structure $(\Omega, \omega)$ and an involution $a$
such that $a^{*}\Omega = \Omega$, $a^{*}\omega = -\omega$.
%Then $\varphi = \Omega + dt \wedge \omega$ is a torsion-free \gstr{} on
%$S^{1} \times X$ which is invariant under the involution $(-1, a)$,
%and therefore induces a torsion-free \gstr{} on $M = S^{1} \times X/(-1, a)$.
%
%The involution acts as a reflection on the $S^{1}$ of parallel spinors on
%$S^{1} \times X$, and the space of parallel spinors on $M$ is the pair of
%fixed points.
The involution acts as a reflection on the $S^{1}$ of torsion-free \gtstr s
defining the product metric on $S^{1} \times X$.
The pair of fixed points descend to torsion-free \gtstr s on the quotient
$M = S^{1} \times X/(-1, a)$.
Thus the moduli space of torsion-free \gtstr s on $M$ is a double
cover of the moduli space of \gtmetric s.
\end{ex}

\begin{rmk}
Let $M^{7}$ be a compact \gtmfd{}.
Hitchin \cite{hitchin00} shows that the Hessian of the volume functional
defines a well-defined non-degenerate symmetric bilinear form on~$\mathcal{M}$.
If we identify $T_{\varphi\mathcal{D}}\mathcal{M} \cong H^{3}(M)$ then
the form is positive-definite on $H^{3}_{1\oplus7}(M)$ and negative-definite
on $H^{3}_{27}(M)$. In particular the signature is
$(1, b^{3}(M) - 1)$ when $M$ is a connected irreducible \gtmfd.

If $g$ is a \gtmetric{} and $\varphi$ is a torsion-free \gtstr{} inducing $g$
then $Dm_{\varphi} : \harm^{3} \to \varepsilon(g)$ gives an isomorphism
$\harm^{3}_{1\oplus27} \cong \varepsilon(g)$ and hence defines
an inner product on $\varepsilon(g)$.
Because the irreducible components of $S^{2}T^{*}M$
under $G_{2}$ are also representations of $SO(7)$ this inner product
is independent of the choice of $\varphi$. Therefore $\defmetgt$ has a
well-defined Lorentzian metric (whether $M$ is irreducible or not),
such that $m : \defstr \to \defmetgt$ is a pseudo-Riemannian submersion.
\end{rmk}


\subsection{The asymptotically cylindrical case}
\label{eacfibresub}

The only part of the proof that does not carry over directly from the
compact case is the identification of the fibres of
$\defstrp \to \defmetg$ in terms of the topology of $M$.
But this only requires us to determine the dimension of the space of parallel
spinors on any EAC \gmfd. For $G = Spin(7)$ this is provided by
proposition \ref{dimharmspprop}. For $G = G_{2}$ the spinor bundle is
$\bbr \oplus TM$, and the dimension of the space of parallel $1$-forms
is determined by Hodge theory (corollary \ref{degonehodgecor}).

\begin{rmk}
Let $G = Spin(7)$ or $G_{2}$ and $H = G_{2}$ or $SU(3)$ respectively,
and let $M$ be an EAC \gmfd{} with cross-section $X$.
Let $B : \defstrp \to \caln$ be the boundary map from the moduli space
of torsion-free EAC \gstr s on $M$ to the moduli space
of torsion-free $H$-structures on $X$
and let $B : \defmetg \to \defmet_{H}$ be the boundary map from
the moduli space of \gmetric s on $M$ to the moduli space of $H$-metrics
on $X$.
The diagram below obviously commutes.
\begin{diagram}[PostScript=Rikicki]
\defstrp &\rTo^{B} & \caln \\
\dTo^{m} & & \dTo_{m} \\
\defmetg &\rTo^{B} & \defmet_{H} 
\end{diagram}
It is easy to see that any tangent to the image $B(\defstrp) \subseteq \caln$
that is also tangent to the fibre of $m : \caln \to \defmet_{H}$ is actually
a tangent to the image of a fibre of $m : \defstrp \to \defmetg$.
Therefore $m : B(\defstrp) \to B(\defmetg)$ is a fibre bundle, whose fibres
have the same dimension as those of $m : \defstrp \to \defmetg$.
\end{rmk}

%\begin{diagram}[PostScript=Rikicki]
%\defstrp &\rTo^{m} & \defmetp \\
%\dTo^{B} & & \dTo_{B} \\
%\caln &\rTo^{m} & \defmet_{X} 
%\end{diagram}



\chapter{Deformations of glued $G_{2}$-manifolds}
\label{gluechap}

Let $M_{\pm}^{7}$ be a pair of EAC \gtmfd s with the same cross-section
$X^{6}$, whose \gtstr s $\varphi_{\pm}$ have matching asymptotic models in a
sense to be made precise below.
$M_{+}$ and $M_{-}$ can be glued along their cylindrical ends
to form a compact smooth manifold~$M$.
Topologically $M$ can be regarded as a generalised connected sum of $M_{+}$
and $M_{-}$.
In \cite{kovalev03} Kovalev explains a gluing construction, which
produces a torsion-free \gtstr{} $\varphi$ on $M$ from $\varphi_{+}$ and
$\varphi_{-}$ together with a gluing parameter $L \in \bbrp$, which
controls the diameter of $M$.
%This is explained in more detail in subsection \ref{glueconstrsec}.
Kovalev constructs examples where the resulting compact \gtmfd{} $M$ has
holonomy exactly~$G_{2}$.
Increasing the gluing parameter in the construction gives a path of
\gtstr s of increasing diameter, which has no limit in the moduli space
of torsion-free \gtstr s on $M$. Intuitively this path corresponds to a
boundary point of the moduli space, where $M$ is `pulled apart' into
the connected summands $M_{\gi}$.

In this chapter we study how the gluing construction behaves under
deformations. In particular
we try to determine whether any sufficiently small torsion-free deformation of
the resulting \gtstr{} $\varphi$ on $M$ arises as the gluing of some
deformations of the structures on $M_{\pm}$. To this end we define
a smooth gluing map $Y$ which takes values in the moduli space $\defstr$ of
torsion-free \gtstr s on $M$, and study its local properties.
This leads to a precise notion of how one can add boundary points to $\defstr$,
representing ways of pulling $M$ apart.

In \S \ref{gluesetupsec} we describe the gluing construction in detail,
and give statements of the main results. As the domain for the gluing
map we take a moduli space $\calg$ of `gluing data', consisting of equivalence
classes of arguments $(\varphi_{+}, \varphi_{-}, L)$ for the gluing
construction with $L$ sufficiently large.
$\calg$ is a smooth manifold, and theorem \ref{maingluethm} states that
\[ Y : \calg \to \calm \]
is a local diffeomorphism. In \S \ref{gluetoposec} we discuss
the topology of the glued manifold $M$. A key result for computing the
derivative of the gluing map is theorem \ref{gluehodgethm}, which is
a Hodge theory gluing result.

The main theorem \ref{maingluethm} is proved in \S \ref{glueproofsec}.
In order to show that $\calg$ is a smooth manifold we make use of
theorem \ref{modsubmersethm} about the boundary map on the moduli
space of torsion-free EAC \gtstr s.

%xxx rewrite this
In \S \ref{glueboundarysec} we look at the consequences of the gluing
construction for the moduli space $\defstr$. $\calg$ can be considered as
a fibre bundle with typical fibre $\bbrp$ over a space $\calb$ of equivalence
classes of matching pairs $(\varphi_{+}, \varphi_{-})$.
We consider how $\defstr$ can be partially compactified by attaching a quotient
of $\calb$ as a boundary, so that the path of torsion-free $\defstr$ of
increasing neck length arising from gluing a matching pair
$(\varphi_{+}, \varphi_{-})$ converges to a boundary point.


\section{Setup}
\label{gluesetupsec}


\subsection{Gluing construction}
\label{glueconstrsub}

Let $M_{\pm}^{7}$ be a pair of oriented manifolds, each with a single
cylindrical end, and the same cross-section $X^{6}$.
We assume that $X$ is oriented so that its orientation
agrees with that defined by $M_{+}$ on its boundary, and is the reverse of that
defined by $M_{-}$ on its boundary. This ensures that the connected sum of
$M_{+}$ and $M_{-}$ obtained by identifying their boundaries at infinity is
oriented.
Let $t_{\pm}$ be cylindrical coordinates on $M_{\pm}$ respectively.
%Let $\sigma : \bbr \to [0,1]$ be a smooth
%function which is $0$ on $(-\infty,0]$ and $1$ on $[1,\infty)$.

\begin{defn}
Let $\varphi_{\gi}$ be torsion-free EAC \gtstr s on $M_{\gi}$. The pair
$(\varphi_{+},\varphi_{-})$ is said to \emph{match} if their asymptotic models
are $\Omega \pm dt_{\pm} \wedge \omega$,
respectively, for some Calabi-Yau structure $(\Omega,\omega)$ on $X$
compatible with the chosen orientation.
%
Let $\ycal{X}$ be the space of such pairs.
\end{defn}


%Let $L > 2$.
%Let $\rho_{\gi} : M_{\gi} \to [0,1], y \mapsto \sigma(t_{\gi}(y))$.

\ignore{
Given $L \in \bbrp$
let $M_{\gi}(L) = \{ y \in M_{\gi} : t_{\gi} < L+2 \}$. If for $x \in X$,
$0 < t < 2$ we identify $(x,L+t) \in M_{+}(L)$ with $(x,L+2-t) \in M_{-}(L)$
then the union of $M_{+}(L)$ and $M_{-}(L)$ forms a smooth manifold $M(L)$.
Let $i_{\gi} : M_{\gi}(L) \to M(L)$ be the inclusion, and include the
cross-section in $M(L)$ by
$j : X \to M_{+}(L) \cap M_{-}(L), \; x \mapsto (x,L+1)$.
}

Given $L \in \bbrp$
let $M_{\gi}(L) = \{ y \in M_{\gi} : t_{\gi} \leq L \}$.
Identify the boundaries of $M_{\gi}(L)$ to form a
compact smooth manifold $M(L)$, and
let $j^{*} : X \into M(L)$ be the inclusion of the common boundary.

\begin{rmk}
$M(L)$ is independent of $L$ up to diffeomorphism, so we will
often refer to it simply as $M$. On the other hand, sometimes we wish to
emphasise that we use a particular description $M(L)$, e.g. if we ever wish
to give a representative for a de Rham cohomology class on $M$ we consider it
as a closed form on $M(L)$ for some~$L$.
\end{rmk}

For notational convenience we suppose that the cylindrical part of
$M_{\gi}$ is given by $t_{\gi} > -2$ rather than $t_{\gi} > 0$. 
By lemma \ref{cylfylem} there is for every closed exponentially asymptotically
translation-invariant $m$-form $\alpha_{\gi}$ on $M_{\gi}$ a decaying
$(m-1)$-form $\eta_{\gi}(\alpha_{\gi})$ which is supported on $t_{\gi} > L-2$,
such that $\alpha_{\gi} + d\eta_{\gi}(\alpha_{\gi})$ is translation-invariant
on $t_{\gi} > L-1$.

For $(\varphi_{+},\varphi_{-}) \in \ycal{X}$
let $\tv_{\gi} = \varphi_{\gi} + d\eta_{\gi}(\varphi_{\gi})$.
Then we can define a \gtstr{} $\tv(\varphi_{+},\varphi_{-}, L)$ on $M(L)$ by
%$i^{*}_{\gi}\tv = \tv_{\gi}$.
$\tv|_{M_{\gi}(L)} = \tv_{\gi}|_{M_{\gi}(L)} $.
Note that the choice of (cut-off function in the definition of)
$\eta_{\gi}$ does not affect the cohomology class of
$\tv(\varphi_{+},\varphi_{-}, L)$.

\begin{prop}
\label{basicglueprop}
There is an upper semi-continuous map $L_{0} : \ycal{X} \to \bbrp$ such
that for any $(\varphi_{+},\varphi_{-}) \in \ycal{X}$ and
$L > L_{0}(\varphi_{+},\varphi_{-})$ there is a unique diffeomorphism class of
torsion-free \gtstr s on $M(L)$ in a small neighbourhood
of $\tv(\varphi_{+},\varphi_{-}, L)$ in its cohomology class.
\end{prop}

\begin{proof}[Sketch proof.]
The idea is that for large $L$ the torsion of $\tv(\varphi_{+},\varphi_{-}, L)$
is very small, and the structure can be perturbed to a torsion-free one using
a contraction-mapping argument. For details see
Kovalev \cite[\S 5]{kovalev03}, in principle inspired by Floer \cite{floer91}.

The result can also be deduced from theorem \ref{joyce2thm} below,
due to Joyce (with the uniqueness part coming from
proposition \ref{unresprop}).
\end{proof}

\begin{rmk}
The resulting \gtmetric{} on $M(L)$ has an almost cylindrical `neck' of length
roughly $2L$. The diameter is $\diam M(L) \sim 2L$ as $L \to \infty$.
\end{rmk}

One might expect to be able to find examples of matching pairs of \gtstr s by
taking $M_{-} = M_{+}$, with the same \gtmetric{} and opposite orientation
and cross-sections identified trivially.
In fact, this only gives trivial examples; it is only possible when
$M_{+}$ has a double cover isometric to a cylinder $X \times \bbr$
(such as example \ref{cyinvex}), and the gluing of $M_{+}$ and $M_{-}$ is then
covered by $X \times S^{1}$ and cannot have holonomy exactly $G_{2}$.

\begin{prop}
Let $M_{-}$ be $M_{+}$ with its orientation reversed and
$(\varphi_{+}, \varphi_{-})$ a matching pair of \gtstr s.
If $\varphi_{+}$ and $\varphi_{-}$ define the same metric then
$M_{+}$ has a double cover isometric to a cylinder.
\end{prop}

\begin{proof}
$-\varphi_{-}$ is a torsion-free \gtstr{} on $M_{+}$ which defines the
same metric as $\varphi_{+}$.
By remark \ref{twistrmk}
this corresponds to a parallel section of $\bbp(\bbr \oplus TM_{+})$. 
The matching condition for $\varphi_{+}$ and
$\varphi_{-}$ implies that the parallel section is asymptotic
to $[\contrat]$. In other words either $M_{+}$ or a double cover of $M_{+}$
has a parallel vector field asymptotic to $\pm\contrat$.
By corollary \ref{degonehodgecor} this is impossible for
a manifold with a single end, so $M_{+}$ has a double cover which is isometric
to a cylinder by theorem \ref{salurthm}. 
\end{proof}

This does not mean that it is impossible to obtain irreducible \gtmfd s
by gluing isometric EAC \gtmfd s, only that the identification of the
boundaries must be distinct from the asymptotic limit of the isometry.
In \S \ref{simplesub} we see examples where an EAC \gtmfd{} is glued to a
copy of itself, using an anti-holomorphic isometry of the boundary.


\subsection{Results}

We need to find an appropriate domain for the gluing map $Y$.
Let $\gical{M}$ denote the moduli space of torsion-free EAC \gtstr s
on $M_{\gi}$, and $\caln$ the moduli space of Calabi-Yau structures on
their common cross-section $X$. We can
define a subset $\ycal{M} \subseteq \onecal{M} \times \twocal{M}$
consisting of pairs which have matching images in $\caln$.
While we can apply our understanding of $\gical{M}$ and their relationship to
$\mathcal{N}$ (particularly theorem \ref{modsubmersethm}) to show that
$\ycal{M}$ is a manifold, it is not an appropriate domain. The reason is that
for a matching pair of points in the moduli spaces $\onecal{M}, \twocal{M}$
there is some ambiguity in how to glue them. This arises both from choosing
representatives of the diffeomorphism classes (this ambiguity in some sense
corresponds to the quotient of the automorphisms of $X$ by a subgroup generated
by elements which extend to automorphisms of $M_{+}$ or $M_{-}$),
and from the dependence of the gluing construction on a gluing parameter $L$.
Instead we define the gluing map on a moduli space of arguments for the gluing
construction.

\begin{defn}
A set of \emph{gluing data} is a triple $(\varphi_{+},\varphi_{-},L)$ such that
$(\varphi_{+},\varphi_{-}) \in \ycal{X}$ and
$L > L_{0}(\varphi_{+},\varphi_{-})$.
Let $G_{0}$ be the space of gluing data.
\end{defn}

$G_{0}$ is an open subset of $\ycal{X} \times \bbr$.
If we let $\mathcal{M}$ be the moduli space of torsion-free \gtstr s
on $M$ then proposition \ref{basicglueprop} provides a well-defined smooth
map
\begin{equation}
\label{upperyeq}
Y : G_{0} \to \mathcal{M} .
\end{equation}

Let $\gical{D}$ be the group of EAC diffeomorphisms of $M_{\gi}$ isotopic to
the identity (cf. definition \ref{cyldiffdef}).
Since the cross-sections of $M_{+}$ and $M_{-}$ have been identified once and
for all we should only expect the gluing map to be invariant under
diffeomorphisms that respect this identification.


\begin{defn}
$(\phi_{+}, \phi_{-}) \in \onecal{D} \times \twocal{D}$
is a \emph{matching pair of EAC diffeomorphisms} if
the asymptotic models of $\phi_{\gi}$ are
$(x,t_{\gi}) \mapsto (\Xi_{\gi}(x), t_{\gi} + h_{\gi})$ with
$\Xi_{+} = \Xi_{-}$. Let $\ycal{D}$ be the identity component
of the group of such pairs.

For  $(\phi_{+}, \phi_{-}) \in \ycal{D}$ let $h = \half(h_{+}+h_{-})$, and
define an action on $\ycal{X} \times \bbr$ by
\begin{equation}
\label{calydacteq}
\phi^{*} : (\varphi_{+},\varphi_{-},L) \mapsto
(\phi_{+}^{*}\varphi_{+}, \phi_{-}^{*}\varphi_{-}, L-h) .
\end{equation}
\end{defn}

The map $L_{0}$ is not $\ycal{D}$-invariant, so nor is the open set
$G_{0} \subseteq \ycal{X} \times \bbr$. Nevertheless we can define

\begin{defn}
The \emph{moduli space of gluing data} is
$\mathcal{G}_{0} = G_{0}\ycal{D}/\ycal{D}$.
\end{defn}

We will show that $\calg_{0}$ is a smooth manifold. Moreover
%We will show in proposition \ref{glueinvprop} that
the gluing map
(\ref{upperyeq}) really is invariant under the action of~$\ycal{D}$, and
therefore descends to a smooth map
\begin{equation}
\label{mainyeq}
Y : \mathcal{G}_{0} \to \mathcal{M} .
\end{equation}

%Proposition \ref{modfibreprop} states that $\mathcal{G}_{0}$ is a smooth
%manifold, so (\ref{mainyeq}) is in fact smooth.

Proposition \ref{yhderprop} computes the derivative of the gluing map
(\ref{mainyeq}). For each matching pair
$(\varphi_{+},\varphi_{-})$ the derivative is invertible at
$(\varphi_{+},\varphi_{-}, L)\ycal{D} \in \calg_{0}$ for all but finitely
many $L > L_{0}(\varphi_{+},\varphi_{-})$.
Therefore $Y$ is a local diffeomorphism on
some open subset $\mathcal{G} \subseteq \mathcal{G}_{0}$ whose gluing
parameters are sufficiently large. This gives the main result of the chapter.

\begin{thm}
\label{maingluethm}
The gluing map $Y : \mathcal{G} \to \mathcal{M}$ is a local diffeomorphism.
\end{thm}

The most important tool in the proof is that the projection
$\pi_{H} : \mathcal{M} \to H^{3}(M)$ to de Rham cohomology is a local
diffeomorphism. This means that we can study the local
properties of the gluing map in terms of what the gluing does to the cohomology
classes. This is discussed in \S \ref{gluetoposec}.
In particular we prove a Hodge theory gluing result.

% xxx
We can restrict (\ref{calydacteq}) to an action of $\ycal{D}$ on
the space $\ycal{X}$ of matching pairs. The quotient $\calb$ is smooth,
and $\calg$ is obviously a fibre bundle over $\calb$ with typical fibre
$\bbrp$. $\calg$ can be included in a fibre bundle
$\calgbar$ over $\calb$ with typical fibre $(0,\infty]$ (so the boundary
is $\calb$).
In \S \ref{glueboundarysec} we deduce from theorem \ref{maingluethm} that
$\defstr$ can be partially
compactified by inclusion in a topological manifold $\defstrbar$ whose
boundary $\hat\calb$ is covered by $\calb$.
%so that the paths defined by gluing a matching pair of EAC \gtstr s with
%increasing gluing parameter converge to the corresponding point in $\calb$.
The boundary points parametrise ways of `pulling apart' $M$
into a pair of EAC connected-summands.

\begin{thm}
\label{gluecptfythm}
$\defstr$ can be identified with the interior of a topological manifold
$\defstrbar$ with boundary~$\hat\calb$, so that the following holds:
If $(\varphi_{+}, \varphi_{-}) \in \ycal{X}$ then the path
$Y(\varphi_{+}, \varphi_{-}, L)$ converges to the image of
$(\varphi_{+}, \varphi_{-})$ in $\hat\calb$ as $L \to \infty$.
\end{thm}

\begin{rmk}
It may be possible to pull apart a compact \gtmfd{} in more than one direction.
Theorem \ref{gluecptfythm} can be used to attach one `face' to the moduli
space for each topologically distinct way of writing the manifold as a union
of a pair of manifolds with cylindrical ends identified at the boundary.
\end{rmk}

%\begin{rmk}
%There is a natural submersion $\calb \to \ycal{M}$.
%The connected components of the fibres are planes of dimension $b^{1}(M)$.
%\end{rmk}


\section{Gluing and topology}
\label{gluetoposec}


\subsection{Topology of the connected sum}

Let $M_{\gi}^{n}$ be oriented manifolds each with a single
cylindrical end, which have common cross-section $X$. As above
we assume that $X$ is oriented compatibly with $M_{+}$
and reverse to $M_{-}$, and we form a generalised connected sum $M$.
We collect here some results about the topology of $M$ that we will use.

As we remarked before, as a smooth manifold $M$ is independent of the choice of
gluing parameter $L$.
Up to isotopy there are natural inclusion maps

\begin{diagram}[notextflow]
& & M & & \\
& \ruTo^{i_{+}} & & \luTo^{i_{-}} & \\
M_{+} & & \uTo^{j} & & M_{-} \\
& \luTo^{j_{+}} & & \ruTo^{j_{-}} & \\
& & X & &
\end{diagram}

%\raggedbottom

A large part of what we need to know about the topology
is contained in the Mayer-Vietoris sequence for $M = M_{+} \cup M_{-}$
and the sequence for the cohomology of $M_{\gi}$ relative to $X$.

\begin{gather}
\label{mayereq}
\cdots \longrightarrow H^{m-1}(X) \stackrel{\delta}{\longrightarrow}
H^{m}(M) \stackrel{i_{+}^{*} \oplus i_{-}^{*}}{\longrightarrow}
H^{m}(M_{+}) \oplus H^{m}(M_{-})
\stackrel{j_{+}^{*} - j_{-}^{*}}{\longrightarrow} H^{m}(X) \longrightarrow
\cdots \\
\label{relexacteq2}
\cdots \longrightarrow H^{m-1}(X) \stackrel{\partial_{\gi}}{\longrightarrow}
H^{m}_{cpt}(M_{\gi}) \stackrel{e_{\gi}}{\longrightarrow} H^{m}(M_{\gi})
\stackrel{j_{\gi}^{*}}{\longrightarrow} H^{m}(X) \longrightarrow \cdots
\end{gather}
%
The inclusions $i_{\gi} : M_{\gi} \hookrightarrow M$ induce maps
$i_{\gi*} : H^{m}_{cpt}(M_{\gi}) \to H^{m}(M)$. Note that
\begin{equation}
\label{dieq}
\delta = i_{+*} \circ \partial_{+} = - i_{-*} \circ \partial_{-} .
\end{equation}

$j_{\gi}^{*} : H^{m}(M_{\gi}) \to H^{m}(X)$ is the Poincar\'e dual of
$\pm \partial_{\gi} : H^{n-m-1}(X) \to H^{n-m}(M)$
(the sign difference comes from our assumption on the orientations of
$M_{\gi}$ and $X$).
The Poincar\'e dual of the Mayer-Vietoris sequence is the sequence for
relative cohomology of $M$ relative to $X$.
%
\begin{equation}
\cdots \; H^{m-1}(X)
\stackrel{\partial_{+} \oplus \partial_{-}}{\longrightarrow}
H^{m}_{cpt}(M_{+}) \oplus H^{m}_{cpt}(M_{-})
\stackrel{i_{+*} + i_{-*}}{\longrightarrow} H^{m}(M)
\stackrel{j^{*}}{\longrightarrow} H^{m}(X) \; \cdots
\end{equation}

%\flushbottom

\begin{lem}
\label{mxinlem}
If $M_{\gi}$ are Ricci-flat EAC manifolds then
$j^{*} : H^{1}(M) \to H^{1}(X)$ is injective.
\end{lem}

\begin{proof}
The kernel of $j^{*}$ is the image of
$H^{1}_{cpt}(M_{+}) \oplus H^{1}_{cpt}(M_{-})$,
which is $0$ by corollary \ref{onecptcor}.
\end{proof}

Denote the image of $j_{\gi}^{*} : H^{m}(M_{\gi}) \to H^{m}(X)$ by
$A^{m}_{\gi}$, and let $A^{m}_{\diag}$ be the image of
$j^{*} : H^{m}(M) \to H^{m}(X)$. By the exactness of the Mayer-Vietoris
sequence $A^{m}_{\diag} =  A^{m}_{+} \cap A^{m}_{-}$.


\subsection{Gluing and cohomology}

We explain how to glue a matching pair of closed forms on $M_{+}, M_{-}$
to a well-defined cohomology class on $M$.

Let $\ycal{Z}^{m}$ be the set of matching pairs of closed
exponentially asymptotically translation-invariant $m$-forms
on $M_{+}, M_{-}$, i.e. $(\psi_{+}, \psi_{-})$ such that $\psi_{\gi}$
is a closed exponentially asymptotically translation-invariant
$m$-form on $M_{\gi}$,
$B_{a}(\psi_{+}) = B_{a}(\psi_{-})$, and $B_{e}(\psi_{+}) = - B_{e}(\psi_{-})$.

If $(\psi_{+}, \psi_{-}) \in \ycal{Z}^{m}$ and $L > 0$ let
$\tp_{\gi} = \psi_{\gi} + d\eta_{\gi}(\psi_{\gi})$.
Choose the cut-off function for the cylinders in the definition (\ref{etaeq})
of $\eta_{\gi}$ to ensure that $\tp_{\gi}$ is translation-invariant
on $t_{\gi} > 0$.
%on $t_{\gi} > 2$.
Then we can define $\tp(\psi_{+}, \psi_{-}, L)$ on $M(L)$ by $i^{*}_{\gi}\tp = \tp_{\gi}$.
We define a gluing map
\begin{equation}
\label{yhdefeq}
Y_{H} : \ycal{Z}^{m} \times \bbrp \to H^{m}(M), \;\:
(\psi_{+}, \psi_{-}, L) \mapsto [\tp] .
\end{equation}
$Y_{H}$ is independent of the choice of $\eta_{\gi}$, so well-defined.
Furthermore, $Y_{H}$ is invariant under the action of the group $\ycal{D}$
of matching diffeomorphisms, defined similarly to (\ref{calydacteq}).

\begin{defn}
For $(\phi_{+}, \phi_{-}) \in \ycal{D}$ with asymptotic models
$(x, t_{\gi}) \mapsto (\Xi(x), t_{\gi} + h_{\gi})$
let $h = \half(h_{+}+h_{-})$, and
define an action on $\ycal{Z}^{m} \times \bbr$ by
\begin{equation}
\label{calydacteq2}
\phi^{*} : (\psi_{+},\psi_{-},L) \mapsto
(\phi_{+}^{*}\psi_{+}, \phi_{-}^{*}\psi_{-}, L-h) .
\end{equation}
\end{defn}

\begin{prop}
\label{yhinvprop}
If $(\psi_{+}, \psi_{-}, L) \in \ycal{Z}^{m} \times \bbrp$ and
$(\phi_{+}, \phi_{-}) \in \ycal{D}$ with $h_{\gi} < L$ then
\[ Y_{H}(\psi_{+}, \psi_{-}, L) = 
Y_{H}(\phi_{+}^{*}\psi_{+}, \phi_{-}^{*}\psi_{-}, L-h)
\in H^{m}(M) . \]
\end{prop}

\begin{proof}[Sketch proof.]
Let $\tp = \tp(\psi_{+}, \psi_{-}, L)$ and $\tp' =
\tp(\phi_{+}^{*}\psi_{+}, \phi_{-}^{*}\psi_{-}, L-h)$.
%The idea is that
$\phi_{+}$ and $\phi_{-}$ can be approximately glued to form
a diffeomorphism $\tilde\phi : M(L-h) \to M(L)$.
This pulls back $[\tp]$ to~$[\tp']$.
%
% ignore full proof
\ignore{
It suffices to prove the statement for $(\phi_{+}, \phi_{-})$ close to the
identity, so we may write $ \phi_{\gi} = \exp V_{\gi}$
with $V_{\gi}$ is exponentially asymptotic to
$W + h_{\gi}\contratgi$ for some vector field $W$
on~$X$. Let
$\tilde V_{\gi} = (1-\rho)V_{\gi} + \rho (W + h_{\gi}\contratgi)$,
where $\rho$ is $0$ for $t < 0$ and $1$ for $t > 1$, and
$\tilde \phi_{\gi} = \exp \tilde V_{\gi}$. $\tilde \phi_{\gi}$ is
translation-invariant on $t_{\gi} > L - |h_{\gi}|$.
If we regard $M(L-h)$ as the gluing of $M_{+}(L-h_{+})$ and $M_{-}(L-h_{-})$
then we can glue 
$\tilde \phi_{\gi} : M_{\gi}(L-h_{\gi}) \to M_{\gi}(L)$ to form
a diffeomorphism $\tilde \phi : M(L-h) \to M(L)$.

$\tilde \phi_{\gi}^{*}\tp_{\gi} = \tp_{\gi} + d\alpha_{\gi}$, where
$\alpha_{\gi}$ has support contained in $t_{\gi} < L + h_{\gi}$.
Hence $\tilde \phi^{*} \tp$ is cohomologous to $\tp'$.
} % end ignore
\end{proof}


\begin{prop}
\label{lderprop}
Let $(\psi_{+}, \psi_{-}) \in \ycal{Z}^{m}$ with
$B_{e}(\psi_{\gi}) = \pm \tau$. If $L, h \in \bbrp$ then
\begin{equation}
\label{yhnegdefeq}
Y_{H}(\psi_{+}, \psi_{-}, L + h) =
Y_{H}(\psi_{+}, \psi_{-}, L) + 2h\delta([\tau]) ,
\end{equation}
where $\delta$ is the boundary homomorphism appearing in the Mayer-Vietoris
sequence \textup{(\ref{mayereq})}.
\end{prop}

\begin{proof}
It suffices to prove the result separately for the cases $B_{a}(\psi) = 0$
and $B_{e}(\psi) = 0$.

If $B_{e}(\psi) = 0$ pick a diffeomorphism $f : (0, L) \to (0,L+h)$ which is
$id$ on $(0,1)$ and $id + h$ on $(L-1,L)$. We can define a diffeomorphism
$M(L) \to M(L+h)$ which is the identity on the images of the compact
pieces $M_{\gi}(0)$ in $M(L)$ and
$(x,t) \mapsto (x, f(t))$ on the cylindrical part. This pulls back
$\tp(\psi_{+}, \psi_{-}, L)$ to $\tp(\psi_{+}, \psi_{-}, L+h)$.

If $B_{a}(\psi) = 0$ then let
$c_{\gi} = \pm \tp_{\gi} - d(\rho_{\gi} t_{\gi} \tau)$,
with $\rho_{\gi}$ chosen so that $c_{\gi}$ is supported in $t_{\gi} < 1$.
By definition of the Mayer-Vietoris boundary map $\delta$ the 
form on $M(L)$ obtained by gluing $d(\rho_{+}t_{+}\tau)$ and
$-d(\rho_{-}t_{-}\tau)$ is cohomologous to
$\delta((t_{+}+t_{-})[\tau]) = 2L\delta([\tau])$ for any~$L$. Hence
for any~$L$
\begin{equation}
\label{yheq}
Y_{H}(\psi_{+}, \psi_{-}, L) = i_{+*}([c_{+}]) + i_{-*}(-[c_{-}]) +
2L\delta([\tau]) .
\end{equation}
Since $i_{\gi*}$ and $\delta$ are independent of $L$ the result follows.
\end{proof}

It is convenient to use proposition \ref{lderprop} to extend
$Y_{H}$ to negative gluing parameters in a well-defined way.

\begin{defn}
\label{yhdefn}
Define
\[ Y_{H} : \ycal{Z}^{m} \times \bbr \to H^{m}(M) \]
as (\ref{yhdefeq}) on $\ycal{Z} \times \bbrp$, and extend
for any $L > 0$ and $h \in \bbr$ by (\ref{yhnegdefeq}).
\end{defn}


\subsection{Hodge theory}
\label{gluehodgesub}

Now suppose that $M_{\gi}^{n}$ have EAC Riemannian metrics whose
cylindrical models match on the cylinders. We wish to consider what the
gluing of closed forms described in the previous subsection does on pairs of
harmonic forms.

Let $\harm^{m}_{\gi,0}$ be the space of bounded harmonic $m$-forms on $M_{\gi}$.
The space of matching pairs of harmonic forms is
\begin{equation*}
\harm^{m}_{y} = (\harm^{m}_{+,0} \times \harm^{m}_{-,0}) \cap\ycal{Z}^{m} .
\end{equation*}
We prove that any cohomology class on $M$ can be obtained
by gluing a matching pair of harmonic forms, except when the gluing parameter
is an eigenvalue of a certain self-adjoint endomorphism
which is to be defined below.

\begin{thm}
\label{gluehodgethm}
Let $M_{\gi}$ have EAC metrics.
Considering $L$ as a parameter, the linear map
\begin{equation}
\label{lfixeq}
Y_{H} : \harm^{m}_{y} \to H^{m}(M), \;\:
(\psi_{+}, \psi_{-}) \mapsto Y_{H}(\psi_{+}, \psi_{-}, L)
\end{equation}
is an isomorphism except when $-2L$ is an eigenvalue of
\begin{equation}
\label{saprotoeq}
\pi_{E}(\partial_{+}^{-1}C_{+} + \partial_{-}^{-1}C_{-}) :
E^{m-1}_{\diag} \to E^{m-1}_{\diag} .
\end{equation}
\end{thm}

We use notation for harmonic forms analogous to that in the summary
in \S \ref{hodgesummary}, and add subscripts $\gi$
to distinguish between objects associated to $M_{+}$ and $M_{-}$.
We write $H^{m}(X)$ as orthogonal direct sums
$A^{m}_{\gi} \oplus E^{m}_{\gi}$, where $A^{m}_{\gi}$ is the image of
$j_{\gi}^{*} : H^{m}(M_{\gi}) \to H^{m}(X)$.
Let $\gical{A}^{m}$ and $\gical{E}^{m}$ be the spaces of harmonic
representatives.

Let $\diagcal{A}^{m} = \onecal{A}^{m} \cap \twocal{A}^{m}$. This is then
the space of harmonic representatives for~$A^{m}_{d}$. Similarly let
$\diagcal{E}^{m} = \onecal{E}^{m} \cap \twocal{E}^{m}$ and denote
the corresponding subspace of $H^{m}(X)$ by $E^{m}_{d}$.
%So $A_{\diag}^{m} = *E_{\diag}^{n-m-1}$.
%Let $\pi_{E} : H^{m}(M) \to E^{m}_{\diag}$ denote the \ltwoorth{} projection.

Recall that $\partial_{\gi}$ denotes the boundary map in the long exact
sequence for relative cohomology (\ref{relexacteq2}).
We define an isomorphism
\[ C_{\gi} : E_{\gi}^{m-1} \to \im \partial_{\gi} \subseteq
H^{m}_{cpt}(M_{\gi})\]
as follows. If $\tau \in \gical{E}^{m-1}$ let $\psi$ be the unique element of
$\harm^{m}_{\gi,E}$ (the bounded exact harmonic forms on $M_{\gi}$)
such that $B_{e}(\psi) = \tau$.
Taking $\eta_{\pm}$ as defined in (\ref{etaeq}),
$\psi + d\eta_{\gi}(\psi) - d(\rho_{\gi}t_{\gi}\tau)$ has
compact support, so represents a class
$C_{\gi}([\tau]) \in H^{m}_{cpt}(M_{\gi})$.
This class is mapped to $0$ by $e_{\gi}$, so lies in the image of
$\partial_{\gi}$.

\begin{rmk}
$C_{\gi}$ is independent of the choice of $\rho_{\gi}$, but depends on both
the metric and the cylindrical coordinate:
replacing $t_{\gi}$ by $t_{\gi} + \lambda$ adds $\lambda \partial_{\gi}$
to $C_{\gi}$.
\end{rmk}

\noindent
Composing $C_{\gi}$ with the inverse of
$\partial_{\gi} : E^{m-1}_{\gi} \to \im \partial_{\gi}$ gives an
endomorphism
$\partial_{\gi}^{-1} C_{\gi}$ of~$E^{m-1}_{\gi}$.
%, which is self-adjoint.

\begin{lem}
\label{pdlem}
$\partial_{\gi}^{-1}C_{\gi} : E^{m}_{\gi} \to E^{m}_{\gi}$ is self-adjoint.
\end{lem}

\begin{proof}
This is really a rephrasing of proposition \ref{bdlagprop}.
If $\chi \in \harm^{m}_{\gi,-}$ with
$BD_{a}(\chi) = \tbomatrixsmall{\alpha}{\beta}
\in (\harm^{m}_{X})^{2}$ then
$d\chi$ is the unique element of $\harm^{n+1}_{\gi, E}$ such that
$B_{e}(d\chi) = \beta$.
\[ C_{\gi}[\beta] = \left[d(\chi + \eta_{\gi}(d\chi) -
\rho_{\gi} t_{\gi}\beta)\right] = \partial_{\gi}[\alpha]
\in H^{m+1}_{cpt}(M_{\gi}) , \]
so $\partial_{\gi}^{-1}C_{\gi}[\beta]$ is the \ltwoorth{} projection of
$[\alpha]$ to $E^{m}_{\gi}$.
The result thus follows from proposition \ref{bdlagprop}.
\end{proof}

\noindent
It follows that the endomorphism (\ref{saprotoeq}) is self-adjoint too.

\begin{proof}[Proof of theorem \ref{gluehodgethm}]
Consider the map
$(i^{*}_{+} \oplus i^{*}_{-}) : H^{m}(M) \to H^{m}(M_{+}) \oplus H^{m}(M_{-})$
in the Mayer-Vietoris sequence.
Recall that $L$ is fixed so that $Y_{H}$ is regarded as a linear map
$\harm^{m}_{y} \to H^{m}(M)$. In order to prove that (\ref{lfixeq})
is an isomorphism
it suffices to show that $\im((i^{*}_{+} \oplus i^{*}_{-}) \circ Y_{H}) =
\im(i^{*}_{+} \oplus i^{*}_{-})$ and that
$Y_{H} : \ker((i^{*}_{+} \oplus i^{*}_{-}) \circ Y_{H}) \to
\ker (i^{*}_{+} \oplus i^{*}_{-})$ is an isomorphism.

$(i^{*}_{+} \oplus i^{*}_{-})Y_{H}(\psi_{+}, \psi_{-}) =
([\psi_{+}], [\psi_{-}])$ and it follows from the exactness of
the Mayer-Vietoris sequence that
$\im((i^{*}_{+} \oplus i^{*}_{-}) \circ Y_{H}) =
\im(i^{*}_{+} \oplus i^{*}_{-})$. Also
$\ker ((i^{*}_{+} \oplus i^{*}_{-}) \circ Y_{H})$ equals $\harm^{m}_{y,E}$, the
pairs of exact forms in $\harm^{m}_{y}$.

Thus the problem reduces to determining whether the restriction
\[ Y_{H} : \harm^{m}_{y,E} \to \ker(i^{*}_{+} \oplus i^{*}_{-}) \]
of (\ref{lfixeq}) is an isomorphism.
Given $\tau \in \diagcal{E}^{m-1}$ let $(\psi_{+},\psi_{-})$ be the unique
element of $\harm^{m}_{y,E}$ such that
$\tau = B_{e}(\psi_{+}) = -B_{e}(\psi_{-})$.
By the definition of $C_{\gi}$ and (\ref{yheq})
\begin{equation*}
Y_{H}(\psi_{+}, \psi_{-}) =
i_{+*}C_{+}([\tau]) + i_{-*}C_{-}([-\tau]) + 2L\delta([\tau]) . 
\end{equation*}
Combining with (\ref{dieq}) we find
\begin{equation}
\label{yharmeeq}
Y_{H}(\psi_{+}, \psi_{-}) =
\delta \left(\partial_{+}^{-1}C_{+}([\tau]) +
\partial_{-}^{-1}C_{-}([\tau]) + 2L[\tau] \right) .
\end{equation}
%
$\delta : H^{m-1}(X) \to H^{m}(M)$ maps
$E^{m-1}_{\diag} \to \ker (i^{*}_{+} \oplus i^{*}_{-})$ isomorphically %bijectively
and vanishes on the orthogonal complement of $E^{m-1}_{\diag}$. 
Hence (\ref{yharmeeq}) is an isomorphism
$\harm^{m}_{y,E} \to \ker (i^{*}_{+} \oplus i^{*}_{-})$ unless
$-2L$ is an eigenvalue of the endomorphism (\ref{saprotoeq}).
\end{proof}


\section{The gluing map}
\label{glueproofsec}

We will now make use of the topological results of the previous subsections to
study the gluing map for torsion-free \gtstr s. As in \S \ref{gluesetupsec},
$M_{+}$ and $M_{-}$ are EAC \gtmfd s
with a common cross-section $X$ and $M$ is their connected sum.
$\mathcal{M}$~denotes the moduli space of torsion-free \gtstr s on $M$ and
$G_{0}$ the space of gluing data.

In order to prove theorem \ref{maingluethm} we need to show that the gluing
map is $\ycal{D}$-invariant so that it is well-defined on
$\calg_{0} = G_{0}\ycal{D}/\ycal{D}$,
to show that $\calg_{0}$ is a smooth manifold
and to compute the derivative of the gluing map.


\subsection{Diffeomorphism invariance}

Note that the composition $\pi_{H} \circ Y : G_{0} \to H^{3}(M)$ of
the gluing map (\ref{upperyeq})
with the local diffeomorphism $\pi_{H} : \mathcal{M} \to H^{3}(M)$
is simply the restriction to $G_{0}$ of the map $Y_{H}$ given by
definition \ref{yhdefn}. We will use this first to show that $Y$
induces a well-defined map on $\mathcal{G}_{0}$, i.e. the quotient of
$G_{0}$ by the group $\ycal{D}$ of matching EAC diffeomorphisms.
Later we will determine the local
properties of $Y : \mathcal{G}_{0} \to \mathcal{M}$ from those of
$Y_{H} : (\ycal{X} \times \bbr)/\ycal{D} \to H^{3}(M)$.

\begin{prop}
\label{glueinvprop}
The map $Y$ is $\ycal{D}$-invariant, so descends to a well-defined
continuous function
\begin{equation}
Y : \mathcal{G}_{0} \to \mathcal{M} .
\end{equation}
\end{prop}

\begin{proof}
We need to show that if $\phi \in \ycal{D}$ and
$(\varphi_{+}, \varphi_{-}, L) \in G_{0}$ such that
$\phi^{*}(\varphi_{+}, \varphi_{-}, L) \in G_{0}$ then
\[ Y(\varphi_{+}, \varphi_{-}, L) = Y(\phi^{*}(\varphi_{+}, \varphi_{-}, L)) .\]
The idea of the proof is to connect $(\varphi_{+}, \varphi_{-}, L)$ and
$\phi^{*}(\varphi_{+}, \varphi_{-}, L)$ by a path in~$G_{0}$. The image
under $Y$ of this path is the lift by the local diffeomorphism
$\pi_{H} : \mathcal{M} \to H^{3}(M)$ of a path in $H^{3}(M)$, which
is determined by propositions \ref{yhinvprop} and \ref{lderprop}.

Let $[0,1] \to \ycal{D}$, $s \mapsto \phi_{s}$ be a path connecting
the identity to $\phi$, and take $k$ sufficiently large that
$\phi_{s}^{*}(\varphi_{+}, \varphi_{-}, L+k) \in G_{0}$ for all $s$.
By proposition \ref{yhinvprop} the path $[0,1] \to \defstr$, 
$s \mapsto Y(\phi_{s}^{*}(\varphi_{+}, \varphi_{-}, L+k))$
is a lift of a constant path in $H^{3}(M)$, so
\begin{equation*}
Y(\phi^{*}(\varphi_{+}, \varphi_{-}, L+k)) =
Y((\varphi_{+}, \varphi_{-}, L+k)) .
\end{equation*}
By proposition \ref{lderprop} the paths $[0,k] \to \defstr$
\begin{gather*}
s \mapsto Y(\phi^{*}(\varphi_{+}, \varphi_{-}, L+k-s)) ,  \\
s \mapsto Y(\varphi_{+}, \varphi_{-}, L+k-s)
\end{gather*}
are both lifts
of $s \mapsto Y_{H}(\varphi_{+}, \varphi_{-}, L+k) - 2s\delta([\omega])$,
so in particular they have the same value at $s = 1$, which gives the result.
\end{proof}


\subsection{A coordinate chart}

Next we describe coordinate charts for $(\ycal{X} \times \bbr)/\ycal{D}$,
which contains $\mathcal{G}_{0}$ as an open subset. This mainly relies on the
pre-moduli spaces of torsion-free EAC \gtstr s that were constructed in
\S \ref{cylgtmodsec}.
The definition (\ref{calydacteq}) of the
action of $\ycal{D}$ on $\ycal{X} \times \bbr$ can be restricted to give
an action on $\ycal{X}$, and $(\ycal{X} \times \bbr)/\ycal{D}$
is obviously a principal $\bbr$-bundle over $\calb = \ycal{X}/\ycal{D}$.
It therefore suffices to find charts for $\calb$.

Let $\gical{M}$ be the moduli space of torsion-free EAC \gtstr s on $M_{\gi}$,
and $\mathcal{N}$ the moduli space of \cystr s on $X$. Let
$\ycal{M} \subseteq \onecal{M} \times \twocal{M}$ be the pairs of
diffeomorphism classes of EAC torsion-free \gtstr s whose boundary images in
$\mathcal{N}$ match. There is a natural projection
\begin{equation}
\label{glueprojeq}
\calb \to \ycal{M}, \;\: (\varphi_{+}, \varphi_{-})\ycal{D} \mapsto
(\varphi_{+}\onecal{D}, \varphi_{-}\twocal{D}) .
\end{equation}
If $b^{1}(M) = 0$ then we will see that this projection is a covering map.
We begin by finding charts for $\ycal{M}$.

\begin{prop}
\label{mymfdprop}
$\ycal{M}$ is a submanifold of $\onecal{M} \times \twocal{M}$.
\end{prop}

Each point in $\ycal{M}$ can be represented by
a matching pair of torsion-free \gtstr s $(\varphi_{+}, \varphi_{-})$,
asymptotic to a Calabi-Yau structure $(\Omega,\omega)$ on $X$.
Let $\gical{R}$ be the pre-moduli space of torsion-free EAC \gtstr s near
$\varphi_{\gi}$. % obtained in section \ref{cylgtdefsec}.

\begin{defn}
The pre-moduli space of matching pairs of torsion-free EAC \gtstr s near
$(\varphi_{+},\varphi_{-})$ is a neighbourhood $\ycal{R}$
of $(\varphi_{+},\varphi_{-})$ in
$\ycal{X} \cap (\onecal{R} \times \twocal{R})$.
\end{defn}

To use $\ycal{R}$ as a coordinate chart we first need to show that its
boundary values form a manifold.
Let $\mathcal{Q}$ be the pre-moduli space of \cystr s near $(\Omega, \omega)$.
%
By theorem \ref{modsubmersethm} the boundary map
\[ B_{\gi} : \gical{R} \to \mathcal{Q} \]
is a submersion onto its image, which is an open subset of a submanifold
$\mathcal{Q}_{\gi,A}$ defined by the equations $[\Omega'] \in A^{3}_{\gi},
[\omega'^{2}] \in A^{4}_{\gi}$. Thus the intersection
\[ \mathcal{Q}_{\diag,A} = \mathcal{Q}_{+,A} \cap \mathcal{Q}_{-,A} \]
is defined as a subset of $\mathcal{Q}$ by
$[\Omega'] \in A^{3}_{\diag}, [\omega'^{2}] \in A^{4}_{\diag}$.

\begin{lem}
\label{qdamfdlem}
$\mathcal{Q}_{\diag,A} \subseteq \mathcal{Q}$ is a submanifold.
\end{lem}

\begin{proof}
We can recycle the proof of proposition \ref{qasubmfdprop}.
\end{proof}

\begin{proof}[Proof of proposition \ref{mymfdprop}.]
%$\onecal{R} \times \twocal{R}$ is a coordinate chart for
%$\onecal{M} \times \twocal{M}$ near
%$(\varphi_{+}\onecal{D}, \varphi_{-}\twocal{D})$.
%Therefore it is enough to show that
%$\ycal{R}$ is a submanifold of $\onecal{R} \times \twocal{R}$.
%
Let $\xcal{D}$ be the group of diffeomorphisms of $X$ isotopic to the identity.
If $(\psi_{+}, \psi_{-}) \in \onecal{R} \times \twocal{R}$ then the boundary
values $B_{\gi}(\psi_{\gi})$ both lie in the pre-moduli space $\calq$.
Therefore proposition
\ref{autinvprop} implies that they are $\xcal{D}$-equivalent if and only if
they are equal.
Hence  $\ycal{R}$ is homeomorphic to a neighbourhood of
$(\varphi_{+}\onecal{D}, \varphi_{-}\twocal{D})$ in $\ycal{M}$,
and it suffices to prove that
$\ycal{R}$ is a submanifold of $\onecal{R} \times \twocal{R}$.

%It remains to show that $\ycal{R}$ is a manifold. 
By lemma \ref{qdamfdlem} the image of $\mathcal{Q}_{\diag,A}$ in
$\mathcal{Q}_{+,A} \times \mathcal{Q}_{-,A}$ under the diagonal map
is a submanifold.
$\ycal{R} \subseteq \onecal{R} \times \twocal{R}$ is the inverse image
of $\mathcal{Q}_{\diag,A} \subseteq \mathcal{Q}_{+,A} \times \mathcal{Q}_{-,A}$
under the submersion $B_{+} \times B_{-} : \onecal{R} \times \twocal{R} \to
\mathcal{Q}_{+,A} \times \mathcal{Q}_{-,A}$,
so it is a submanifold.
\end{proof}

\begin{rmk}
This argument shows also that for $(\varphi_{+}, \varphi_{-}) \in \ycal{X}$
the tangent spaces to the corresponding pre-moduli spaces $\calq_{d, A}$ and
$\ycal{R}$ are what one would naively expect. In particular
\begin{equation}
\label{typreeq}
T_{\varphi}\ycal{R} = \harm^{3}_{y, cyl} ,
\end{equation}
the subspace of matching pairs in $\harm^{3}_{+,cyl} \times \harm^{3}_{-,cyl}$.
\end{rmk}

Now consider the fibres of the projection (\ref{glueprojeq}).
Given $(\psi_{+}, \psi_{-}) \in \ycal{R}$ any point in the fibre above
$(\psi_{+}, \psi_{-})\ycal{D}$ is represented
by a matching pair $(\phi_{+}^{*}\psi_{+}, \phi_{-}^{*}\psi_{-})$ with
$\phi_{\gi} \in \gical{D}$. The pair $(\phi_{+}, \phi_{-})$ need not match,
and even if it does it need not lie in the identity component $\ycal{D}$
of the group of matching pairs of EAC diffeomorphisms.
However, if $(x,t) \mapsto (\Xi_{\gi}(x), t + h_{\gi})$ denotes the
the asymptotic limit of $\phi_{\gi}$ then
$\chi = \Xi_{-}^{-1} \circ \Xi_{+} \in \cali_{X}$,
where $\cali_{X} \subset \xcal{D}$ is the group of automorphisms of the
Calabi-Yau manifold $X$ that are isotopic to the identity.
Because $\phi_{+}$ and $\phi_{-}$ are isotopic to the identity there is
a path $h : \bbr \to \xcal{D}$ which
is $id$ for $t \leq -1$ and $\phi$ for $t \geq 0$ such that
$(\phi_{+}, \phi_{-}h) \in \ycal{D}$
(regarding $h$ as an EAC diffeomorphism of $M_{-}$).

\begin{defn}
\label{txcalidef}
Let $\txcali$ be the group of paths in $\xcal{D}$ from the identity to elements
of $\cali_{X}$ modulo homotopies relative to the endpoints.
\end{defn}

\begin{rmk}
\label{autinvrmk}
Proposition \ref{autinvprop} implies that $\cali_{X}$ and $\txcali$
are independent of of the Calabi-Yau structure in $\calq$.
So are $\cala^{1}_{\diag}$ and $\cala^{1}_{\gi}$, the harmonic representatives
of $A^{1}_{\diag}$ and $A^{1}_{\gi}$.
Similarly the automorphism group $\cali_{M_{\pm}}$ of the
EAC \gtmfd{} $M_{\pm}$ is independent of the choice of \gtstr{} in $\gical{R}$.
\end{rmk}

If $\tilde\chi \in \txcali$ then a representative for $\tilde\chi$ can
be considered as a diffeomorphism $h_{\gi} \in \gical{D}$.
This is determined by $\tilde\chi$ up to multiplication by
diffeomorphisms asymptotic to the identity.
For $(\psi_{+}, \psi_{-}) \in \ycal{R}$ let
\begin{equation}
\label{txcaliacteq}
\tilde\chi^{*}((\psi_{+}, \psi_{-})\ycal{D}) =
(\psi_{+}, h_{-}^{*}\psi_{-})\ycal{D} \in \calb.
\end{equation}
This is well-defined, independent of the choice of $h_{-}$.
Any point in the pre-image of $(\psi_{+}, \psi_{-})$ in $\calb$ can be written
in this form, so essentially the points of the fibre correspond to different
ways of identifying the cylinders of the \gtstr s $\psi_{+}$ and $\psi_{-}$.

The identity component of $\txcali$ is the universal cover of the identity
component of $\xcal{I}$. Proposition \ref{killingprop} implies that
$T_{id}\xcal{I}$ is the Abelian Lie algebra $(\harm^{1}_{X})^{\sharp}$.
That is therefore the universal cover of the identity
component of $\xcal{I}$, so we can identify the identity component
of $\txcali$ with $H^{1}(X)$.

For $(\psi_{+}, \psi_{-}) \in \ycal{R}$ let $S \subseteq \txcali$ be the
stabiliser of  $(\psi_{+}, \psi_{-})\ycal{D}$ under the
action (\ref{txcaliacteq}). Then
\begin{equation}
\label{loctriveq}
\txcali/S \times \ycal{R} \to \calb, \;\:
(\tilde\chi S, \psi_{+}, \psi_{-}) \mapsto
\tilde\chi^{*}((\psi_{+}, \psi_{-})\ycal{D})
\end{equation}
is a homeomorphism onto a neighbourhood.
$\tilde\chi \in S$ means that $\tilde\chi = \tilde\chi_{+} \tilde\chi_{-}$
for some $\tilde\chi_{\gi} \in \txcali$ such that the corresponding EAC
diffeomorphisms $h_{\gi} \in \gical{D}$ are equivalent to automorphisms of
$M_{\gi}$ modulo diffeomorphisms asymptotic to the identity.
The Lie algebra of $\cali_{M_{\gi}}$ corresponds
to the bounded harmonic $1$-forms $\harm^{1}_{\gi,0}$,
whose image under the boundary map $B_{\gi}$ is
$\cala^{1}_{\gi} \subseteq \harm^{1}_{X}$.
Therefore the identity component of $S$ is $A^{1}_{+} \oplus A^{1}_{-}$.
Each $A^{1}_{\gi} \subseteq H^{1}(X)$ is a half-dimensional subspace according
to proposition \ref{realsubprop}, while their intersection is
$A^{1}_{d} \cong H^{1}(M)$.
Thus the components of $\txcali/S $ are planes of dimension $b^{1}(M)$.

The most interesting case is when $M$ has holonomy exactly $G_{2}$. Then
$b^{1}(M) = 0$, so $\calb \to \ycal{M}$ is a covering map.
In general one can still show that transition function between the maps of
the form (\ref{loctriveq}) are smooth, and give $\calb$ the structure of
a smooth manifold.

Finally, let us write down a convenient coordinate chart for
$(\ycal{X} \times \bbr)/\ycal{D}$ based on (\ref{loctriveq}).
First, for $\tilde\chi \in \txcali$ we define
\begin{equation}
\tilde\chi^{*}((\psi_{+}, \psi_{-}, L)\ycal{D}) =
(\psi_{+}, h_{-}^{*}\psi_{-}, L)\ycal{D} \in (\ycal{X} \times \bbr)/\ycal{D}.
\end{equation}
like (\ref{txcaliacteq}). Note that if $\tilde\chi$
is a loop in $\xcal{D}$ based at the identity then the glued \gtstr s
$Y(\psi_{+}, \psi_{-}, L)$ and $Y(\tilde\chi^{*}(\psi_{+}, \psi_{-}, L))$
on $M$ are certainly diffeomorphic, but not necessarily by a diffeomorphism
isotopic to the identity.

Let $E^{1}_{\diag, \psi}$ be the orthogonal complement of
$A^{1}_{+} + A^{1}_{-}$ in $H^{1}(X)$ with respect to the Kähler
metric on $X$ defined by the boundary value of
$(\psi_{+}, \psi_{-}) \in \ycal{R}$.
By lemma \ref{mxinlem}, $j^{*} : H^{1}(M) \to H^{1}(X)$ is injective and
the image is precisely $A^{1}_{\diag}$.
It follows from proposition \ref{realsubprop} that the complex structure
$J_{\psi}$ maps $A^{1}_{\diag} \to E^{1}_{\diag,\psi}$
bijectively. Composition gives a homomorphism %isomorphism $\beta_{\psi}$ from
%$H^{1}(M)$ to the identity component of $\txcali/S$.
\begin{equation}
\label{cxisoeq}
\beta_{\psi} : H^{1}(M) \to \txcali/S ,
\end{equation}
mapping onto the identity component.
%Hence if $U \subset H^{1}(M)$ is a neighbourhood of $0$ and $I$ is an interval
%containing $L$ then
Hence
\begin{equation}
\label{gluecharteq}
\ycal{R} \times H^{1}(M) \times \bbr \to (\ycal{X} \times \bbr)/\ycal{D}, \;\:
(\psi_{+}, \psi_{-}, a, L) \mapsto 
\beta_{\psi}(a)^{*}(\psi_{+}, \psi_{-}, L)\ycal{D}
%\ycal{R} \times U \times I \to \mathcal{G}_{0}, \;\:
%(\psi_{+}, \psi_{-}, a, L+k) \mapsto 
%(\psi_{+}, (\exp \rho\beta_{\psi}^{\sharp}(a))^{*}\psi_{-}, L+k)\ycal{D}
\end{equation}
parametrises a neighbourhood of $(\varphi_{+}, \varphi_{-}, L)\ycal{D}$
in $(\ycal{X} \times \bbr)/\ycal{D}$.


\subsection{The derivative of the gluing map}

Since $\pi_{H} : \mathcal{M} \to H^{3}(M)$ is a local diffeomorphism
the local behaviour of the gluing map $Y : \mathcal{G}_{0} \to \mathcal{M}$
is determined by that of $Y_{H} = \pi_{H} \circ Y$. $Y_{H}$ is just
the gluing map for cohomology of definition \ref{yhdefn}, so can be defined
on all of $(\ycal{X} \times \bbr)/\ycal{D}$.
We will compute the derivative.

Let $(\varphi_{+}, \varphi_{-}) \in \ycal{X}$, and let $\ycal{R}$
be the pre-moduli space of matching pairs of torsion-free EAC \gtstr s
near it.
In the chart (\ref{gluecharteq}), $Y_{H}$ is affine linear in the $\bbr$ factor
by proposition \ref{lderprop}. The same is true for the $H^{1}(M)$ factor.

\begin{lem}
\label{autderlem}
Let $(\psi_{+}, \psi_{-}) \in \ycal{R}$ with common boundary $(\Omega,\omega)$,
and let $\delta$ be the boundary map in the Mayer-Vietoris sequence
\textup{(\ref{mayereq})}.
Let $\alpha \in \harm^{1}_{X}$, and let $\tilde\chi$ be the corresponding
element of the identity component of $\txcali$. Then
\begin{equation*}
Y_{H}(\tilde\chi^{*}(\psi_{+}, \psi_{-}, L))
= Y_{H}(\psi_{+}, \psi_{-}, L) - \delta([\alpha^{\sharp} \lrcorner \Omega]) .
\end{equation*}
\end{lem}

\begin{proof}
Let $\rho : \bbr \to [0,1]$ be a smooth function such that $\rho(t) = 0$ for
$t \leq -1$ and $\rho(t) = 1$ for $t \geq 0$.
Then $h : \bbr \to \xcal{D}, \; t \mapsto \exp (\rho(t)\alpha^{\sharp})$
is a representative for $\tilde\chi$. $h$ can be considered as
a diffeomorphism of $M_{-}$, and
\begin{equation*}
Y_{H}(\tilde\chi^{*}(\psi_{+}, \psi_{-}, L)) =
Y_{H}((\psi_{+}, h^{*}\psi_{-}, L)).
\end{equation*}
For $s \in [0,1]$
\[ \frac{d}{ds}Y_{H}\left(\varphi_{+},
(\exp(\rho s \alpha^{\sharp}))^{*} \varphi_{-}, L\right) = 
Y_{H}\left(0, d(\rho \alpha^{\sharp} \lrcorner
(\exp(\rho s \alpha^{\sharp}))^{*} \varphi_{-}), L \right) = 
-\delta \left( [\alpha^{\sharp} \lrcorner \Omega] \right) . \qedhere \]
\end{proof}

Recall that $E^{2}_{\diag} = *A^{4}_{\diag}$, the Hodge dual of the
image of $j^{*} : H^{4}(M) \to H^{4}(X)$.
Proposition \ref{aeisoprop} implies that $E^{2}_{\diag}$ splits by type:
\begin{equation*}
E^{2}_{\diag}  = E^{2}_{\diag,1} \oplus E^{2}_{\diag,6} \oplus E^{2}_{\diag,8} .
\end{equation*} 
Let $\pi_{E} : H^{2}(X) \to E^{2}_{\diag}$ denote the orthogonal projection.
%and $\pi_{8} : H^{2}(X) \to H^{2}_{8}(X)$ the type component projection.
In \S \ref{gluehodgesub} we described the self-adjoint endomorphism
$\pi_{E}(\partial_{+}^{-1}C_{+} + \partial_{-}^{-1}C_{-})$ of $E^{2}_{\diag}$.
By an argument similar to the proof of theorem \ref{gluehodgethm}
we show that the gluing map has invertible derivative at
$(\varphi_{+}, \varphi_{-}, L)\ycal{D} \in \calg_{0}$ unless $-2L$
is an eigenvalue of $\pi_{E}(\partial_{+}^{-1}C_{+} + \partial_{-}^{-1}C_{-})$
for some eigenvector in $E^{2}_{\diag,8}$.
In particular the derivative is invertible for large $L$.

\begin{prop}
\label{yhderprop}
Given $(\varphi_{+}, \varphi_{-}) \in \ycal{X}$ the derivative of
\[ Y_{H} : (\ycal{X} \times \bbr)/\ycal{D} \to H^{3}(M) \]
at $(\varphi_{+}, \varphi_{-}, L)\ycal{D}$ is bijective for all sufficiently
large $L$.
%, except when $-2L$ is an
%eigenvalue of the endomorphism
%\begin{equation}
%\label{saeq}
%\pi_{E}(\partial_{+}^{-1}C_{+} + \partial_{-}^{-1}C_{-}) :
%E^{2}_{\diag,8} \to E^{2}_{\diag,8} .
%\end{equation}
\end{prop}

\begin{proof}
Let
\begin{equation*}
Y'_{H} : \ycal{R} \times H^{1}(M) \times \bbr \to H^{3}(M)
\end{equation*}
be the representation of $Y_{H}$ in the coordinate chart (\ref{gluecharteq}).
%
By (\ref{typreeq}) the tangent space of $\ycal{R}$ at
$(\varphi_{+},\varphi_{-})$ is a space of matching pairs of
harmonic forms $\harm^{3}_{y,cyl}$.
It can be described as the subspace of $\harm^{3}_{y}$ consisting of
pairs $(\psi_{+}, \psi_{-})$ such that the corresponding cylindrical
form $B(\psi) = \sigma + dt \wedge \tau$ is in $\harm^{3}_{SU}$
(i.e. satisfies (\ref{leqs2})).

Now consider the derivative
\[ (DY'_{H})_{(\varphi_{+},\varphi_{-},0,L)} :
\harm^{3}_{y,cyl} \times H^{1}(M) \times \bbr \to H^{3}(M) . \]
As in theorem \ref{gluehodgethm} we consider the map
$(i^{*}_{+} \oplus i^{*}_{-}) : H^{3}(M) \to H^{3}(M_{+}) \oplus H^{3}(M_{-})$
in the Mayer-Vietoris sequence. To show that $DY'_{H}$ is an isomorphism it
suffices to show that $\im((i^{*}_{+} \oplus i^{*}_{-}) \circ DY'_{H}) =
\im(i^{*}_{+} \oplus i^{*}_{-})$, and that
$DY'_{H} : \ker((i^{*}_{+} \oplus i^{*}_{-}) \circ DY'_{H}) \to
\ker (i^{*}_{+} \oplus i^{*}_{-})$ is an isomorphism.

The restriction of $DY'_{H}$ to $\harm^{3}_{y,cyl} \times 0 \times 0$ is just
$Y_{H}$. %, and as before we note that $(i^{*}_{+} \oplus
%i^{*}_{-})Y_{H}(\psi_{+}, \psi_{-}) = ([\psi_{+}], [\psi_{-}])$.
We first show that
\[ (i^{*}_{+} \oplus i^{*}_{-}) \circ Y_{H} : \harm^{3}_{y,cyl}
\to \im(i^{*}_{+} \oplus i^{*}_{-}) \]
is surjective.
If $[\psi]$ is a class in $H^{3}(M)$, let
$\psi_{\gi}$ be the unique representative of $i_{\gi}^{*}[\psi]$
in $\harm^{3}_{\gi,cyl}$ such that $\pi_{8}B_{e}(\psi_{\gi}) = 0$
(cf. corollary \ref{hsuacor}).
Then $(\psi_{+}, \psi_{-})$ lies in $\harm^{3}_{y,cyl}$ and satisfies
$(i^{*}_{+} \oplus i^{*}_{-})Y_{H}(\psi_{+}, \psi_{-}) =
(i_{+}^{*}[\psi], i_{-}^{*}[\psi])$.

It remains to determine whether
\[ DY'_{H} : \ker ((i^{*}_{+} \oplus i^{*}_{-}) \circ DY'_{H}) \to
\ker (i^{*}_{+} \oplus i^{*}_{-}) \]
is an isomorphism.
$\delta : H^{2}(X) \to H^{3}(M)$ restricts to an isomorphism
$E^{2}_{\diag} \to \ker (i^{*}_{+} \oplus i^{*}_{-})$.
The derivative $DY'_{H}$ maps $\bbr$ isomorphically to
$\delta(E^{2}_{\diag,1})$ by proposition \ref{lderprop}.
Also $\alpha \mapsto *[\alpha \wedge \Omega]$ defines an isomorphism
$A^{1}_{d} \to E^{2}_{\diag,6}$ by proposition \ref{aeisoprop},
so lemma \ref{autderlem} implies that $DY'_{H}$ maps $H^{1}(M)$ isomorphically
to $\delta(E^{2}_{\diag,6})$.

The kernel of $(i^{*}_{+} \oplus i^{*}_{-}) \circ Y_{H}$
in $\harm^{3}_{y,cyl}$ is
$\harm^{3}_{y, E, cyl} = \harm^{3}_{y,cyl} \cap \harm^{3}_{y,E}$.
If we identify $\harm^{3}_{y,E}$ with $E^{2}_{\diag}$ by
$(\psi_{+}, \psi_{-}) \mapsto B(\psi)$ then
$\harm^{3}_{y,E,cyl}$ is identified with $E^{2}_{\diag,8}$
(cf. corollary \ref{hsuacor}).
By (\ref{yharmeeq}), $DY'_{H}$ restricted to
$\harm^{3}_{y,E,cyl} \times 0 \times 0$ is identified with
\[ E^{2}_{\diag,8} \to H^{3}(M), \;\: [\alpha] \mapsto
\delta\left(\pi_{E}(\partial_{+}^{-1}C_{+} +
\partial_{-}^{-1}C_{-})[\alpha] + 2L[\alpha]\right) . \]
Hence the derivative is bijective unless 
$\pi_{E}(\partial_{+}^{-1}C_{+} + \partial_{-}^{-1}C_{-})$
has a non-zero eigenvector in $E^{2}_{\diag,8}$ with eigenvalue $-2L$.
\end{proof}


\begin{rmk}
\label{pdcprmk}
It is possible to show that
$\pi_{E}(\partial_{+}^{-1}C_{+} + \partial_{-}^{-1}C_{-})$
maps $E^{2}_{\diag,8}$ to itself.
\end{rmk}

%\begin{proof}
%It suffices to show that $\partial_{\gi}^{-1}C_{\gi}$
%preserves the decomposition
%$E^{2}_{\gi} = E^{2}_{\gi,8} \oplus E^{2}_{\gi, 6\oplus1}$. If
%$\beta \in \harm^{2}_{X}$ is the harmonic representative of a class
%$[\beta] \in E^{2}_{\gi, 6\oplus1}$ then by arguments similar to lemma
%\ref{aemapslem} there
%is a $\chi \in \harm^{1}_{\gi,-}$ such that
%$\beta = B_{e}\left(d(\chi^{\sharp} \lrcorner \varphi_{\gi})\right)$.
%If $BD_{a}(\chi^{\sharp} \lrcorner \varphi_{\gi}) = \tbomatrix{\alpha}{\beta}$
%then clearly $\alpha$ also has type $6\oplus1$.
%Hence (cf. lemma \ref{pdlem}) $\partial_{\gi}^{-1}C_{\gi}$ preserves
%$E^{2}_{\gi, 6\oplus1}$ and its orthogonal complement $E^{2}_{\gi,8}$.
%\end{proof}

Lemma \ref{pdlem} implies that
$\pi_{E}(\partial_{+}^{-1}C_{+} + \partial_{-}^{-1}C_{-})$ has only
real eigenvalues. Therefore
\[ \{ (\varphi_{+}, \varphi_{-}, L) \in \ycal{X} \times \bbr :
-2L \textrm{ is an eigenvalue of }
\pi_{E}(\partial_{+}^{-1}C_{+} + \partial_{-}^{-1}C_{-}) \} \]
is locally a union of graphs of smooth functions $\ycal{X} \to \bbr$.
The image of this set in the quotient $(\ycal{X} \times \bbr)/\ycal{D}$
is locally a union of graphs of smooth sections over $\calb$.
In view of proposition \ref{yhderprop},
$Y_{H} : (\ycal{X} \times \bbr)/\ycal{D} \to H^{3}(M)$ is a local
diffeomorphism on the complement of this subset.
%
We therefore define
\[ G = \{ (\varphi_{+}, \varphi_{-}, L) \in G_{0} : % \ycal{X} \times \bbr :
-2L < \textrm{all eigenvalues of }
\pi_{E}(\partial_{+}^{-1}C_{+} + \partial_{-}^{-1}C_{-}) \} . \]
$G \subseteq G_{0}$ is the subset of gluing data for which the gluing
parameter is large enough to guarantee that the derivative of the gluing map is
surjective. The quotient $\mathcal{G} = G\ycal{D}/\ycal{D}$ is an open
subset of $\mathcal{G}_{0}$, and $Y : \mathcal{G} \to \mathcal{M}$ is
a local diffeomorphism.
This completes the proof of theorem \ref{maingluethm}.


\section{Boundary points of the moduli space}
\label{glueboundarysec}

Let the compact manifold $M^{7}$ be the gluing of two
EAC \gtmfd s $M_{\gi}^{7}$ as before and
\[ Y : \calg \to \defstr \]
the gluing map for \gtstr s.
The gluing space $\calg$ is a fibre bundle over $\calb$ with typical
fibre~$\bbrp$. It can be considered as the interior of a
topological manifold $\calgbar$ with boundary
$\calb$ `at infinity' by adding a limit point to each of the fibres.
Proposition \ref{yhderprop} means that each boundary point in $\calgbar$
has an open neighbourhood $\bar U$ such that $Y$ is a diffeomorphism on the
interior of $\bar U$ (i.e. $\bar U \cap \calg$).
Therefore we can use $Y : \bar U \to \defstr \cup \calb$
as coordinate charts for the disjoint union of $\defstr$ and $\calb$
as a topological manifold with boundary.
The only problem is that the induced topology on $\defstr \cup \calb$ is not
Hausdorff, because points of $\calb$ need not be separated by
open sets; when we attach $\calb$ as a boundary of $\defstr$ distinct points
of $\calb$ may become limits of the same path in $\defstr$. We then say that
they \emph{define the same boundary point} for $\defstr$.

To get around this problem we will instead attach a quotient of $\calb$
as a boundary of~$\defstr$. To do this we need to check that the relation
of defining the same boundary point is an equivalence relation on $\calb$,
and let $\hat\calb$ be the quotient. We then verify that the quotient map
$\calb \to \hat\calb$ is a covering map, so that $\hat\calb$ is a smooth
manifold.
Let $\defstrbar = \defstr \cup \hat\calb$, and use coordinate charts
\begin{equation}
\label{barcharteq}
Y : \bar U \to \defstrbar
\end{equation}
to give $\defstrbar$ the structure of a (Hausdorff)
topological manifold with boundary, and thus prove theorem \ref{gluecptfythm}.

As a first step we show that elements of $\calb$ that define the same boundary
point must have the same image in $\ycal{M}$.
If $b^{1}(M) = 0$ then $\calb \to \ycal{M}$ is a covering map, and we will
find that $\hat\calb$ is an intermediate cover. As usual, the case
$b^{1}(M) = 0$ is most interesting since it is a necessary condition for
the glued manifold $M$ to have holonomy exactly $G_{2}$.
The details of the proof are more complicated in general, so in the
$b^{1}(M) > 0$ case we will only point out what the correct statements are.
One still finds
that $\hat\calb$ is obtained by identifying some connected components
of the fibres of the submersion $\calb \to \ycal{M}$.
The upshot is that whether $b^{1}(M) = 0$ or not,
$\hat\calb$~is smooth manifold covered by $\calb$,
there is a natural submersion $\hat\calb \to \ycal{M}$,
the connected components of the fibres of the submersion look like those
for $\calb \to \ycal{M}$, and theorem \ref{gluecptfythm} holds.

%Let us also observe that if $(\varphi_{+}, \varphi_{-})$ and
%$(\psi_{+}, \psi_{-})$ define the same boundary point of $\defstr$
%then they have the same image in $\ycal{M}$. Thus (\ref{glueprojeq})
%descends to a submersion $\hat\calb \to \ycal{M}$.

\begin{prop}
\label{hatbfibreprop}
Suppose that $(\varphi_{+}, \varphi_{-}), (\psi_{+}, \psi_{-}) \in \ycal{X}$
define the same boundary point for $\defstr$.
Then $\psi_{\gi}$ is $\gical{D}$-equivalent to $\varphi_{\gi}$.
%suppose that the there is a sequence in $\defstr$
%which converges to both 
%$(\varphi_{+}, \varphi_{-})\ycal{D}$ and
%$(\psi_{+}, \psi_{-})\ycal{D}$ in $\defstr \cup \calb$.
\end{prop}

\begin{proof}
Pick some $R$ greater than the diameter of both of the compact pieces
$M_{\pm}(0)$. % (for both the metrics defined by $\varphi$ and $\psi$).
For any $L$ the set of points $x \in M(L)$ for which the complement of
the ball $B(x,2R)$ is connected is contained in the compact subset
$M_{+}(3R) \cup M_{-}(3R) \subset M(L)$.

Let $\ycal{R}$ be the pre-moduli space of matching pairs of torsion-free
\gtstr s near~$\varphi$. The hypothesis means that
there are sequences $L_{i}, L'_{i} > 0$, and
$(\varphi_{i,+}, \varphi_{i,-})$, $(\psi_{i,+}, \psi_{i,-}) \in \ycal{R}$
such that $L_{i}, L'_{i} \to \infty$, $\varphi_{i,\gi} \to \varphi_{\gi}$ and
$\psi_{i,\gi} \to \psi_{\gi}$, together with a sequence of
diffeomorphisms $\phi_{i} : M(L_{i}) \to M(L_{i}')$ which pull back
$Y(\psi_{i,+}, \psi_{i,-}, L_{i})$ to
$Y(\varphi_{i,+}, \varphi_{i,-}, L'_{i})$.

Pick a pair of points $x_{\pm} \in M_{\pm}(0)$. Then
$\phi_{i}(x_{\pm}) \in M_{\pm}(3R)$ for all $i$. ($\phi_{i}(x_{\pm})$ cannot
lie in $M_{\mp}(3R)$ because $\phi_{i}$ is isotopic to the identity.)
By passing to a subsequence we may assume that $\phi_{i}(x_{\pm})$ converges to
some $x'_{\pm} \in M_{\pm}(3R)$. Moreover, the derivatives
$D\phi_{i}|_{T_{x_{\pm}}M}$ can be regarded as maps defined on
$T_{x_{\pm}}M_{\pm}$, and we may assume that they converge to isometries
$A_{\pm} : (T_{x_{\pm}}M_{\pm}, g_{\varphi_{\pm}}) \to
(T_{x'_{\pm}}M_{\pm}, g_{\psi_{\pm}})$.
Now for any $v \in T_{x_{\pm}}M$
\[ \phi_{i}(\exp_{\varphi} v) \to \exp_{\psi} A_{\pm}v \]
as $i \to \infty$. We can therefore define diffeomorphisms $\phi_{\pm}$ of
$M_{\pm}$ by
$\phi_{\pm}(\exp_{\varphi} v) = \exp_{\psi} (A_{\pm}v)$
for all $v \in T_{x_{\pm}}M_{\pm}$. The convergence of
$\phi_{i} \to \phi_{\gi}$ is uniform on $M_{\gi}(L)$ for each $L$
(regarded as a compact subset of either
$M_{\gi}$ or $M(L')$ for all $L' > L$), and $\phi_{\pm}$ pulls back
$\psi_{\pm}$ to $\varphi_{\pm}$.
It is and is smooth and EAC by proposition \ref{eacisomprop}.
\end{proof}

We now make the simplifying assumption that $b^{1}(M) = 0$.

\begin{lem}
\label{conelem}
Suppose that $(\varphi_{+}, \varphi_{-}), (\psi_{+}, \psi_{-}) \in \ycal{X}$
define the same boundary point for~$\defstrbar$.
Then the path $Y(\psi_{+}, \psi_{-}, L)$ converges to
$(\varphi_{+}, \varphi_{-})\ycal{D}$ in $\defstr \cup \calb$ as $L \to \infty$.
\end{lem}

\begin{proof}
We only prove the lemma for $b^{1}(M) = 0$.

Proposition \ref{hatbfibreprop} implies that
$(\psi_{+}, \psi_{-}) = \tilde\chi^{*}(\varphi_{+}, \varphi_{-})$
for some $\tilde\chi \in \txcali$, in the sense of (\ref{txcaliacteq}).
In particular 
$i_{\gi}^{*}[\psi_{\gi}] = i_{\gi}^{*}[\varphi_{\gi}] \in H^{3}(M_{\gi})$,
and the paths
$Y_{H}(\psi_{+}, \psi_{-}, L)$ and $Y_{H}(\varphi_{+}, \varphi_{-}, L)$
both lie in the affine space $K = [\varphi] + \delta(H^{2}(X))$.
They are both affine rays with slope $2\delta([\omega])$.

Now consider the image in $H^{3}(M)$ of the interior $U$ of a small
neighbourhood of $(\varphi_{+}, \varphi_{-})\ycal{D}$ in $\calgbar$.
When $b^{1}(M) = 0$ the image contains an affine cone in $K$.
In particular it contains all but a finite part of the ray 
$Y_{H}(\psi_{+}, \psi_{-}, L)$, and from this we will deduce the result.

If $b^{1}(M) = 0$ then we may take $U$ to be represented by
$\ycal{R} \times (L_{1}, \infty)$, where $\ycal{R}$ is a pre-moduli space
of matching pairs of torsion-free EAC \gtstr s near 
$(\varphi_{+}, \varphi_{-})$ and $L_{1} \in \bbr$ is large.
The projection of the gluing to $H^{3}(M)$ can be written as
\begin{equation}
\label{coneeq}
Y_{H} : \ycal{R} \times (L_{1}, \infty) \to H^{3}(M), \;\:
(\varphi'_{+}, \varphi'_{-}, L) =
f(\varphi'_{+}, \varphi'_{-}) + 2L\delta([\omega']) ,
\end{equation}
where $i^{*}_{\gi}f(\varphi'_{+}, \varphi'_{-}) = [\varphi'_{\gi}]$,
and $\omega'$
is the Kähler form of the common boundary value of~$\varphi'_{\gi}$.
If we let
\[ \ycal{R}' = \{ (\varphi'_{+}, \varphi'_{-}) \in \ycal{R} :
i^{*}_{\gi}\varphi'_{\gi} = i^{*}_{\gi}\varphi_{\gi} \} \]
then the restriction of $Y_{H}$ maps $\ycal{R}' \times (L_{1}, \infty)$
locally diffeomorphically to $K$.
The computation of the derivative in proposition \ref{yhderprop}
shows that the map
\[ \ycal{R}' \to \delta(H^{2}(X)), \;
(\varphi'_{+}, \varphi'_{-}) \mapsto \delta([\omega']) \]
is an embedding, and that the tangent space to the image is a direct complement
to the radial line.
The image of $\ycal{R}' \times (L_{1}, \infty)$ under
$Y_{H}$ therefore contains an open affine cone in~$K$, and in particular
it contains the ray $Y_{H}(\psi_{+}, \psi_{-}, L)$.

$\tilde\chi^{*}U$ is the interior of a neighbourhood of
$(\psi_{+}, \psi_{-})\ycal{D}$ in $\calgbar$. By lemma \ref{autderlem},
$Y_{H}(\tilde\chi^{*}U)$ is a translate of $Y_{H}(U)$, so their intersection
also contains an open affine cone in $K$. In particular the intersection is
simply-connected. Therefore if $Y(U)$ and $Y_{H}(\tilde\chi^{*}U)$ intersect
in $\defstr$ then any pair of points in $Y(U)$ and $Y_{H}(\tilde\chi^{*}U)$
which represent the same cohomology class must be equal. 
Hence $Y(U)$ contains $Y(\psi_{+}, \psi_{-}, L)$ for all sufficiently
large $L$. As this holds for the interior of any small neighbourhood
of $(\varphi_{+}, \varphi_{-})\ycal{D}$ in $\calgbar$ the result follows.
\end{proof}

The lemma implies that the relation on $\calb$ of defining the same boundary
point for $\defstr$ is transitive. Let $\hat\calb$ be the quotient.
Proposition \ref{hatbfibreprop} implies that there is a natural projection
$\hat\calb \to \ycal{M}$. 
It is clear that the quotient map $\calb \to \hat\calb$ has discrete fibres,
because points in a coordinate neighbourhood of $\calb$ can always be separated
by open sets in $\defstr \cup \calb$.
Next we check (when $b^{1}(M) = 0$) that the quotient map is a covering map.

\begin{lem}
\label{hatbcoverlem}
Assume $b^{1}(M) = 0$.
Let $(\varphi_{+}, \varphi_{-}) \in \ycal{X}$, and let $\ycal{R}$ be the
pre-moduli space of matching pairs of torsion-free EAC \gtstr s near $\varphi$.
Suppose that $\varphi$ and $\tilde\chi^{*}\varphi$ define the same
boundary point for some $\tilde\chi \in \txcali$.
Then $\varphi'$ and $\tilde\chi^{*}\varphi'$ define the same boundary point
for each $\varphi' \in \ycal{R}$.
\end{lem}

\begin{proof}
The path $Y_{H}(\varphi',L)$ is a translation of
$Y_{H}(\tilde\chi^{*}\varphi',L)$ by an element in $\delta(H^{2}(X))$,
so the result follows by the argument of lemma \ref{conelem}.
\end{proof}

Thus $\hat\calb$ is an intermediate cover between $\calb$ and $\ycal{M}$.
In particular it is a manifold, so if we let
$\defstrbar = \defstr \cup \hat\calb$ then we can use the gluing maps
(\ref{barcharteq}) to give $\defstrbar$ the structure of a
topological manifold with boundary. This completes the proof of theorem
\ref{gluecptfythm} for the case when $b^{1}(M) = 0$.

\begin{rmk}
We could give $\calgbar$, and hence $\defstrbar$, a smooth structure
by choosing a homeo\-morphism of the fibre $(0, \infty]$ with
a half-open interval $[0, 1)$ that is a diffeomorphism on the interior,
but it is not clear if there is a natural choice.
\end{rmk}

Finally let us say something about the case $b^{1}(M) > 0$,
starting with a sketch proof of lemma \ref{conelem}.
$(\varphi_{+}, \varphi_{-}) \in \ycal{X}$ determines a splitting
$E^{2} = E^{2}_{1} \oplus E^{2}_{6} \oplus E^{2}_{8}$, which in turn gives
\begin{equation}
\label{deltaspliteq}
\delta(H^{2}(X)) =
\delta(E^{2}_{1}) \oplus \delta(E^{2}_{6}) \oplus \delta(E^{2}_{8}) .
\end{equation}
The image in $H^{3}(M)$ of the interior of a neighbourhood of 
$(\varphi_{+}, \varphi_{-})$ in $\calgbar$ contains a cone in $K$,
whose tangent space is $\delta(E^{2}_{1\oplus 8})$
at points of the ray $Y_{H}(\varphi_{+}, \varphi_{-}, L)$.
If $\psi = \tilde\chi^{*}\varphi$ determines the same boundary point as
$\varphi$ for some $\tilde\chi \in \txcali$ then necessarily
\begin{equation*}
%\label{raynecceq}
Y_{H}(\varphi, L) - Y_{H}(\psi, L) \in  \delta(E^{2}_{1\oplus 8}) .
\end{equation*}
We can then deduce that the image in $H^{3}(M)$ of the interior of any small
neighbourhood of $\varphi\ycal{D}$ in $\calgbar$ contains
$Y_{H}(\psi_{+}, \psi_{-}, L)$ for all sufficiently large $L$,
and proceed similarly to the $b^{1}(M) = 0$ case.

The quotient map $\hat\calb \to \calb$ has discrete fibres as before.
To check that it is a covering map one proves a more general version of
lemma \ref{hatbcoverlem}.
Let $(\varphi_{+}, \varphi_{-}) \in \ycal{X}$ and $\ycal{R}$ the
pre-moduli space of matching pairs of torsion-free EAC \gtstr s near $\varphi$,
and suppose that $\varphi$ and $\tilde\chi^{*}\varphi$ define the same
boundary point for some $\tilde\chi \in \txcali$.
What we can say when $b^{1}(M) > 0$ is that there is a smooth map
$\tilde\chi' : \ycal{R} \to \txcali$ such that
$\tilde\xi^{*}\varphi'$ and $(\tilde\xi\tilde\chi')^{*}\varphi'$ define
the same boundary point for each $\varphi' \in \ycal{R}$
and $\tilde\xi$ in the identity component of $\txcali$.
Thus $\hat\calb$ is smooth manifold covered by $\calb$, and theorem
\ref{gluecptfythm} holds.

%Thus $\hat\calb$ is obtained from $\calb$ by identifying some of the connected
%components of the fibres of $\calb \to \ycal{M}$.
%The upshot is that whether $b^{1}(M) = 0$ or not,
%$\hat\calb$ is smooth manifold covered by $\calb$,
%there is a natural submersion $\hat\calb \to \ycal{M}$,
%the connected components of the fibres of the submersion look like those
%for $\calb \to \ycal{M}$, and theorem \ref{gluecptfythm} holds.


\chapter{Pulling apart $G_{2}$-manifolds}
\label{pullchap}

In this chapter we attempt to reverse the gluing construction for
compact \gtmfd s from \cite{kovalev03} summarised
in \S \ref{gluesetupsec},
and show that some of the compact \gtmfd s produced
by Joyce's Kummer-type construction can be deformed to the result of
gluing a matching pair of EAC \gtmfd s.
A key technical result that is needed is a generalisation to the EAC
case of Joyce's theorems for perturbing \gtstr s with small torsion to
torsion-free \gtstr s.
We are able to cut some of Joyce's compact examples into two parts
such that we can find torsion-free EAC \gtstr s on each half,
and apply deformation results from \S \ref{gluechap} to show that the
gluing of the two halves can be deformed to the original manifold.

In the process we find some examples of irreducible EAC \gtmfd s.
We compute their Betti numbers and exhibit some examples of asymptotically
cylindrical coassociative submanifolds.


\section{Results}

In \S \ref{joycesub} we review how Joyce \cite{joyce00} constructs
examples of compact \gtmfd s $M$ by desingularising the quotient
of a flat torus $T^{7}$ by a finite group $\Gamma$ preserving the standard
flat \gtstr. The construction has two main steps. First, the singularities of
$T^{7}/\Gamma$ are resolved to obtain a smooth manifold $M$ with a family of
\gtstr s with closed defining $3$-form $\tv$. These have arbitrarily small
torsion in a suitable sense; one can estimate norms of a $3$-form $\psi$
such that $d\Theta(\tv) = d\!*\!\psi$.
When the torsion is sufficiently small $\tv$ can then be perturbed to a
torsion-free \gtstr{} $\varphi$ in the same cohomology class.

Suppose that the torsion-free \gtstr{} $\varphi$ on $M$ can be obtained by
perturbing a closed \gtstr{} $\tv$ with small torsion that has a cylindrical
neck, in the following sense.  There is a Calabi-Yau $3$-fold $X^{6}$ with
structure $(\Omega, \omega)$ and an interval $I = (-\epsilon,\epsilon)$ such
that $M$ has an open subset $N \cong X \times I$ with
\begin{equation}
\label{neckeq}
\tv|_{N} = \Omega + dt \wedge \omega ,
\end{equation}
such that the complement of $N$ in $M$ has exactly $2$ connected components.
Furthermore we require that the $3$-form $\psi$ controlling the torsion
vanishes on the neck.  Let $M_{\gi}(0)$ denote the closures of the connected
components of the complement in $M$ of the hyper\-surface $X \times \{0\}
\subset N$.

Let $M_{\gi}$ be the manifolds with cylindrical ends obtained by gluing
$X \times [0,\infty)$ to $M_{\gi}(0)$ along the common boundary $X$.
$\tv|_{M_{\gi}(0)}$ and $\Omega + dt \wedge \omega$ can obviously be glued
to define a closed \gtstr{} $\tv_{\gi}$ on $M$ which is exactly cylindrical
and torsion-free on the cylindrical end.
This has small torsion in the same sense as~$\tv$.
Joyce's perturbation result for \gtstr s (cf. theorem \ref{joyce2thm})
is carefully phrased so that it does not intrinsically rely on the underlying
manifold being compact (e.g. it does not involve any volume estimates).
It is therefore relatively straight-forward to generalise it to the EAC
setting.

\begin{thm}
\label{eacperturbthm}
Let  $\mu, \nu, \lambda$ positive constants.
Then there exist positive constants $\kappa, K$ such that whenever
$0 < t < \kappa$ the following is true.

Let $M^{7}$ be a manifold with cylindrical ends and cross-section $X^{6}$,
and let $\tv$ be a closed \gtstr{} on $M$ that is exactly cylindrical
and torsion-free on the cylindrical end.
Suppose that $\psi$ is a smooth compactly supported $3$-form on $M$ satisfying
$d^{*}\psi = d^{*}\tv$, and let $r(\tv)$ and $R(\tv)$ be the injectivity
radius and Riemannian curvature of the EAC metric $g_{\tv}$ on~$M$. If
\begin{enumerate}
\item \label{psiprioritem} $\norm{\psi}_{L^{2}} < \lambda t^{4}$,
$\norm{\psi}_{C^{0}} < \lambda t^{1/2}$,
$\norm{d^{*}\psi}_{L^{14}} < \lambda$,
\item $r(\tv) > \mu t$,
\item \label{curvestitem} $\norm{R(\tv)}_{C^{0}} < \nu t^{-2}$,
\end{enumerate}
then there is a smooth exact $3$-form $d\eta$ on $M$, exponentially decaying
with all derivatives, such that
\begin{equation}
\label{eacdetasmalleq}
\norm{d\eta}_{L^{2}} < K t^{4}, \;\:
\norm{d\eta}_{C^{0}} < K t^{1/2}, \;\:
\norm{\nabla d\eta}_{L^{14}} < K,
\end{equation}
and $\varphi = \tv + d\eta$ is a torsion-free \gtstr.
\end{thm}

\begin{rmk}
The fact that $d\eta$ is exponentially decaying is more important than
its precise rate of decay. We will need to choose the rate
$\delta > 0$ so that $\delta^{2}$ is smaller than any non-zero
eigenvalue of the Hodge Laplacian on $X$.
It should be easy to modify the proof of the theorem to allow $\tv$ to be EAC
and $\psi$ to be exponentially decaying. In that case one would also need
$\delta$ to be smaller than the decay rates of $\tv$ and~$\psi$.
\end{rmk}

Applying theorem \ref{eacperturbthm} to $\tv_{\gi}$ gives a matching pair
of torsion-free EAC \gtstr s $(\varphi_{+}, \varphi_{-})$ on $M_{\gi}$.
As discussed in \S \ref{gluechap}, these can be glued with an additional
sufficiently large parameter $L$ to give a torsion-free \gtstr{}
$Y(\varphi_{+}, \varphi_{-}, L)$ on $M$.
Varying the parameter $L$ defines
a path in the moduli space $\defstr$ whose image in $H^{3}(M)$ is an
affine line with slope $\delta([\omega])$. In the partial compactification
$\defstrbar$ defined in \S \ref{glueboundarysec} the path converges to
%the point $(\varphi_{+}, \varphi_{-})\ycal{D}$ in the boundary~$\calb$.
a point in the boundary $\hat\calb$.

Restricting the glued structures to $M_{\gi}$ defines the same cohomology class
as the original \gtstr{} $\varphi$, but it is hard to relate them directly
beyond that.
We can, however, define another path of \gtstr s on $M$ by `stretching'
$\varphi$. Let $M(L)$ denote the manifold obtained by increasing the length
of the cylindrical neck of $M$ by $2L$.
$\tv$ can be stretched to define a \gtstr{} $\tv(L)$ on
$M(L)$. This too has small torsion, so can be perturbed to a torsion-free
\gtstr{} $\varphi(L)$ by Joyce's results.
As $L \to \infty$ the path $\varphi(L)$ converges to a boundary point in the
partial compactification $\defstrbar$.
%When $M$ is an irreducible \gmfd{}
%the limit of $\varphi(L)$ as $L \to \infty$ corresponds to the matching pair
%$(\varphi_{+}, \varphi_{-})$.

\begin{thm}
\label{pullthm}
Let $M^{7}$ be a compact manifold with a closed \gtstr{} $\tv$.
Suppose that $\psi$ is a $3$-form such that $d^{*}\tv = d^{*}\psi$, that
the estimates \ref{psiestitem}-\ref{curvestitem} in theorem
\ref{eacperturbthm} are satisfied, and that $t$ is sufficiently small.
Suppose further that $M$ has a cylindrical neck in the sense of
\textup{(\ref{neckeq})}, and that $\psi$ vanishes on the neck.
Define the path $\varphi(L)$ as above.
Let $\varphi_{\gi}$ be torsion-free EAC \gtstr s on $M_{\gi}$ that are close
to $\tv_{\gi}$ in the sense of \textup{(\ref{eacdetasmalleq})}, which
exist by theorem \ref{eacperturbthm}.

Then $\varphi(L)$ is in the image of the gluing map for sufficiently large $L$.
In the sense of theorem \ref{gluecptfythm},
the path $L \mapsto \varphi(L)\cald$ converges to a point in the boundary
$\hat\calb$ of $\defstrbar$, whose image under the projection
$\hat\calb \to \ycal{M}$ is $(\varphi_{+}\onecal{D}, \varphi_{-}\twocal{D})$.
\end{thm}

\ignore{
In fact, we also prove that the path of representatives $\varphi(L)$ converges
`geometrically'. Let $M_{\gi}(L)$ denote
the gluing of $M_{\gi}(0)$ and $X \times [0,L]$, which can be regarded as
a compact subset of either $M_{\gi}$ or $M(L')$ for $L' > L$.

\begin{prop}
%\label{strprop}
$\varphi(L')|_{M_{\gi}(L)} \to \varphi_{\gi}|_{M_{\gi}(L)}$ uniformly
as $L' \to \infty$ with $L$ fixed.
\end{prop}

It is appealing that torsion-free EAC \gtstr s can be recovered in this way 
from compact \gtstr s. But note that the argument for convergence
is very indirect. It relies on first constructing the limits $\varphi_{\gi}$
by application of theorem \ref{eacperturbthm} and then applying
deformation theory for the matching pairs of torsion-free EAC \gtstr s near
$(\varphi_{+}, \varphi_{-})$ that was developed in chapter \ref{gluechap}.
}

We prove the results in \S \ref{pullmainsec}.
For them to be meaningful we also need examples to which they can be applied.
In \S \ref{pullexsec} we show how some of Joyce's examples can be obtained
by performing the resolution of orbifold singularities in two steps so that
the intermediate resolution has a cylindrical neck in the sense
of (\ref{neckeq}), and theorem \ref{pullthm} applies.
Some of the EAC \gtmfd s obtained this way have holonomy exactly $G_{2}$.
We explain how to compute their Betti numbers and find some examples of
asymptotically cylindrical coassociative submanifolds.

Joyce's construction actually gives a path of torsion-free \gtstr s, whose
limit can be thought of as a boundary point of the moduli space $\defstr$
of torsion-free \gtstr s, defined by the orbifold \gtstr{} the
construction started from.
When theorem \ref{pullthm} can be applied we therefore find that a component of
$\defstr$ has boundary points of two different types: both orbifold and
connected-sum.


\section{Main arguments}
\label{pullmainsec}


\subsection{Summary of Joyce's construction}
\label{joycesub}

We sketch the setup used by Joyce \cite[\S 11]{joyce00} to construct
examples of compact \gtmfd s.
Let $T^{7}$ be the quotient of $\bbr^{7}$ by a lattice, equipped with the
standard flat \gtstr{}, and let $\Gamma$ be a finite group of automorphisms
of $T^{7}$. The quotient $T^{7}/\Gamma$ is a $G_{2}$-orbifold.

$T^{7}/\Gamma$ has singularities coming from the fixed point sets of elements of
$\Gamma$. For each component of the fixed point sets we choose a topological
resolution of the singularity and a family of \gtstr s with small torsion on
the resolution. Since different parts of the fixed point set may intersect
or map to the same singularity in $T^{7}/\Gamma$ these choices have to
satisfy some compatibility conditions if they are to
be used to resolve the singularities of~$T^{7}/\Gamma$. Joyce calls a
suitable choice of resolutions a set of \emph{R-data}.
For simplicity we only sketch the definition of R-data and the resolution
process for the case when the fixed point sets do not intersect.

For concreteness we assume also
that the elements of $\Gamma$ are all of order $2$, since those are the only
examples we consider.
Then each component $F_{\alpha}$ of the fixed point set is a
flat torus $T^{3}$. A tubular neighbourhood of $F_{\alpha}$ is isomorphic to
a neighbourhood of $T^{3} \times \{0\}$ in $T^{3} \times \bbc^{2}$, equipped
with a flat product \gtstr. In suitable coordinates ($\theta^{i}$ on $T^{3}$,
$z^{j}$~on $\bbc^{2}$) the product structure is of the form
\[ d\theta^{1} \wedge d\theta^{2} \wedge d\theta^{3}
+ d\theta^{1} \wedge \omega_{0}
+ d\theta^{2} \wedge \beta_{0}
+ d\theta^{3} \wedge \beta_{0}' , \]
where $\omega_{0} = -{\textstyle\frac{i}{2}}
(dz^{1} \wedge d \bar z^{1} +dz^{2} \wedge d \bar z^{2})$
and $\beta_{0} + i\beta_{0}' =  dz^{1} \wedge dz^{2}$.

Let $\gamma_{\alpha}$ be the element of $\Gamma$ that fixes $F_{\alpha}$.
The action of $\gamma_{\alpha}$ on the tubular neighbourhood of $F_{\alpha}$
corresponds to multiplication by $-1$ on $\bbc^{2}$.
Choose a resolution $\pi_{\alpha} : W_{\alpha} \to \bbc^{2}/\{\pm 1\}$,
together with an asymptotically locally Euclidean (ALE) Calabi-Yau metric
with Kähler form $\omega_{\alpha}$. This means that the restriction
$\pi_{\alpha} : W_{\alpha} \setminus \pi_{\alpha}^{-1}(0) \to
(\bbc^{2} \setminus \{0\})/\{\pm 1\}$ is an isomorphism, and that the
difference between $\omega_{0}$ and the push-forward of $\omega_{\alpha}$
is of order $O(r^{-4})$, where $r$ is the distance to the origin in $\bbc^{2}$.
Then $W_{\alpha}$ has a parallel $(2,0)$-form $\beta + i \beta'$,
and
\[ \varphi_{\alpha} = d\theta^{1} \wedge d\theta^{2} \wedge d\theta^{3}
+ d\theta^{1} \wedge \omega_{\alpha}
+ d\theta^{2} \wedge \beta + d\theta^{3} \wedge \beta' \]
is a torsion-free \gtstr{} on $T^{3} \times W_{\alpha}$.
We require that there is an equivariance condition with respect to $\Gamma$:
if $\gamma \in \Gamma$ maps $F_{\alpha}$ to $F_{\beta}$ then there
is an identification $W_{\alpha} \leftrightarrow W_{\beta}$ such that
the corresponding map $T^{3} \times W_{\alpha} \to T^{3} \times W_{\beta}$
pulls back $\varphi_{\beta}$ to $\varphi_{\alpha}$. In particular the stabiliser
of $F_{\alpha}$ acts by automorphisms on $T^{3} \times W_{\alpha}$.
The R-data consists of choosing the resolutions $W_{\alpha}$ for each fixed
set component $F_{\alpha}$ and the isomorphisms
$W_{\alpha} \leftrightarrow W_{\beta}$ for each~$\gamma$.

Once the R-data is fixed, we obtain a compact resolution $M$ of $T^{7}/\Gamma$.
For each component of the singular set in $T^{7}/\Gamma$ we cut out a
neighbourhood, pick a pre-image $F_{\alpha}$ and glue in a neighbourhood of
$T^{3} \times W_{\alpha}$ divided by the stabiliser of $F_{\alpha}$
(the stabiliser acts freely on $T^{3} \times W_{\alpha}$ because of the
simplifying assumption that different pieces of the fixed point set do not
intersect).
We can use a (fixed) smooth cut-off function to interpolate between the flat
\gtstr{} $\varphi_{0}$ on $T^{7}$ and the torsion-free \gtstr s
$\varphi_{\alpha}$ on the resolving neighbourhoods,
and define a \gtstr{} $\tv$ on $M$
with $d\tv = 0$. The equivariance condition for the R-data with
respect to $\Gamma$ ensures that
both $M$ and $\tv$ are independent of the choice of pre-images $F_{\alpha}$.

Although both $\varphi_{0}$ and the
$\varphi_{\alpha}$ are torsion-free, $\tv$ has some torsion in the
region where the cut-off function used is non-constant.
The magnitude of the torsion depends on the derivatives of $\varphi_{\alpha}$
in the cutting-off region. This can be reduced by composing the resolutions
$W_{\alpha} \to \bbc^{2}/\{\pm 1\}$ with $\bbc^{2} \to \bbc^{2}, z \mapsto tz$
for small $t > 0$.
We then need to rescale the metric $\omega_{\alpha}$ by $t^{2}$ to keep it
ALE. This increases the Riemannian curvature of $\omega_{\alpha}$ by a
factor~$t^{-2}$. Using these resolutions instead will thus decrease the torsion
of $\tv$ at the cost of increasing the curvature of the metric it defines.

\hfuzz=2pt
\begin{thm}[{\cite[Theorem 11.5.7]{joyce00}}]
\label{joyce1thm}
Let $T^{7}/\Gamma$ be an orbifold of $T^{7}$ with flat \gtstr{} $\varphi_{0}$.
Suppose we are given a set of R-data for $T^{7}/\Gamma$, and $M$
is the corresponding resolution. Then we can write down the following data on
$M$ explicitly:
\begin{itemize}
\item constants $\epsilon \in (0,1]$, and $\lambda, \mu, \nu > 0$,
\item a \gtstr{} $\tv$ on $M$ with $d\tv = 0$ for each
$t \in (0,\epsilon]$,
\item a smooth $3$-form $\psi$ on $M$ with $d^{*}\psi = d^{*}\tv$ for each
$t \in (0,\epsilon]$.
\end{itemize}
\hfuzz=0.1pt
These satisfy the three conditions
\begin{enumerate}
\item
\label{psiestitem}
$\norm{\psi}_{L^{2}} < \lambda t^{4}$,
$\norm{\psi}_{C^{0}} < \lambda t^{3}$,
$\norm{d^{*}\psi}_{L^{14}} < \lambda t^{16/7}$,
\item the injectivity radius is $r(\tv) > \mu t$,
\item the Riemannian curvature $R(\tv)$ satisfies
$\norm{R}_{C^{0}} < \nu t^{-2}$.
\end{enumerate}
\end{thm}

These conditions ensure that for sufficiently small $t$ the \gtstr{} $\tv$
can be perturbed within its cohomology class to a torsion-free \gtstr{} on $M$.

\begin{thm}[cf. {\cite[Theorem 11.6.1]{joyce00}}]
\label{joyce2thm}
Let $\mu, \nu, \lambda$ positive constants.
Then there exist positive constants $\kappa, K$ such that whenever
$0 < t < \kappa$ the following is true.

Let $M^{7}$ be a compact manifold, and $\tv$ a closed \gtstr{} on $M$.
Suppose $\psi$ is a smooth $3$-form on $M$ satisfying
$d^{*}\psi = d^{*}\tv$, and
\begin{enumerate}
\item $\norm{\psi}_{L^{2}} < \lambda t^{4}$,
$\norm{\psi}_{C^{0}} < \lambda t^{1/2}$,
$\norm{d^{*}\psi}_{L^{14}} < \lambda$,
\item the injectivity radius is $> \mu t$,
\item the Riemannian curvature $R$ satisfies $\norm{R}_{C^{0}} < \nu t^{-2}$.
\end{enumerate}
Then there is a smooth exact $3$-form $d\eta$ on $M$
with
\begin{equation}
\label{detasmalleq}
\norm{d\eta}_{L^{2}} < K t^{4}, \;\:
\norm{d\eta}_{C^{0}} < K t^{1/2}, \;\:
\norm{\nabla d\eta}_{L^{14}} < K,
\end{equation}
such that $\varphi = \tv + d\eta$ is torsion-free.
\end{thm}

Thus we obtain a family of torsion-free \gtstr s on $M$. Note that as $t \to 0$
both $\tv$ and $\varphi$ resemble the singular \gtstr{} on $T^{7}/\Gamma$.
We can think of the family $\varphi$ as converging to a boundary
point of the moduli space of torsion-free \gtstr s on $M$, defined by
the singular \gtstr.

Also note that there may be many different choices of R-data for a single
orbifold $T^{7}/\Gamma$, which can give rise to many topologically
inequivalent resolutions $\pi : M \to T^{7}/\Gamma$ and many different
diffeomorphism classes for $M$.

We will often informally refer to a form satisfying estimates like
(\ref{detasmalleq}) as \emph{small}.
For future reference we point out that the uniform estimate in
(\ref{detasmalleq}) follows from the $L^{2}$ and $\sobf{1}$ estimates by
a version of Sobolev embedding.

\begin{thm}[{\cite[Theorem G1]{joyce00}}]
\label{g1thm}
Let $\mu$, $\nu$ and $t$ be positive constants, and suppose $M$ is a complete
Riemannian $7$-manifold, whose injectivity radius $\delta$ and
Riemannian curvature $R$ satisfy $\delta \geq \mu t$ and
$\norm{R}_{C^{0}} \leq \nu t^{-2}$. Then there exists $C > 0$ depending only
on $\mu$ and $\nu$, such that if
$\chi \in \sobf{1}(\Lambda^{3}) \cap L^{2}(\Lambda^{3})$ then
\begin{equation*}
\norm{\chi}_{C^{0}} \leq
C(t^{1/2}\norm{\nabla\chi}_{L^{14}} + t^{-7/2}\norm{\chi}_{L^{2}}) .
\end{equation*}
\end{thm}

\ignore{
The core of the proof of theorem \ref{joyce2thm} can be phrased as a
contraction-mapping argument. First Joyce proves by integrating by parts that
$d\Theta(\tv + d\eta) = 0$ if $d\eta$ is uniformly small and
\begin{equation}
%\label{suff1eq}
\triangle \eta = d^{*}\psi + d^{*}(f\psi) + *dF(d\eta) ,
\end{equation}
where $f\tv = \frac{7}{3} \pi_{1}d\eta$ and $F(d\eta)$ is the quadratic part
of $\Theta(\tv + d\eta)$. In other words, a sufficient condition for
$\tv + d\eta$ to be torsion-free is that $d\eta$ is a fixed point of the
map $d\eta \mapsto d\eta'$, where $\eta'$ is a solution of
$\triangle \eta' = $ RHS of (\ref{suff1eq}).
For $K > 0$ and $t$ sufficiently small (depending on $K$) this map is a
contraction on the ball $\norm{d\eta}_{t} < K$,
with respect to a norm
$\norm{d\eta}_{t} = \max\{C_{1}t^{-4}\norm{d\eta}_{L^{2}},
C_{2}t^{-1/2}\norm{d\eta}_{C^{0}}, C_{3}\norm{\nabla d\eta}_{L^{14}}\}$ for
appropriate constants $C_{i}$.
This is proved in \cite[Proposition 11.8.1]{joyce00},
though without using the word ``contraction''.
One therefore easily obtains a uniqueness result, not stated explicitly
in \cite{joyce00}.

\begin{prop}
%\label{contractprop}
Let $\mu, \nu, \lambda, K$ be positive constants.
Then there exists a positive constant $\kappa$ such that whenever
$0 < t < \kappa$ the following is true.

Suppose $M^{7}$ and $\tv$ satisfy the conditions of theorem \ref{joyce2thm}.
If there is a solution of \textup{(\ref{suff1eq})} such that $d\eta$ satisfies
the bounds \textup{(\ref{detasmalleq})} then it is unique.
\end{prop}
}

The torsion-free \gtstr{} $\varphi$ given by theorem \ref{joyce2thm} depends
continuously on the input $\tv$ and $\psi$. In a certain sense it represents
a unique diffeomorphism class of torsion-free \gtstr s near $\tv$.
In particular, the resolution $\varphi$ of $(\tv, \psi)$ is independent
of $\psi$ up to diffeomorphism, and two families of torsion-free \gtstr s
which are cohomologous and ``close'' must be diffeomorphic.

\begin{prop}
\label{unresprop}
For $i = 0,1$ let $\tv_{i}$ be a closed \gtstr{} and $\psi_{i}$ a smooth
$3$-form satisfying the hypotheses of theorem \ref{joyce2thm}. Suppose
moreover that the difference between $\tv_{0}$ and $\tv_{1}$ is exact,
and that it satisfies
\[ \norm{\tv_{0}-\tv_{1}}_{L^{2}} < \lambda t^{4}, \quad
\norm{\tv_{0}-\tv_{1}}_{C^{0}} < \lambda t^{1/2}, \quad
\norm{\tv_{0}-\tv_{1}}_{\sobf{1}} < \lambda  \]
(measured in the metric defined by $\tv_{0}$).
If $t$ is sufficiently small then the torsion-free \gtstr s $\varphi_{i}$
produced from $(\tv_{i}, \psi_{i})$ by theorem \ref{joyce2thm} are
diffeomorphic.
\end{prop}

\begin{proof}
Connect $\tv_{0}$ and $\tv_{1}$ by an affine line segment of closed \gtstr s
$\tv_{s}$, $s \in [0,1]$.
Let $\psi'_{s} = \tv_{s} + *_{s}(*_{0}\psi_{0} - \Theta(\tv_{0}))$,
where $*_{s}$ denotes the Hodge star of the metric defined by~$\tv_{s}$.
Then $(\tv_{s}, \psi'_{s})$ satisfies the
hypotheses of theorem \ref{joyce2thm}, so when $t$ is sufficiently small
it defines a torsion-free perturbation $\varphi'_{s}$ of $\tv_{s}$ for each
$s \in [0,1]$.

Then define a line segment $\psi''_{s}$ connecting $\psi_{1}$ to $\psi'_{1}$.
Theorem \ref{joyce2thm} gives a torsion-free perturbation $\varphi''_{s}$
of $\tv_{1}$ for each input $(\tv_{1}, \psi''_{s})$. The concatenation
of the paths $\varphi'_{s}$ and $\varphi''_{s}$ is a continuous path
connecting $\varphi_{0}$ and $\varphi_{1}$, staying in a single cohomology
class.
By theorem \ref{maincptthm} the image of the path in the moduli space of
torsion-free \gtstr s is constant, so the end-points are diffeomorphic.
\end{proof}


\subsection{Proof of theorem \ref{eacperturbthm}}
\label{eacperturbsub}

Theorem \ref{eacperturbthm} is an extension of theorem \ref{joyce2thm} to
the EAC setting. We wish to find an exact exponentially asymptotically decaying
$3$-form $d\eta$ such that $\tv + d\eta$ is torsion-free.
The proof consists of three parts. First we show that for $\tv + d\eta$ to
be torsion-free it suffices that $\eta$ is a solution of a certain elliptic
equation, the same that Joyce used in the proof of theorem \ref{joyce2thm}.
Then we find a solution for this equation by a contraction-mapping argument.
The details of this are complicated, but largely carry over from the
compact case as solved by Joyce.
Elliptic regularity shows that solutions are smooth and uniformly decaying.
In the next subsection we prove that the solutions decay exponentially,
which completes the proof.

\begin{prop}
\label{suffprop}
There is an absolute constant $\epsilon_{1}$ such that the following holds.
Let $M^{7}$ be an EAC manifold, $\tv$ a closed EAC \gtstr{} on $M$ and $\psi$
an exponentially decaying $3$-form such that
$\norm{\psi}_{C^{0}} < \epsilon_{1}$ and $d^{*}\psi = d^{*}\tv$.
Suppose that $\eta$ is $2$-form %exponentially
asymptotic to a translation-invariant harmonic form, and that
$\norm{d\eta}_{C^{0}} < \epsilon_{1}$. Suppose further that
\begin{equation}
\label{suff2eq}
\triangle \eta = d^{*}\psi + d^{*}(f\psi) + *dF(d\eta) ,
\end{equation}
where the function $f$ is defined by
$f\tv = \frac{7}{3} \pi_{1}(d\eta)$ and $F$ is
the quadratic part of the non-linear fibre-wise map
$\Theta : \Lambda^{3}_{G_{2}} \to \Lambda^{4}, \;
\varphi \mapsto *_{\varphi}\varphi$.
Then $\tv + d\eta$ is a torsion-free EAC \gtstr{} on~$M$.
\end{prop}

\begin{proof}
This is an asymptotically cylindrical version of
\cite[Theorem $10.3.7$]{joyce00}. The proof relies on integrating by parts.
It is easy to check that, in the asymptotically cylindrical setting,
the necessary integrals still converge provided that $\eta$ is bounded
and $d\eta$ decays, so we can still use
(\ref{suff2eq}) as a sufficient condition for the torsion to vanish.
\end{proof}

A key part in the proof of theorem \ref{joyce2thm} is the contraction-mapping
result \cite[Proposition 11.8.1]{joyce00}. We observe that this can easily be
adapted to the EAC case.

\begin{prop}
\label{estprop}
Let $(\Omega,\omega)$ be a \cystr{} on a compact manifold $X^{6}$
and $\mu, \nu, \lambda$ be positive constants.
Then there exist positive constants $\kappa, K, C_{1}$
such that whenever $0 < t < \kappa$ the following is true.

Let $M^{7}$ be a manifold with cylindrical ends and cross-section $X$,
and $\tv$ a closed EAC \gtstr{} on $M$ with asymptotic
limit $\Omega + dt \wedge \omega$. Suppose that $\psi$ is a smooth
exponentially decaying $3$-form on $M$
satisfying $d^{*}\psi = d^{*}\tv$, and that
\begin{enumerate}
\item \label{lambdaitem} $\norm{\psi}_{L^{2}} < \lambda t^{4}$,
$\norm{\psi}_{C^{0}} < \lambda t^{1/2}$,
$\norm{d^{*}\psi}_{L^{14}} < \lambda$,
\item the injectivity radius is $> \mu t$,
\item the Riemannian curvature $R$ satisfies $\norm{R}_{C^{0}} < \nu t^{-2}$.
\end{enumerate}
Then there is a sequence $\eta_{j}$ of smooth exponentially asymptotically
translation-invariant $2$-forms with $\eta_{0} = 0$ satisfying the equation
\begin{equation}
\label{inducteq}
\triangle \eta_{j} = d^{*}\psi + d^{*}(f_{j-1}\psi) + *dF(d\eta_{j-1}),
\end{equation}
where $f_{j}\tv = {\textstyle\frac{7}{3}}\pi_{1}(d\eta_{j})$ for each
$j > 0$. The solutions satisfy the inequalities
\begin{enumerate}
\item \label{l2item}
$\norm{d\eta_{j}}_{L^{2}} < 2\lambda t^{4}$,
\item \label{l14item}
$\norm{\nabla d\eta_{j}}_{L^{14}} < 4C_{1}\lambda$,
\item \label{c0item}
$\norm{d\eta_{j}}_{C^{0}} < K t^{1/2}$,
\item \label{l2bitem}
$\norm{d\eta_{j+1} - d\eta_{j}}_{L^{2}} < 2^{-j}\lambda t^{4}$,
\item \label{l14bitem}
$\norm{\nabla (d\eta_{j+1} - d\eta_{j})}_{L^{14}} < 4 \cdot 2^{-j}C_{1}\lambda$,
\item \label{c0bitem}
$\norm{d\eta_{j+1} - d\eta_{j}}_{C^{0}} < 2^{-j}K t^{1/2}$.
\end{enumerate}
\end{prop}

\begin{proof}
We prove that the sequence exists using the Hodge theory in \S \ref{cylsec}.
As before we let $\harm^{m}_{\infty}$ denote the space of
translation-invariant harmonic $m$-forms on the cylinder $X \times \bbr$ and
$\rho$ a cut-off function for the cylinder on $M$.
Take $\delta > 0$ smaller than the decay rates of $\tv$ and $\psi$
such that $\delta^{2}$ is smaller than any positive
eigenvalue of the Hodge Laplacian on~$X$.
Inductively, the RHS of (\ref{inducteq}) is $d^{*}$ of a $3$-form that decays
exponentially with rate $\delta$.
The EAC Hodge decomposition theorem \ref{hodgedecompthm}
implies that there is a unique coexact solution
$\eta_{j} \in \holdad{k}(\Lambda^{2}) \oplus \rho \harm^{2}_{\infty}$
for all $k \geq 2$.

The inequalities can be proved inductively,
using exactly the same argument as in \cite[Theorem 11.8.1]{joyce00}.
\ref{l2item} and \ref{l2bitem} are proved using an integration by parts
argument, and since each $d\eta_{j}$ decays exponentially this is still 
justified when $M$ has cylindrical ends.

\ref{l14item}, \ref{c0item}, \ref{l14bitem} and \ref{c0bitem} are proved using
interior estimates, which do not require compactness.
%\ref{l14item} and \ref{l14bitem} rely on the local estimate
%estimate \cite[Theorem G1]{joyce00}, which implies that
%\begin{equation}
%\label{interioreq}
%\norm{\nabla d\eta}_{L^{14}} < 
%C_{1}(\norm{d\eta}_{L^{14}} + t^{-4}\norm{d\eta}_{L^{2}}) ,
%\end{equation}
%where $C_{1}$ is a constant determined by $\lambda, \mu$ and $\nu$,
%and in particular independent of $t$.
\end{proof}

It follows that $d\eta_{j}$ is a Cauchy sequence and has a limit $\chi$ with
\begin{equation}
\label{chismalleq}
\norm{\chi}_{L^{2}} < K t^{4}, \;\:
\norm{\chi}_{C^{0}} < K t^{1/2}, \;\:
\norm{\nabla \chi}_{L^{14}} < K,
\end{equation}
for some $K > 0$. $\chi$ is closed, \ltwoorth{} to the decaying harmonic forms
$\harm^{3}_{+}$ and satisfies the equation
\begin{equation}
\label{chi1eq}
d^{*}\chi = d^{*}\psi + d^{*}(f\psi) + *dF(\chi) ,
\end{equation}
where $f\tv = {\textstyle\frac{7}{3}}\pi_{1}(\chi)$.
We do not know \emph{a priori} that $\chi$ is the exterior derivative of a
bounded form, so we cannot yet apply proposition \ref{suffprop} to show that
$\tv + \chi$ is torsion-free.  We first show by elliptic regularity that $\chi$
is smooth and uniformly decaying.

%What makes this proposition much
%easier then the preceding one is that we do not require the estimate of
%the $\sobfd{1}$-norm to be independent of $t$.

\begin{prop}
If $t$ is sufficiently small then $\chi \in \sobf{k}(\Lambda^{3})$ for all
$k \geq 0$.
\end{prop}

\begin{proof}
Since $F(\chi)$ depends only point-wise on $\chi$ and is of quadratic order
we can write
\begin{equation}
\label{expandfeq}
*dF(\chi) = P(\chi, \nabla \chi) + Q(\chi) ,
\end{equation}
where $P(u, v)$ is linear in $v$ and smooth of linear order in $u$, while
$Q(u)$ is smooth of quadratic order in $u$ for $u$ small. We can then rephrase
(\ref{chi1eq}) as stating that $\beta = \chi$ is a solution of

\begin{equation}
\label{chi2eq}
\begin{gathered}
d^{*}\beta - P(\chi, \beta) - d^{*}(f(\beta)\psi) = d^{*}\psi + Q(\chi) , \\
d \beta = 0 ,
\end{gathered}
\end{equation}
where $f(\beta)\tv = \frac{7}{3}\pi_{1}(\beta)$. The LHS is a linear
partial differential operator acting on $\beta$. Its symbol depends
on $\chi$ and $\psi$, but not on their derivatives.
By taking $t$ small we can ensure that $\chi$ and $\psi$ are both
small in the uniform norm (see (\ref{chismalleq}) and hypothesis
\ref{lambdaitem} in proposition \ref{estprop}) so that the equation is
elliptic.

Now suppose that $\chi$ has regularity $\sobf{k}$. Then so do the coefficients
and the RHS of (\ref{chi2eq}). Because
$\beta = \chi \in \sobf{1}(\Lambda^{3})$ is a solution of (\ref{chi2eq})
standard interior estimates
(a Sobolev version of theorem \ref{intestthm}) imply that
it must have regularity $\sobf{k+1}$ locally.
Moreover, because the metric is asymptotically cylindrical the local
bounds are actually uniform (cf. theorem \ref{cylregthm}), so in fact
$\chi$ is globally $\sobf{k+1}$. The result follows by induction on $k$.
\end{proof}

\begin{cor}
\label{uniformcor}
If $t$ is sufficiently small then $\chi$ decays uniformly with all derivatives.
\end{cor}

\begin{proof}
Because $M$ is EAC, standard Sobolev embedding results imply that
we can pick $r > 0$ such that $M$ is covered
by balls $B(x_{i}, r)$ with the following property:
\[ \norm{\chi|_{B(x_{i}, r)}}_{C^{k}} <
C \norm{\chi|_{B(x_{i}, 2r)}}_{L^{14}_{k+1}} , \]
where the constant $C > 0$ is independent of $x_{i} \in M$.
If we ensure that each point of $M$ is contained in no more than $N$ of the
balls $B(x_{i}, 2r)$ then
\[ \sum_{i} \norm{\chi|_{B(x_{i}, r)}}_{C^{k}}^{14} <
NC^{14} \norm{\chi}_{L^{14}_{k+1}}^{14} . \]
As the sum is convergent the terms tend to $0$, i.e. the $k$th derivatives
of $\chi$ decay uniformly.
\end{proof}

If we can show that $\chi$ decays exponentially then, because $\chi$ is
closed and \ltwoorth{} to the decaying harmonic forms $\harm^{3}_{+}$,
$\chi = d\eta$ for some exponentially asymptotically translation-invariant
$\eta$ by the Hodge decomposition theorem \ref{hodgedecompthm}.
Proposition \ref{suffprop} then implies that $\tv + \chi$ is torsion-free,
and the proof of theorem \ref{eacperturbthm} would be complete.
In the next subsection we prove the required exponential decay result.


\subsection{Exponential decay}

%To complete the proof of conjecture \ref{eacperturbconj} it remains to
%prove that the rate of decay of $\chi$ is exponential.
By hypothesis $\tv$ is exactly cylindrical on the cylindrical end $t > 0$
of $M$, and $\psi$ is supported in $t \leq 0$.
%This holds in the cases for which we wish to apply theorem \ref{eacperturbthm}.
Thus on the cylindrical end the equation (\ref{chi1eq}) for $\chi$ simplifies
to
\begin{equation}
\label{chi3eq}
d^{*}\chi = *dF(\chi) .
\end{equation}
On the cylindrical end $t > 0$ we can write
\begin{align*}
\chi &= \sigma + dt \wedge \tau , \\
F(\chi) &= \beta + dt \wedge \gamma ,
\end{align*}
where $\tau \in \Omega^{2}(X)$, $\sigma, \gamma \in \Omega^{3}(X)$ and
$\beta \in \Omega^{4}(X)$ are forms on the cross-section $X$ depending on the
parameter $t$. Let $\dsubx$ denote the exterior derivative on $X$.
Then the conditions $d\chi = 0$ and (\ref{chi3eq}) are
equivalent to
\begin{subequations}
\begin{align}
\label{cylchi1eq} \dsubx\sigma &= 0 , \\
\label{cylchi2eq} \contrat \sigma &= \dsubx\tau , \\
\label{cylchi3eq} \dsubx\!*\!\tau &= -\dsubx\beta , \\
\label{cylchi4eq} \contrat\!*\!\tau &=
- \dsubx\!*\!\sigma - \contrat \beta + \dsubx\gamma .
\end{align}
\end{subequations}
(\ref{cylchi2eq}) implies that $\sigma(t_{1}) - \sigma(t_{2})$ is exact for
any $t_{1}, t_{2} > 0$. Since the exact forms form a closed subspace of
the space of $3$-forms on $X$ (in the $L^{2}$ norm) and $\sigma \to 0$ as
$t \to \infty$ it follows that $\sigma$ is exact for all $t > 0$.
Similarly (\ref{cylchi4eq}) implies that $*\tau - \beta$ is exact for
all $t > 0$.
(The equations (\ref{cylchi1eq}) and (\ref{cylchi3eq}) are thus redundant.)
The path $(\sigma, \tau)$ is therefore constrained to lie in the space
\begin{equation*}
\calf = \{ (\sigma, \tau) \in \dsubx\sob{1}(\Lambda^{2}T^{*}X) \times
L^{2}(\Lambda^{2}T^{*}X) : *\tau - \beta \textrm{ is exact} \} .
\end{equation*}

\begin{rmk}
We have not assumed that $\chi$ is $L^{1}$ on $M$.
\end{rmk}

$\beta$ is a function of $\sigma$ and $\tau$, and it is of quadratic order.
The implicit function theorem applies to show that if we replace $\calf$ with
a small neighbourhood of $0$ then it is a Banach manifold with tangent space
\begin{equation*}
T_{0}\calf = B =  \dsubx\sob{1}(\Lambda^{2}T^{*}X) \times
\dcsubx\sob{1}(\Lambda^{3}T^{*}X) .
\end{equation*}

We can now interpret (\ref{cylchi2eq}) and (\ref{cylchi4eq}) as a flow
on $\calf$, or equivalently near the origin in $B$.
By the chain rule we can write $\contrat \beta$ as
\begin{equation*}
\contrat \beta = A_{2} \contrat\tau + A_{3} \contrat\sigma + \beta',
\end{equation*}
where $A_{m}$ is a linear map from $\Lambda^{m}T^{*}X$ to $\Lambda^{2}T^{*}X$,
determined point-wise by $\sigma$ and $\tau$ and of linear order, while
$\beta'$ is a $2$-form determined point-wise by $\sigma$ and $\tau$ and of
quadratic order. In particular, for large $t$ the norm of $A_{2}$ is small,
and (\ref{cylchi2eq}) and (\ref{cylchi4eq}) are equivalent to
\begin{equation}
\label{floweq}
\begin{aligned}
\contrat \sigma &= \dsubx\tau , \\
\contrat\tau  &= (id + A_{2})^{-1}
(\dcsubx \sigma - *A_{3}\dsubx\tau - *\beta' + *\dsubx\gamma) .
\end{aligned}
\end{equation}
The origin is a stationary point for the flow, and the linearisation of the
flow near the origin is given by the linear operator
$L = \tbtmatrix{\dcsubx}{0}{0}{\dsubx}$ on $B$.
Because $L$ is formally self-adjoint $B$ has an orthonormal basis of
eigenvectors. $L$ is injective on $B$, so $B$ can be written as
a direct sum of subspaces with positive and negative eigenvalues,
\[ B = B_{+} \oplus B_{-} . \]
$\{e^{\mp t L} : t \geq 0 \}$ defines a continuous semi-group of bounded
operators on $B_{\pm}$.
If we let $\mu$ denote the smallest absolute value of the eigenvalues of $L$
then $e^{t\mu}e^{\mp t L}$ is uniformly bounded on $B_{\pm}$ for $t \geq 0$, so
the origin is a hyperbolic fixed point.
By analogy with the finite-dimensional case we expect that any flow line
approaching the origin must do so at an exponential rate.

A similar problem of exponential convergence for an infinite-dimensional flow
is considered by Mrowka, Morgan and Ruberman \cite[Lemma 5.4.1]{mmr94}.
Their problem is more general in that the linearisation of their flow has
non-trivial kernel, so that they need to consider convergence to a `centre
manifold' rather than to a well-behaved isolated fixed point. As a simple
special case we can prove $L^{2}$ exponential decay for $\chi$.

\begin{prop}
Let $\delta > 0$ such that $\delta^{2}$ is smaller than any positive eigenvalue
of the Hodge Laplacian on $X$. Then $\chi$ is $L^{2}_{\delta}$.
\end{prop}

\begin{proof}
Identify $\calf$ with a neighbourhood of the origin in the tangent space $B$,
and let $x$ be the path in $B$ corresponding to $(\sigma, \tau)$ in $\calf$.
Then (\ref{floweq}) transforms to a differential equation for $x$,
\begin{equation*}
\frac{dx}{dt} = Lx + Q(x) ,
\end{equation*}
where $L$ is the linearisation of (\ref{floweq}) as above,
and $Q$ is the remaining quadratic part.
Let $x = x_{+} + x_{-}$ with $x_{\pm} \in B_{\pm}$. 
If we let $\mu$ denote the smallest absolute value of the eigenvalues of $L$
then
\begin{equation*}
\lnorm{Lx_{+}} \geq \mu \lnorm{x_{+}}, \quad
\lnorm{Lx_{-}} \leq -\mu \lnorm{x_{-}} .
\end{equation*}
Applying the chain rule to the quadratic part gives
\begin{equation*}
\lnorm{Q(x)} < O(\lnorm{x})\norm{x}_{\sob{1}} + O(\lnorm{x}^{2}).
\end{equation*}
By corollary \ref{uniformcor}, $x$ converges uniformly to $0$ with all
derivatives as $t \to \infty$. Therefore for any fixed $k > 0$ we can find
$t_{0}$ such that
\begin{equation*}
\lnorm{Q(x)} < k\lnorm{x}
\end{equation*}
for any $t > t_{0}$. As $\mu^{2}$ is an eigenvalue for the
Hodge Laplacian on $X$ we may fix $k$ so that $\mu - 2k > \delta$.

We thus obtain that for $t > t_{0}$
\begin{subequations}
\begin{gather}
\dddt \lnorm{x_{+}} \geq \phantom{-} \mu \lnorm{x_{+}} - k\lnorm{x}, \\
\label{xmeq}
\dddt \lnorm{x_{-}} \leq -\mu \lnorm{x_{-}} + k\lnorm{x} .
\end{gather}
\end{subequations}
In particular $\lnorm{x_{+}} - \lnorm{x_{-}}$ is an increasing function of $t$.
Because it converges to $0$ as $t \to \infty$,
\begin{equation*}
\lnorm{x_{+}} \leq \lnorm{x_{-}}
\end{equation*}
for all $t > t_{0}$. Substituting into (\ref{xmeq})
\begin{equation*}
\dddt \lnorm{x_{-}} \leq -\mu \lnorm{x_{-}} + 2k \lnorm{x_{-}} ,
%< \delta \lnorm{x_{-}},
\end{equation*}
so $\lnorm{x_{-}}$ is of order $e^{(-\mu+2k)t}$.
Hence so is $\lnorm{x}$, so $e^{\delta t}\chi$ is $L^{2}$-integrable on $M$.
\end{proof}

\begin{cor}
$\chi$ decays exponentially with rate $\delta$.
\end{cor}

\begin{proof}
We prove by induction that $\chi$ is $\sobd{k}$ for all $k \geq 0$.
Interior estimates for the elliptic operator $d + d^{*}$ on $M$ imply that
we can fix some $r > 0$ and cover the cylindrical part of $M$ with open balls
$U = B(x,r)$ such that
\begin{equation*}
\norm{\chi}_{\sob{k+1}(U)} <
C_{1}(\norm{d\chi}_{\sob{k}(U)} + \norm{d^{*}\chi}_{\sob{k}(U)}) +
C_{2}\norm{\chi}_{L^{2}(U)} .
\end{equation*}
The constants $C_{1}$ and $C_{2}$ depend on the local properties of the metric
and the volume of $U$.
Since $M$ is EAC we can take the constants to be independent of $U$.
Recall that, on the cylinder, $d\chi = 0$ and $d^{*}\chi = *dF(\chi)$.
In view of the chain rule expression (\ref{expandfeq}) there is a constant
$C_{3} > 0$ such that
\begin{equation*}
\norm{dF(\chi)}_{\sob{k}(U)} < C_{3} \norm{\chi}_{C^{k}(U)}
\left(\norm{\nabla\chi}_{\sob{k}(U)} + \norm{\chi}_{\sob{k}(U)}\right) .
\end{equation*}
As $\chi$ decays uniformly we can ensure that
$\norm{\chi}_{C^{k}(U)} < 1/2C_{1}C_{3}$ by taking $U$ to be sufficiently far
along the cylindrical end. Then
\begin{equation*}
\norm{\chi}_{\sob{k+1}(U)} < \norm{\chi}_{\sob{k}(U)} +
2C_{2} \norm{\chi}_{L^{2}(U)} .
\end{equation*}
Hence $\chi$ is $\sobd{k}$ for all $k \geq 0$.
\end{proof}

This completes the proof of theorem \ref{eacperturbthm}.

\begin{rmk}
\label{kickrmk}
We did not obtain any bounds on the weighted norms in terms of $t$.
Thus, while we have estimates for the non-weighted norms
of $d\eta$ and know that $d\eta$ decays exponentially, we have no idea how
far down the cylinder we need to go before the exponential decay kicks in.
\end{rmk}


\subsection{Proof of theorem \ref{pullthm}}
\label{pullproofsub}

Recall that the torsion-free \gtstr s $\varphi(L)$ are obtained by
perturbing the closed \gtstr s $\tv(L)$ with small torsion, which are
in turn defined by stretching the cylindrical neck $X \times I$ of $\tv$
by a length $2L$. The cohomology class
$[\varphi(L)] = [\varphi] + 2L\delta([\omega])$, where $\omega$ is the Kähler
form on $X$, so the image of the path $\varphi(L)$ in $H^{3}(M)$ is an
affine line with slope $2\delta([\omega])$.

We also defined torsion-free EAC \gtstr s $\varphi_{\gi}$ on $M_{\gi}$
by perturbing the \gtstr s $\tv_{\gi}$ obtained by cutting $\tv$ in half.
Gluing $\varphi_{+}$ and $\varphi_{-}$ defines a path
$Y(\varphi_{+}, \varphi_{-}, L)$ of torsion-free \gtstr s on $M$.
The restrictions
$i_{\gi}^{*}[Y(\varphi_{+}, \varphi_{-}, L)] = i_{\gi}^{*}[\varphi]$, so
the image of the path in $H^{3}(M)$ lies in
the affine space $K = [\varphi] + \delta(H^{2}(X))$.
It too is an affine line with slope $2\delta([\omega])$.

We first prove the result using the simplifying assumption that
$b^{1}(M) = 0$.  Let $\ycal{R}$ be the pre-moduli space 
of matching pairs of torsion-free \gtstr s near $(\varphi_{+}, \varphi_{-})$,
and
\[ \ycal{R}' = \{ (\psi_{+}, \psi_{-}) \in \ycal{R} :
i^{*}_{\gi}\psi_{\gi} = i^{*}_{\gi}\varphi_{\gi} \} . \]
As in the proof of lemma \ref{conelem}, for large $L_{1} \in \bbr$ the image of
$Y_{H} : \ycal{R}' \times (L_{1}, \infty) \to K$  contains an
open affine cone in $K$.

The difference between $[\varphi(L)]$ and $[Y(\varphi_{+}, \varphi_{-}, L)]$
is constant in $L$.
%Therefore there is some $L_{1}$ such that for any $L > L_{1}$ there is
Therefore for sufficiently large $L$ there is
an $L'(L)$ close to $L$ and a matching pair
$(\varphi_{+}(L), \varphi_{-}(L)) \in \ycal{R}'$
such that $\varphi(L)$ is cohomologous to
the glued structure $Y(\varphi_{+}(L), \varphi_{-}(L), L'(L))$.
In fact, because the RHS of (\ref{coneeq}) is dominated by the
$2L\delta([\omega])$ term for large $L$, the distance between
$(\varphi_{+}(L), \varphi_{-}(L))$ and $(\varphi_{+}, \varphi_{-})$ is of
order $1/L$, measured in the $C^{1}$ norm (since $\ycal{R}$ has finite
dimension all sensible norms are Lipschitz equivalent).
Hence the difference between $Y(\varphi_{+}, \varphi_{-}, L)$ and
$Y(\varphi_{+}(L), \varphi_{-}(L), L)$ is of order $1/L$ in $C^{0}$ norm.
As the volume is of order $L$ it follows also that the difference is of
order $L^{-1/2}$ in $L^{2}$-norm, and order $L^{-13/14}$ in $L^{14}_{1}$-norm.

Now $\varphi(L)$ and $Y(\varphi_{+}(L), \varphi_{-}(L), L'(L))$ are both
torsion-free perturbations of $\tv(L)$ within its cohomology class,
so we can try to use proposition
\ref{unresprop} to show that they are diffeomorphic.
For large $L$ the difference between $Y(\varphi_{+}(L), \varphi_{-}(L), L'(L))$
and $\tv(L)$ is dominated by the difference between $\tv_{\gi}$
and $\varphi_{\gi}$, which is estimated in terms of $t$
in (\ref{eacdetasmalleq}). Therefore if $t$ is sufficiently small
then for all sufficiently large $L$ the estimates required to
apply proposition \ref{unresprop} are satisfied, and
\[ Y(\varphi_{+}(L), \varphi_{-}(L), L'(L)) \cong \varphi(L) . \]
This proves theorem \ref{pullthm}.

\begin{rmk}
\label{geomconvrmk}
Looking more closely at the contraction-mapping argument in the proof
of theorem \ref{joyce2thm}, one can use
$Y(\varphi_{+}(L), \varphi_{-}(L), L'(L))$ as an ansatz for $\varphi(L)$ and
deduce that the distance between the two is in fact controlled by $L$ alone,
and not by $t$. Therefore the representatives $\varphi(L)$ converge in
`geometric' sense. Let $M_{\gi}(L)$ denote
the gluing of $M_{\gi}(0)$ and $X \times [0,L]$, which can be regarded as
a compact subset of either $M_{\gi}$ or $M(L')$ for $L' > L$. Then
$\varphi(L')|_{M_{\gi}(L)} \to \varphi_{\gi}|_{M_{\gi}(L)}$ uniformly
as $L' \to \infty$ with $L$ fixed.
This way torsion-free EAC \gtstr s can be recovered from compact \gtstr s.
%
%It is appealing that torsion-free EAC \gtstr s can be recovered in this way 
%from compact \gtstr s. But note that the argument for convergence
%is very indirect. It relies on first constructing the limits $\varphi_{\gi}$
%by application of theorem \ref{eacperturbthm} and then applying
%deformation theory for the matching pairs of torsion-free EAC \gtstr s near
%$(\varphi_{+}, \varphi_{-})$ that was developed in chapter \ref{gluechap}.
\end{rmk}

Some minor changes to the above argument are needed to deal with the case
$b^{1}(M) > 0$. Then (cf. (\ref{deltaspliteq})) $\delta(H^{2}(X))$ splits as
\begin{equation*}
\delta(H^{2}(X)) =
\delta(E^{2}_{1}) \oplus \delta(E^{2}_{6}) \oplus \delta(E^{2}_{8}) .
\end{equation*}
The affine space $K$ is modelled on $\delta(H^{2}(X))$ so we may consider
the quotient $K/\delta(E^{2}_{6})$.
The image of  
$Y_{H} : \ycal{R}' \times (L_{1}, \infty) \to K/\delta(E^{2}_{6})$ contains an
open cone.
Therefore for sufficiently large $L$ there is
an $L'(L)$ close to $L$ and a matching pair
$(\varphi_{+}(L), \varphi_{-}(L)) \in \ycal{R}'$
such that
\[ Y_{H}(\varphi_{+}(L), \varphi_{-}(L), L'(L)) - [\varphi(L)] \in
\delta(E^{2}_{6}) . \]
Lemma \ref{autderlem} implies that there is $\tilde\chi(L)$ in the identity
component of $\txcali$ such that
$\varphi(L)$ is cohomologous to
the glued structure $Y(\tilde\chi^{*}(\varphi_{+}(L), \varphi_{-}(L), L'(L)))$.
As $L \to \infty$, we have the same bounds as before
for the difference between $\tv(L)$ and
$Y(\varphi_{+}(L), \varphi_{-}(L), L)$.

Meanwhile, $\tilde\chi(L)$ converges as $L \to \infty$.
We can therefore choose the representative for 
$Y(\tilde\chi^{*}(\varphi_{+}(L), \varphi_{-}(L), L'(L)))$ so that it is
close to $Y(\varphi_{+}(L), \varphi_{-}(L), L'(L))$ in $C^{0}$,
$L^{2}$ and $L^{14}_{1}$-norm. For one is obtained from the other by `twisting'
the cylindrical part by a path of automorphisms of the cross-section. If
the twisting is stretched out to take place over a length $L$ of the cylinder
then the difference in $L^{1}$-norm is kept approximately constant, but the
$C^{0}$ norm is of order $1/L$, the $L^{2}$ norm of order $L^{-1/2}$,
and the $L^{14}_{1}$-norm of order $L^{-13/14}$. One can then complete
the proof as in the $b^{1}(M) = 0$ case.


\section{Examples}
\label{pullexsec}

We now show how theorem \ref{pullthm} can be applied to pull apart some of
Joyce's examples of compact \gtmfd s into pairs of EAC \gtmfd s.
In the process we find some examples of EAC manifolds with holonomy
exactly $G_{2}$. We compute the Betti numbers of the manifolds,
and find some examples of EAC coassociative submanifolds.

\subsection{A preliminary example}
\label{prelimsub}

We begin with an example which does not require any preparation
(but for which the glued \gtmfd{} is reducible).
Let $\gamma : T^{7} \to T^{7}$ be the involution acting by
\[ (x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}) \mapsto
(-x_{1}, x_{2}, -x_{3}, x_{4}, -x_{5}, x_{6}, -x_{7}) . \]
The singularities of $T^{7}/\gen{\gamma}$ can be resolved
with the classical Kummer construction to give $M = T^{3} \times K3$.
If we resolve the singularities using Joyce's results then we find that
the conditions of theorem \ref{pullthm} are satisfied, so that we can pull
apart $M$.

$\gamma$ preserves the standard flat \gtstr{} on $T^{7}$ and
theorem \ref{joyce1thm} provides a family of closed \gtstr s $\varphi$ with
small torsion on the resolution $M$.

Now let $I \subset (0, \half)$ be an open interval and $X$ the flat torus
$T^{6}$ with the standard \cystr.
$I \times X \subset T^{7}$ maps injectively to its image
$N \subseteq T^{7}/\gen{\gamma}$.
If $I$ is not too close to $0$ or $\half$ then $N$ is
not affected when the singularities are resolved, so can be regarded as
a subset of $M$.
\[ \varphi|_{N} = \Omega + dt \wedge \omega , \]
so $N \subset M$ can be taken as a cylindrical neck.
The family of torsion-free \gtstr s on $M$ together with the neck $N$
satisfies the hypotheses of theorem \ref{pullthm}
(with an appropriate choice of~$\lambda$).
The resulting EAC \gtmfd s are of the form $T^{3} \times S_{\pm}$,
where $S_{\pm}$ are EAC manifolds of holonomy $SU(2)$.

\begin{rmk}
\label{halfk3rmk}
$S_{+}$ and $S_{-}$ are equivalent as complex surfaces, and can be obtained as
the complement of an anti-canonical divisor of the blow-up of $\bbc P^{2}$ at
$9$ points. Since $S_{+}$ and $S_{-}$ can be glued to form a $K3$ we will
refer to them as \emph{half-$K3$ surfaces}.
\end{rmk}


\subsection{Preparing the examples}
\label{preparesub}

We describe how to cast some of Joyce's examples into a form to which theorem
\ref{pullthm} applies by resolving only some orbifold singularities in
a first step.
This idea is similar to an outline argument for how to construct irreducible
quasi-asymptotically locally Euclidean \gtmfd s in
Joyce \cite[p. 277]{joyce00}.

Assume that the flat torus from which the construction starts has 
the form $S^{1} \times T^{6}$, where $T^{6}$ is a torus with a flat \cystr.
Let $\Gamma$ be a finite group of $G_{2}$ automorphisms of
$S^{1} \times T^{6}$.
Choose R-data and form the resolution $M$ of
$(S^{1} \times T^{6})/\Gamma$. Theorem \ref{joyce1thm} provides a family
of closed \gtstr s $\tv$ with small torsion on $M$, which can be perturbed
to torsion-free \gtstr s. We would like to claim that these torsion-free
\gtstr s can also be obtained by perturbing a \gtstr{} with small torsion
that satisfies the hypotheses of theorem \ref{pullthm}.

Suppose that the elements of $\Gamma$ act by products of isometries of
$S^{1}$ and $T^{6}$ and let $\Gamma'$ be the subgroup acting trivially on
the $S^{1}$ factor. We assume that some elements of $\Gamma$ act by reflection
on~$S^{1}$ (as has to be the case if $b^{1}(M) = 0$).
If $\theta \in S^{1}$ is not a fixed point of any of the
reflections then $\{\theta\} \times T^{6}$ does not meet the
fixed point set of $\Gamma \setminus \Gamma'$, and its image divides
$(S^{1} \times T^{6})/\Gamma$ into exactly two connected components.
In order to make things work we assume furthermore that the fixed point set
of $\Gamma \setminus \Gamma'$ does not meet that of $\Gamma'$.

The R-data for $(S^{1} \times T^{6})/\Gamma$ can be restricted to give R-data
for $S^{1} \times T^{6}/\Gamma'$. Let $M'$ be the corresponding resolution.
Theorem \ref{joyce1thm} gives closed \gtstr s $\tv'$ with small torsion
and by theorem \ref{joyce2thm} they can be perturbed to torsion-free \gtstr s
$\varphi'$.
$M'$~is homeomorphic to $S^{1} \times X^{6}$ for a compact manifold $X$.

The lemma below can be thought of as a simple version of the Cheeger-Gromoll
line splitting theorem (cf. lemma \ref{riccoverlem})
and ensures that there is a \cystr{} on $X$ such that $M'$ is
isomorphic to $S^{1} \times X$ as a \gtmfd{}
(cf. Chan \cite[p. 15]{chan06}).

\begin{lem}
\label{chanlem}
Let $T^{m}$ be a torus and $X$ a compact manifold with $b^{1}(X) = 0$.
If $g$ is a Ricci-flat metric on $T^{n} \times X$ that is invariant under
translations of the torus factor then there is a function
$f : X \to \bbr^{n}$ such that the graph diffeomorphism
\[ T^{n} \times X \to T^{n} \times X, \;\: (t,x) \mapsto (t + f(x), x) \]
pulls $g$ back to a product metric.
\end{lem}

\begin{proof}[Sketch proof.]
Let $\contrath{1}, \ldots, \contrath{n}$ be the unit coordinate vector fields
on $T^{n}$, and set $\alpha_{i} = \contrath{i}^{\flat}$.
$\contrath{i}$ is a Killing vector field, so
$\alpha_{i}$ is harmonic by proposition \ref{killingprop}.
Since $b^{1}(X) = 0$ the closed forms $\alpha_{i}|_{X}$ are exact.
Define $f : X \to \bbr^{n}$ by picking $f_{i}$ such that
$\alpha_{i} = df_{i}$.
\end{proof}

The compatibility conditions for the R-data ensure that the quotient
group $\Psi = \Gamma/\Gamma'$ acts in a well-defined way on $M'$.
Moreover $\tv'$ is invariant under this action and hence $\varphi'$ is too.
We can use the R-data to resolve the singularities of
$M'/\Psi$ and topologically this gives~$M$.
If $I \subset S^{1}$ is an interval not containing any fixed points of
the reflections then $I \times X$ maps homeomorphically onto its image $N$ in
$M$ and is a candidate for a cylindrical neck.
We wish to to define a closed \gtstr{} %$\tv$
on $M$ with small torsion, whose restriction to $N$ is $\varphi'$.

The issue is that theorem \ref{joyce2thm} relies on the orbifold
singularities that are to be resolved with small torsion being modelled on a
quotient of the flat \gtstr. But $\varphi'$ need not be flat near the
fixed point set $F$ of $\Psi$.

Let $S$ be a tubular neighbourhood of $F$. 
Recall that $\varphi' = \tv' + d\eta'$ for some $d\eta'$ satisfying the
estimates (\ref{detasmalleq}).
The assumption that the fixed point sets of $\Gamma'$
and $\Gamma \setminus \Gamma'$ are disjoint ensures that $\tv'|_{S}$ is flat.
In order to `restore' the flatness near $F$ we wish to define an exact form
that is supported on $S$, equal to $d\eta'$ near $F$, and small in the same
sense that $d\eta'$ is small. To do this we use a version of the classical
Poincar\'e lemma.

\begin{lem}
\label{poincarelem}
Let $F$ be a compact Riemannian manifold and $I$ a bounded open interval.
For any $n \geq 0$, $k \geq 0$ and $p \geq 1$ there is a constant $C > 0$
such that for any exact $\sobp{k}$ $m$-form $d\eta$ on the Riemannian product
$X = F \times I^{n}$ there is an $(m-1)$-form
$\chi$ with $d\chi = d\eta$ and
\begin{equation}
\label{poinesteq}
\norm{\chi}_{\sobp{k+1}} < C \norm{d\eta}_{\sobp{k}} .
\end{equation}
\end{lem}

\begin{proof}
The proof is by induction on $n$. The result holds for $n = 0$ by usual
Hodge theory.
For the inductive step, we show that if a manifold $X$ satisfies the conclusion
of the theorem, then so does $X \times I$ (with the product metric).

For $s \in I$ let $X_{s}$ denote the hypersurface $X \times \{s\}$.
Let $t$ denote the coordinate on $I$, and write
\[ d\eta = \alpha + dt \wedge \beta , \]
with $\alpha$ and $\beta$ sections of the pull-back of $\Lambda^{*}T^{*}X$
to $X \times I$.
Write $\alpha(s)$, $\beta(s)$ for the corresponding forms on $X_{s}$.
%
%Let $\dsubx$ and $\nsubx$ denote the components of the exterior and covariant
%derivatives in the $X$ directions (so $d = dt \wedge \contrat + \dsubx$
%and $\nabla = dt \otimes \contrat + \nsubx$).
Fix $s_{0} \in I$ and let
\[ \chi_{1}(s) = \int_{s_{0}}^{s} \beta(t) dt . \]
Let $\nabla$ denote the covariant derivative on $X \times I$, and consider
$\chi_{1}$ as a form on $X \times I$. For any $0 \leq i \leq k$ and $s \in I$
\begin{multline*}
\norm{(\nabla^{i}\chi_{1})(s)}^{p}_{L^{p}(X_{s})} =
\int_{X} \left\Vert\int_{s_{0}}^{s} (\nabla^{i}\beta)(t) dt \right\Vert ^{p}
vol_{X} \\ \leq
V^{p-1} \int_{X} \int_{s_{0}}^{s} \norm{(\nabla^{i}\beta)(t) dt}^{p} dt \,vol_{X}
\leq
V^{p-1} \norm{\nabla^{i}\beta}^{p}_{L^{p}(X \times I)} ,
\end{multline*}
where $V$ is the length of $I$. Hence
\[ \norm{\nabla^{i}\chi_{1}}^{p}_{L^{p}(X \times I)} \leq
\int_{I} \norm{(\nabla^{i}\chi_{1})(s)}^{p}_{L^{p}(X_{s})} ds
\leq V^{p} \norm{\nabla^{i}\beta}^{p}_{L^{p}(X \times I)} , \]
and
\[ \norm{\chi_{1}}_{\sobp{k}(X \times I)} \leq
V \norm{d\eta}_{\sobp{k}(X \times I)} . \]
$d(\eta - \chi_{1})$ has no $dt$-component, so the $dt$-component of
$d^{2}(\eta - \chi_{1})$ is $\contrat d(\eta - \chi_{1}) = 0$.
Hence $d(\eta - \chi_{1})$ is the pull-back to $X \times I$ of an exact
form on $X$. By the inductive hypothesis there is a form $\chi_{2}$ such that
$d\chi_{2} = d(\eta - \chi_{1})$ and $\chi = \chi_{1} + \chi_{2}$ satisfies
(\ref{poinesteq}) for some $C$ independent of $d\eta$.
\end{proof}

Therefore there is a $2$-form $\chi$ on $S$ such that
\[ d\chi = d\eta'|_{S} \]
and $\chi$ satisfies estimates proportional to (\ref{detasmalleq})
(the uniform estimate comes from theorem \ref{g1thm}).
%with proportionality constant independent of $t$.
If $\rho$ is a cut-off function which is $1$ near $F$ and $0$ outside $S$ then
\begin{equation}
\label{drcsmalleq}
\norm{d(\rho\chi)}_{L^{2}} < K' t^{4}, \;\:
\norm{d(\rho\chi)}_{C^{0}} < K' t^{1/2}, \;\:
\norm{\nabla d(\rho\chi)}_{L^{14}} < K',
\end{equation}
with $K'$ independent of $t$.
$\tv' + d(\eta' -\rho\chi)$ is a family of closed \gtstr s %with small torsion,
which are flat near $F$. It is clear from the chain rule that torsion
is small, but we need to take care to choose the small form $\psi'$
such that $d\!*\!\psi' = d\Theta(\tv' + d(\eta' -\rho\chi))$ in such a way
that it vanishes not only on the cylindrical neck region, but also near $F$.
%
%To claim that $\tv' + d(\eta' -\rho\chi)$ has
%small torsion we could let
%$\psi' = *(\Theta(\tv' + d(\eta' -\rho\chi)) - \Theta(\tv' + d\eta'))$.
%However, for the next step to work we need to ensure also that $\psi'$ is $0$
%on the singular set $F$.
Because $F$ has dimension $3$ any closed $4$-form on
the tubular neighbourhood $S$ is exact. By lemma \ref{poincarelem} we can write
\begin{equation*}
(\Theta(\tv' + d\eta') - \Theta(\tv'))|_{S} = d\chi'
\end{equation*}
for some $3$-form $\chi'$ on $S$, so that %. By the same argument as above
$d(\rho\chi')$ satisfies estimates of the form (\ref{drcsmalleq}).
We can then take
\begin{equation*}
\psi' =
*(\Theta(\tv' + d(\eta' -\rho\chi))- \Theta(\tv' + d\eta') + d(\rho\chi')) ,
\end{equation*}
which is supported in $S$ but $0$ near $F$.

We can ensure that all forms are invariant under $\Psi$, so
$d(\eta' -\rho\chi)$ descends to an exact $3$-form $\beta$ on the orbifold
$M'/\Psi$.
As it is supported away from the singular set, $\beta$ is also defined on
the resolution $M$. In the same way $\psi'$ descends to a small $3$-form
$\psi_{1}$ on $M$.

Recall that we denoted by $\tv$ the \gtstr{} on $M$ with
small torsion obtained by resolving $(S^{1} \times T^{6})/\Gamma$.
$\tv' + d(\eta' -\rho\chi)$ descends to an orbifold \gtstr{} on $M'/\Psi$ with
small torsion. Its orbifold singularities are modelled on quotients
of the flat \gtstr{}, so the singularities can be resolved like in theorem
\ref{joyce1thm} to define a closed \gtstr{} on $M$.
This is precisely $\tv + \beta$. The torsion introduced by the resolution is
small, in the sense that there is a smooth $3$-form $\psi_{2}$ on $M$,
supported near the pre-image $F'$ of the singular set, such that
$d^{*}\psi_{2} = d^{*}\tv$ near $F'$ and $\psi_{2}$ satisfies the estimate
\ref{psiprioritem} in theorem \ref{joyce1thm}.
Now $(\tv + \beta, \psi_{1} + \psi_{2})$ satisfies
the hypotheses of theorem \ref{pullthm} (for $t$ sufficiently small).
%
Moreover, proposition \ref{unresprop} shows that the torsion-free \gtstr s
obtained by perturbing the `one-step' resolution $\tv$ and the `two-step'
resolution $\tv + \beta$ are diffeomorphic.


\subsection{A simple example}
\label{simplesub}

We now discuss the example of \cite[\S 12.2]{joyce00}.
This is a resolution of the quotient of $T^{7}$ by the group
$\Gamma \cong \bbz^{3}_{2}$ generated by
\begin{equation}
\label{simplegammaeq}
\begin{aligned}
\alpha &: (x_{1}, \ldots, x_{7}) \mapsto
(x_{1}, x_{2}, x_{3}, -x_{4}, -x_{5}, -x_{6}, -x_{7}) , \\
\beta &: (x_{1}, \ldots, x_{7}) \mapsto
(x_{1}, -x_{2}, -x_{3}, x_{4}, x_{5}, \half - x_{6}, -x_{7}) , \\
\gamma &: (x_{1}, \ldots, x_{7}) \mapsto
(-x_{1}, x_{2}, -x_{3}, x_{4}, \half-x_{5}, x_{6}, \half-x_{7}) .
\end{aligned}
\end{equation}
The fixed point set of each of $\alpha, \beta$ and $\gamma$ consists of $16$
copies of $T^{3}$ and these are all disjoint. $\alpha\beta$, $\beta\gamma$,
$\gamma\alpha$ and $\alpha\beta\gamma$ act freely on $T^{7}$.
Furthermore $\gen{\beta, \gamma}$ acts freely on the set of $16$ $3$-tori
fixed by $\alpha$, so they map to $4$ copies of $T^{3}$ in the singular set
of $T^{7}/\Gamma$. Similarly $\gen{\alpha,\gamma}$ and $\gen{\alpha,\beta}$
acts freely on the $16$ $3$-tori fixed by $\beta$ and $\gamma$, respectively.
Thus the singular set of $T^{7}/\Gamma$ consists of $12$ copies of $T^{3}$.

At each $T^{3}$ in the singular set the normal space is $\bbc^{2}/\{\pm 1\}$.
To form a set of R-data we need to choose a resolution of $\bbc^{2}/\{\pm 1\}$
for each component of the fixed point set. Topologically the essentially
unique resolution is the blow-up $Y$ of $\bbc^{2}/\{\pm 1\}$ at the origin.
This has a family of ALE $SU(2)$-metrics called Eguchi-Hansen metrics.
So for each of the $48$ copies of $T^{3}$ in the fixed point set we use
$Y$ with an Eguchi-Hansen metric as our resolution of $\bbc^{2}/\{\pm 1\}$.
The equivariance condition on the R-data then means that we must actually
choose the same Eguchi-Hansen metric for each quadruple of $3$-tori which map
to the same component of the singular set.

Once R-data have been chosen we can apply theorems \ref{joyce1thm} and
\ref{joyce2thm} to give torsion-free \gtstr s on a resolution $M^{7}$ of
$T^{7}/\Gamma$. This \gtmfd{} can be pulled apart in several different ways.
In order to say something about the topology of the resulting EAC manifolds
let us first recall the technique for computing the Betti numbers of the
resolutions from \cite[\S 12.1]{joyce00}.
We consider first the compact \gtmfd{} $M$.

The cohomology of $T^{7}/\Gamma$ is just the $\Gamma$-invariant part of
the cohomology of $T^{7}$, so $b^{2}(T^{7}/\Gamma) = 0$ while
$b^{3}(T^{7}/\Gamma) = 7$. For each of the $12$ copies of $T^{3}$ in the
singular set we cut out a tubular neighbourhood, which deformation retracts
to~$T^{3}$, and glue in a resolution $T^{3} \times Y$.
$Y$ is the blow-up of $\bbc^{2}/\{\pm 1\}$ at the singular point,
which is biholomorphic to $T^{*}\bbc P^{1}$.
$T^{3} \times Y$ therefore deformation retracts to $T^{3} \times S^{2}$.
Each of the operations increases the Betti numbers of $M$ by the difference
between the Betti numbers of $T^{3} \times Y$ and $T^{3}$.
This is justified using the long exact sequences for the cohomology of
$T^{7}/\Gamma$ relative to its singular set and $M$ relative to the resolving
neighbourhoods.
Hence
\begin{equation*}
\begin{aligned}
b^{2}(M) &= 12 \cdot 1 = 12, \\
b^{3}(M) &= 7 + 12 \cdot 3 = 43 .
\end{aligned}
\end{equation*}
To compute the Betti numbers of the EAC \gtmfd s we consider them as
resolutions of $(T^{6} \times \bbr)/\Gamma$.
In the cases where the holonomy is exactly $G_{2}$
we find that the Betti numbers of $(T^{6} \times \bbr)/\Gamma$ are $b^{2}= 0$,
$b^{3}= 4$, $b^{4}= 3$, $b^{5}= 0$.
There are singular sets not only of the form $T^{3}$, but also
$T^{2} \times \bbr$. Resolving these adds $1$, $2$ and $1$ to $b^{2}$,
$b^{3}$ and $b^{4}$, respectively.

\begin{enumerate}
\item
\label{ex1item}
To begin with, choose $\Gamma'$ to be the stabiliser of the $S^{1}$ factor
corresponding to the $x_{1}$ coordinate, i.e. $\Gamma' = \gen{\alpha,\beta}$.
The quotient $S^{1} \times T^{6}/\Gamma'$ is isomorphic to
$S^{1} \times X_{19}$, for a simply-connected Calabi-Yau $3$-fold $X_{19}$.
The fixed points of the reflections on $S^{1}$ are $0$ and $\half$,
so if we take $I \subset (0, \half)$ then the image $N$ in $M$ of
$I \times X_{19}$ can be used as a cylindrical neck for some
closed \gtstr{} $\tv$ with small torsion.
The manifolds $M_{\gi}$ with cylindrical ends that result from pulling apart
along $N$ are simply connected with a single end.
Therefore by theorem \ref{topcondthm} they have holonomy exactly~$G_{2}$.

The \gtmfd s $M_{+}$ and $M_{-}$ are isometric. Their boundaries are
identified by the anti-holomorphic involution of $X_{19}$ induced by the
action of $\gamma$ on $T^{6}$.

The singular set in $(\bbr \times T^{6})/\Gamma$ consists of
$8$ copies of $T^{2} \times \bbr$ and $2$ copies of $T^{3}$.
Therefore
\begin{equation*}
\begin{aligned}
b^{2}(M_{\gi}) &=  8 \cdot 1 + 2 \cdot 1 = 10 , \\
b^{3}(M_{\gi}) &= 4 + 8 \cdot 2 + 2 \cdot 3 = 26 , \\
b^{4}(M_{\gi}) &= 3 + 8 \cdot 1 + 2 \cdot 3 = 17 , \\
b^{5}(M_{\gi}) &= 2 \cdot 1 = 2 .
\end{aligned}
\end{equation*}
We can also compute the Betti numbers of the cross-section $X_{19}$, and
find that $b^{2}(X_{19}) = 19$, $b^{3}(X_{19}) = 40$. Therefore its Hodge
numbers are
\[ h^{1,1}(X_{19}) = h^{1,2}(X_{19}) = 19 . \]

If instead we pull $M$ apart in the $x_{2}$ or $x_{4}$ direction
we get essentially the same result. We just need to use
$\gen{\gamma, \alpha}$ or $\gen{\beta, \gamma}$ as $\Gamma'$ to define
the intermediate resolution.

\item
\label{ex2item}
If we pull apart along the $x_{3}$ direction we get a slightly
different result. In this case $\Gamma' = \gen{\alpha, \beta\gamma}$,
which only contains one element with fixed points.
The cross-section of the neck is a resolution $X_{11}$ of
$T^{6}/\Gamma'$. The first Betti number $b^{1}(X_{11})$ vanishes,
but $X_{11}$ is doubly covered by $T^{2} \times K3$,
so $Hol(X_{11}) = \bbz_{2} \ltimes SU(2)$.
The EAC \gtmfd s $M_{\gi}$ are however simply-connected without cylindrical
double covers.
They are thus examples of irreducible EAC \gtmfd s with locally reducible
cross-section.

In this case the singular set in each half is $4$ copies of $T^{3}$ and $4$
copies of $T^{2} \times \bbr$. The Betti numbers are therefore
\begin{equation*}
\begin{aligned}
b^{2}(M_{\gi}) &= 4 \cdot 1 + 4 \cdot 1 = 8 , \\
b^{3}(M_{\gi}) &= 4 + 4 \cdot 2 + 4 \cdot 3 = 24 , \\
b^{4}(M_{\gi}) &= 3 + 4 \cdot 1 + 4 \cdot 3 = 19 , \\
b^{5}(M_{\gi}) &= 4 \cdot 1 = 4 .
\end{aligned}
\end{equation*}
The Hodge numbers of $X_{11} = (T^{2} \times K3)/\bbz_{2}$ are
\[ h^{1,1}(X_{11}) = h^{1,2}(X_{11}) = 11 . \]

\item
\label{ex3item}
If we pull apart along the $x_{5}$ or $x_{6}$ direction then
$\Gamma'$ is $\gen{\beta}$ or $\gen{\gamma}$ respectively. The
resulting EAC \gtmfd s are of the form $S^{1} \times M'_{\pm}$, where $M'_{\pm}$
is a simply-connected EAC Calabi-Yau $3$-fold and the cross-section is
$T^{2} \times K3$. This case therefore looks similar to the twisted connected
sums of reducible EAC \gtmfd s used by Kovalev in \cite{kovalev03}.
However, this example cannot arise from Kovalev's method for generating
matching pairs of reducible EAC \gtmfd s, since the connected sum of such
pairs always has $b^{2} \leq 9$.

\item
\label{ex4item}
Finally, if we pull apart along the $x_{7}$ direction then
$\Gamma' = \gen{\alpha\beta}$. This has no fixed points, so the cross-section
$X$ of the neck is flat. It has $b^{1}(X) = 2$.
Like in \S \ref{prelimsub} there is no need for an intermediate
resolution in this case.
Of the two EAC \gtmfd s one is a product of $S^{1}$ with a simply-connected
EAC Calabi-Yau $3$-fold, and the other is a quotient of the product
of $T^{3}$ with a half-$K3$ surface, as described in remark \ref{halfk3rmk}.
\end{enumerate}

\subsection{EAC coassociative submanifolds}
\label{coasssub}

Let us also give some examples of EAC coassociative submanifolds of EAC
manifolds with holonomy exactly $G_{2}$. The deformation problem for
such submanifolds has been consider by Joyce and Salur \cite{joyce05}.

Coassociative submanifolds are instances of calibrated
submanifolds as introduced by Harvey and Lawson \cite{harvey82}.
They call a closed $m$-form $\alpha$ on a Riemannian manifold a calibration if
$\alpha|_{V} \leq vol_{V}$ for any oriented $m$-dimensional tangent space $V$.
An oriented $m$-dimensional
submanifold $C$ is said to be calibrated by $\alpha$ if $\alpha|_{C} = vol_{C}$.
It is elementary to deduce from the definition that any calibrated submanifold
is volume-minimising in its homology class.
If $M^{7}$ is a \gtmfd{} with \gtstr{} $\varphi$ then $*\varphi$ is a
calibration.

\begin{defn}
A $4$-dimensional submanifold $C \subset M$ is called \emph{coassociative}
if it is calibrated by $*\varphi$.
\end{defn}

\noindent
The next two propositions are elementary to prove by working in the point-wise
model for the \gtstr.

\begin{prop}[{\cite[Lemma 10.8.2]{joyce00}}]
A $4$-dimensional submanifold $C \subset M$ is orientable and coassociative
(with respect to one of its orientations) if and only if $\varphi|_{C} = 0$.
\end{prop}

\begin{prop}[{\cite[Proposition 10.8.5]{joyce00}}]
\label{coassprop}
Let $\sigma : M \to M$ be an involution such that $\sigma^{*}\varphi = \varphi$.
Then each connected component of the fixed point set of $\sigma$ is either
a coassociative $4$-manifold or a single point.
\end{prop}

Let $M^{7}$ be the compact \gtmfd{} constructed in \S \ref{simplesub},
and denote its torsion-free \gtstr{} by $\varphi$.
Joyce \cite[\S 12.6]{joyce00} applies proposition \ref{coassprop} to find
examples of coassociative submanifolds of $M$.
We consider just the two simplest examples, where the
coassociative $4$-manifolds are diffeomorphic to $T^{4}$ or $K3$.

\begin{ex}
\label{coass1ex}
Define an orientation-reversing isometry of $T^{7}$
as in \cite[Example 12.6.4]{joyce00}.
\begin{equation}
\label{sigmaeq}
\sigma : (x_{1}, \ldots, x_{7}) \mapsto
(\half-x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, \half-x_{6}, \half-x_{7}) .
\end{equation}
Then $\sigma$ commutes with the action of $\Gamma$ defined
by (\ref{simplegammaeq})
and pulls back $\varphi_{0}$ to $-\varphi_{0}$. When the singularities
of $T^{7}/\Gamma$ are resolved to form the compact \gtmfd{} $M$ one can ensure
that $\sigma$ lifts to an involution of $M$ such that
$\sigma^{*}\varphi = -\varphi$. It is easy to see that the fixed point set of
$\sigma$ in $M$ consists of 16 isolated points and one copy of $T^{4}$,
which is a coassociative submanifold of $M$.

We can also consider (\ref{sigmaeq}) to define an involution $\sigma$ of
$T^{6} \times \bbr$. Provided that the $\bbr$ factor corresponds to the
$x_{2}$, $x_{3}$ or $x_{4}$ coordinate this again commutes with the
action of~$\Gamma$. When we pull apart $M$ in
the $x_{2}$, $x_{3}$ or $x_{4}$ direction the resulting irreducible
EAC \gtmfd s $M_{\pm}$
are resolutions of $(T^{6} \times \bbr)/\Gamma$, so $\sigma$ lifts to an
involution of $M_{\gi}$ that reverses the torsion-free \gtstr.
The fixed point set in each half $M_{\gi}$ consists of $8$ isolated points
and one copy of $T^{3} \times \bbr$, which is an asymptotically cylindrical
coassociative submanifold of $M_{\gi}$.
\end{ex}

\begin{ex}
\label{coass2ex}
Define an orientation-reversing isometry of $T^{7}$
as in \cite[Example 12.6.4]{joyce00}.
\begin{equation*}
%\label{sigmaeq}
\sigma : (x_{1}, \ldots, x_{7}) \mapsto
(\half-x_{1}, \half-x_{2}, \half-x_{3}, x_{4}, x_{5}, x_{6}, x_{7}) .
\end{equation*}
Again, $\sigma$ lifts to an involution of $M$
such that $\sigma^{*}\varphi = -\varphi$ and whose fixed point set
in $T^{7}/\Gamma$ consists of 16 isolated points and two copies of
$T^{4}/\{\pm1\}$. The corresponding coassociative submanifolds in $M$
are two copies of the usual Kummer resolution of $T^{4}/\{\pm1\}$, which is
a $K3$ surface.

If we pull apart $M$ in the $x_{4}$ direction then $\sigma$ defines also
involutions of the resulting irreducible EAC \gtmfd s $M_{\pm}$.
In each half the fixed point set has two $4$-dimensional components,
which are resolutions of $(T^{3} \times \bbr)/\{\pm1\}$. These are
asymptotically cylindrical coassociative submanifolds of $M$, diffeomorphic
to the half-$K3$ from remark \ref{halfk3rmk}.
\end{ex}



\begin{rmk}
Compact coassociative submanifolds have a well-behaved deformation theory.
For any coassociative submanifold $C \subset M$ the normal bundle of $C$ is
isomorphic to $\Lambda^{2}_{+}T^{*}C$.
McLean \cite{mclean98} shows that if $C$ is compact then the
moduli space of coassociative submanifolds isotopic to $C$ is a smooth
manifold of dimension $b^{2}_{+}(C)$.
In particular, if $C$ is diffeomorphic to $T^{4}$ or $K3$ then the dimension
of the deformation space equals the codimension of $C$ in $M$. They
also have trivial self-dual bundle, so they are candidates for fibres of
fibrations by coassociative submanifolds.

Joyce and Salur \cite{joyce05} prove an EAC analogue of McLean's result.
Let $M^{7}$ be an EAC \gtmfd{} with cross-section $X^{6}$ and
$C \subset M$ an asymptotically cylindrical coassociative submanifold
with cross-section $L \subset X$ ($L$ is then a special Lagrangian submanifold
of the Calabi-Yau $3$-fold $X$, i.e. $L$ is calibrated by the real part
$\Omega$ of the holomorphic volume form on $X$).
Then the space of deformations of $C$ asymptotic to the fixed boundary $L$
is a smooth manifold and its dimension is $b^{2}_{+}(C)$,
the dimension of the positive part
of the compactly supported subspace $H^{2}_{0}(C) \subseteq H^{2}(C)$.
For $T^{3} \times \bbr$ or the half-$K3$ surface this vanishes,
so the coassociative submanifolds in example \ref{coass1ex} and \ref{coass2ex}
are rigid if the boundary is kept fixed.
\end{rmk}


\subsection{Another example}
\label{anothersub}

Now let us consider the example of \cite[\S 12.3]{joyce00}. This is a slight
modification of the previous example. The components of the singular set still
do not intersect each other, but some of the components have two different
possible resolutions.

Let $T^{7}$ be the standard torus with the standard flat \gtstr{}
$\varphi_{0}$.
Let $\Gamma \cong \bbz^{3}_{2}$ be the group of automorphisms of $T^{7}$
generated by
\begin{align*}
\alpha &: (x_{1}, \ldots, x_{7}) \mapsto
(x_{1}, x_{2}, x_{3}, -x_{4}, -x_{5}, -x_{6}, -x_{7}) , \\
\beta &: (x_{1}, \ldots, x_{7}) \mapsto
(x_{1}, -x_{2}, -x_{3}, x_{4}, x_{5}, \half - x_{6}, -x_{7}) , \\
\gamma &: (x_{1}, \ldots, x_{7}) \mapsto
(-x_{1}, x_{2}, -x_{3}, x_{4}, -x_{5}, x_{6}, \half-x_{7}) .
\end{align*}
The fixed point set of each of $\alpha, \beta$ and $\gamma$ consists of $16$
copies of $T^{3}$ and these are all disjoint. $\alpha\beta$, $\beta\gamma$,
$\gamma\alpha$ and $\alpha\beta\gamma$ act freely on $T^{7}$.
Furthermore $\gen{\beta, \gamma}$ and $\gen{\alpha,\gamma}$ act freely on the
set of $16$ $3$-tori fixed by $\alpha$ and $\beta$, respectively.
However, $\alpha\beta$ maps each component of the fixed point set of $\gamma$
to itself. Therefore the singular set of $T^{7}/\Gamma$ consists of
$8$ copies of $T^{3}$ and $8$ copies of $T^{3}/\bbz_{2}$.
The $T^{3}$ fixed by $\gamma$ has a natural choice of coordinates
$x_{2}$, $x_{4}$, $x_{6}$. In these coordinates the action of $\alpha\beta$ is
\[ \alpha\beta : T^{3} \to T^{3}, \;\:
(x_{2}, x_{4}, x_{6}) \mapsto (-x_{2}, -x_{4}, \half + x_{6}) . \]
%(\cite[(12.7)]{joyce00} is a misprint).

To form R-data we can first use the blow-up $Y$ of $\bbc^{2}/\{\pm 1\}$ with
an Eguchi-Hansen metric for each of the $8$ $3$-tori in the singular set,
just as in \S \ref{simplesub}.
For each of the $8$ copies of $T^{3}/\bbz_{2}$ there are, however, two
non-equivalent choices $Y_{+}$ and $Y_{-}$ of resolution of $\bbc/\{\pm 1\}$.
This is because $\alpha\beta$ acts on the normal space $\bbc^{2}$ to
the $T^{3}$ fixed by $\gamma$, and the R-data must include a compatible
automorphism of the resolution. Both $Y_{+}$ and $Y_{-}$ are diffeomorphic
to~$Y$, but $\alpha\beta$ acts differently on them. In fact the induced
action on the second cohomology $H^{2}(Y_{\pm}) \cong \bbr$ is
$(\alpha\beta)^{*} = \pm 1$. $Y_{+}$ and $Y_{-}$ both have ALE
$SU(2)$-metrics which are $\alpha\beta$-invariant.

The upshot is that for each of the $8$ copies of $T^{3}/\bbz_{2}$ there are two
topologically distinct choices of resolutions. The different R-data will
therefore give $2^{8}$ topologically distinct resolutions
$\pi : M \to T^{7}/\Gamma$ (however some of the manifolds $M$ are
diffeomorphic).


We can pull the resulting \gtmfd{} $M$ apart in several ways as 
in \S \ref{simplesub}.
To help compute the Betti numbers of the parts let us tabulate the
contributions from the different types of resolutions used.
\[ \begin{array}{c|l|l|c|c|c|c}
 & \textrm{Singularity} & \textrm{Resolution} &
b^{2} & b^{3} & b^{4} & b^{5} \\ \hline
A & \phantom{(} T^{3} \times (\bbc^{2}/\{\pm1\}) & \phantom{(} T^{3} \times Y &
1 & 3 & 3 & 1 \\ \hline
B & (T^{3} \times (\bbc^{2}/\{\pm1\}))/\bbz_{2} & (T^{3} \times Y_{+})/\bbz_{2}
 & 1 & 1 & 1 & 1 \\ \hline
C & (T^{3} \times (\bbc^{2}/\{\pm1\}))/\bbz_{2} & (T^{3} \times Y_{-})/\bbz_{2}
 & 0 & 2 & 2 & 0 \\ \hline
D & \phantom{(} T^{2} \times \bbr \times (\bbc^{2}/\{\pm1\}) &
\phantom{(} T^{2} \times \bbr \times Y & 1 & 2 & 1 & 0 \\ \hline
E & (T^{2} \times \bbr \times (\bbc^{2}/\{\pm1\}))/\bbz_{2} &
(T^{2} \times \bbr \times Y_{+})/\bbz_{2}  & 1 & 1 & 0 & 0 \\ \hline
F & (T^{2} \times \bbr \times (\bbc^{2}/\{\pm1\}))/\bbz_{2} &
(T^{2} \times \bbr \times Y_{-})/\bbz_{2}  & 0 & 1 & 1 & 0
\end{array} \]
For the compact \gtmfd{} $M$ one uses $8$ $A$-resolutions, and a total of
$8$ $B$- or $C$-resolutions. If $0 \leq k \leq 8$ is the number
of $B$-resolutions used then
\begin{equation*}
\begin{aligned}
b^{2}(M) &= 8 \cdot 1 + k \cdot 1 = 8+k, \\
b^{3}(M) &= 7 + 8 \cdot 3 + k \cdot 1 + (8-k)2 = 47-k .
\end{aligned}
\end{equation*}

\begin{enumerate}

\item
Pulling apart $M$ in the $x_{1}$ direction gives a pair of simply-connected
EAC manifolds $M_{\gi}$ with holonomy exactly $G_{2}$ and cross-section
$X_{19}$, similar to \S \ref{simplesub}\ref{ex1item}.
The resolved singular set in each half consists of $8$ copies of
$T^{2} \times \bbr$ and $4$ copies of $T^{3}/\bbz_{2}$.
If we use $k_{\gi}$ $B$-resolutions and $4-k_{\gi}$ $C$-resolutions
(so $k = k_{+} + k_{-}$) then
\begin{equation*}
\begin{aligned}
b^{2}(M_{\gi}) &=  8 \cdot 1 + k_{\gi} \cdot 1 = 8 + k_{\gi} , \\
b^{3}(M_{\gi}) &= 4 + 8 \cdot 2 + k_{\gi} \cdot 1 + (4-k_{\gi}) 2 =
28- k_{\gi} , \\
b^{4}(M_{\gi}) &= 3 + 8 \cdot 1 + k_{\gi} \cdot 1 + (4-k_{\gi}) 2 =
19 - k_{\gi} , \\
b^{5}(M_{\gi}) &= k_{\gi} \cdot 1 = k_{\gi} .
\end{aligned}
\end{equation*}

\item
Pulling apart $M$ in the $x_{2}$ or $x_{4}$ directions still gives
simply-connected EAC manifolds $M_{\gi}$ with holonomy exactly $G_{2}$,
and cross-section $X_{19}$, but the topology is different.
The resolutions used are $2 \times A$, $4 \times D$, $k \times E$ and
$(8-k) \times F$.
\begin{equation*}
\begin{aligned}
b^{2}(M_{\gi}) &=  4 \cdot 1 + 2 \cdot 1 + k \cdot 1 = 6 + k , \\
b^{3}(M_{\gi}) &= 4 + 4 \cdot 2 + 2 \cdot 3 + k \cdot 1 + (8-k) 1 =
26 , \\
b^{4}(M_{\gi}) &= 3 + 4 \cdot 1 + 2 \cdot 3 + (8-k) 1 =
21 - k , \\
b^{5}(M_{\gi}) &= 2 \cdot 1 = 2 .
\end{aligned}
\end{equation*}

\item
Pulling apart $M$ in the $x_{3}$ or $x_{5}$ direction gives a pair of
simply-connected EAC manifolds $M_{\gi}$ with holonomy exactly $G_{2}$ and
cross-section $X_{11}$, similar to \S \ref{simplesub}\ref{ex2item}.
Now one uses resolutions
$2 \times A$, $4 \times D$, $k_{\gi} \times B$ and $(4-k_{\gi}) \times C$.
\begin{equation*}
\begin{aligned}
b^{2}(M_{\gi}) &=  4 \cdot 1 + 2 \cdot 1 + k_{\gi} \cdot 1 = 6 + k , \\
b^{3}(M_{\gi}) &= 4 + 4 \cdot 2 + 2 \cdot 3 + k_{\gi} \cdot 1 + (4-k_{\gi}) 2 =
26 - k_{\gi} , \\
b^{4}(M_{\gi}) &= 3 + 4 \cdot 1 + 2 \cdot 3 + k_{\gi} \cdot 1 + (4-k_{\gi}) 2 =
21 - k_{\gi} , \\
b^{5}(M_{\gi}) &= 2 \cdot 1 + k_{\gi} \cdot 1 = 2 + k_{\gi} .
\end{aligned}
\end{equation*}

\item
Pulling apart $M$ in the $x_{6}$ direction one obtains EAC \gtmfd s
which are products of $S^{1}$ with simply-connected EAC Calabi-Yau $3$-folds,
topologically the same as those in \S \ref{simplesub}\ref{ex3item}.
% conjugate by (x_{5}, x_{7} |-> (x_{5}+x_{7}, x_{7})

\item
Pulling apart $M$ in the $x_{6}$ direction produces one EAC \gtmfd s
which is a product of $S^{1}$ with a simply-connected EAC Calabi-Yau $3$-fold
and one which is doubly covered by the product of $T^{3}$ and half-$K3$,
similar to \S \ref{simplesub}\ref{ex4item}.
\end{enumerate}


\section{Concluding remarks}

We have explained the pulling-apart argument for \gtmfd s only in some simple
cases and many natural questions remain.
One obvious question is how to deal with those examples
obtained by desingularising quotients of tori where different parts
of the singular set intersect.
In these cases it is more difficult to write down an intermediate resolution
which has both a torsion-free cylindrical neck and suitably small torsion.

One way to deal with this difficulty would be to implement a method for
constructing \gtstr s with small torsion proposed by Joyce in
\cite[p. 304]{joyce00}. Let $X^{6}$ be a Calabi-Yau $3$-fold with Calabi-Yau
structure $(\Omega, \omega)$ and suppose that $a$ is an anti-holomorphic
involution of $X$ with $a^{*}\Omega = \Omega$, $a^{*}\omega = -\omega$.
Then $\Omega + dt \wedge \omega$ is a torsion-free \gtstr{} on
$S^{1} \times X$, which descends to a well-defined \gtstr{} on the quotient
$(S^{1} \times X)/(-1, a)$ (cf. example \ref{cyinvex}). If the fixed
point set $L$ of $a$ is non-empty then the quotient is singular.
In this case $L$ is a real $3$-dimensional submanifold of $X$ (in fact it is
special Lagrangian).
The singular set of $(S^{1} \times X)/(-1, a)$ is $\{0, \half \} \times L$
and locally each point should be resolved like $\bbr^{3} \times Y$, where
$Y$ is an Eguchi-Hansen space. One would need to vary the choice of metric on
$Y$ smoothly. Joyce suggests that a suitable choice can be made to obtain
a \gtstr{} with small torsion on the resolution provided that $L$ has
a non-vanishing harmonic $1$-form.
The main purpose of this argument would be to produce new examples of compact
\gtmfd s. However, finding Calabi-Yau $3$-folds to which it can be applied is
a non-trivial task.

Examples produced by this method would automatically satisfy the conditions
of theorem \ref{pullthm}, so they could be pulled apart. If the construction
were made rigorous then it could almost certainly be applied to give
intermediate resolutions for many of Joyce's examples.

For many cases it would suffice to carry the argument through for the special
case when $X = T^{2} \times K3$, with $a$ acting by $(x,y) \mapsto (x,-y)$
on $T^{2}$ and by an anti-holomorphic involution on $K3$. By applying
a hyper-Kähler rotation to $K3$ this problem is equivalent to resolving the
singularities of a quotient $(T^{2} \times K3)/\gen{b}$ within holonomy $SU(3)$,
where $b$ acts by $-1$ on $T^{2}$ and by a holomorphic involution on $K3$.
The quotient is then a singular analytic variety, and can be resolved by
blowing up the singular set.
One would then need to find a Calabi-Yau metric on the blow-up
and give estimates for its distance to the $T^{2} \times K3$ metric away
from the singularities.

Resolving quotients of $T^{2} \times K3$ within holonomy $SU(3)$ should also be
sufficient to construct some examples of EAC manifolds with holonomy
exactly $Spin(7)$. One reason why the method in \S \ref{pullexsec} for
constructing EAC manifolds with holonomy exactly $G_{2}$ does not carry over
directly is that in the Kummer-type construction for compact \spmfd s there are
\emph{always} intersecting singular sets. The simplest example Joyce gives is
in \cite[\S 14.2]{joyce00}. This starts from the singular quotient of $T^{8}$
by a group $\Gamma \cong \bbz_{2}^{4}$ generated by
\begin{align*}
\alpha &: (x_{1}, \ldots, x_{8}) \mapsto
(-x_{1}, -x_{2}, -x_{3}, -x_{4}, x_{5}, x_{6}, x_{7}, x_{8}) , \\
\beta &: (x_{1}, \ldots, x_{8}) \mapsto
(x_{1}, x_{2}, x_{3}, x_{4}, -x_{5}, -x_{6}, -x_{7}, -x_{8}) , \\
\gamma &: (x_{1}, \ldots, x_{8}) \mapsto
(\half-x_{1}, \half-x_{2}, x_{3}, x_{4}, \half-x_{5}, \half-x_{6}, x_{7}, x_{8}) , \\
\delta &: (x_{1}, \ldots, x_{8}) \mapsto
(-x_{1}, x_{2}, \half-x_{3}, x_{4}, -x_{5}, x_{6}, \half-x_{7}, x_{8}) .
\end{align*}
The quotient can be resolved in an essentially unique way to produce
a compact \spmfd{} $M^{8}$. Pulling apart $M$ in the $x_{4}$ (or $x_{8}$)
direction yields halves which have the right topology to be EAC manifolds
with holonomy exactly $Spin(7)$, but the argument from \S \ref{preparesub}
cannot be applied directly to produce an intermediate resolution
with a cylindrical neck because the fixed point sets of $\alpha$ and $\beta$
intersect at the $256$ fixed points of $\alpha\beta$.
One could get around this problem by performing the resolution in three steps.
First resolve the quotient $T^{8}/\gen{\beta}$ to obtain $T^{4} \times K3$.
Then resolve $(T^{4} \times K3)/\gen{\gamma,\delta}$ to get
$S^{1} \times Y^{7}$, where $Y$ is a \gtmfd. Hopefully the second
resolution can be made in such a way that a neighbourhood of the singular set
of $\alpha$ remains very close to a product $\bbr^{4} \times K3$, with $\alpha$
acting as $-1$ on $\bbr^{4}$ and trivially on the $K3$ factor.
Finally the singularities of $(S^{1} \times Y)/\gen{\alpha}$ could then be
resolved to give a \spstr{} with small torsion and a neck region isomorphic
to the product of $Y$ with an interval.
An analogue of theorem \ref{eacperturbthm} should then
provide torsion-free EAC \spstr s on the two halves.

If one succeeds in constructing EAC \spmfd s this way one could ask if the
compact \spmfd{} can be deformed to a glued manifold. This would require
a set-up for a gluing construction for EAC \spmfd s and a study of its 
deformation properties, similar to that for \gtmfd s in \S \ref{gluechap}.
The technical details are likely to be more complicated.
%In the $G_{2}$ case we chose to define
%the gluing construction in such a way that increasing the neck length parameter
%corresponds to an affine line in $H^{3}(M)$.
%In the $Spin(7)$ case that will not work,
%because the bundle of point-wise \spstr s $\spspace$ is not open in
The fact that the image in $H^{4}(M)$ of the moduli space of torsion-free
\spstr s is a submanifold rather than an open subset makes it harder to
write down a local expression for the gluing map.

Another remaining question concerns the boundary of the moduli space $\defstr$
of torsion-free \gtstr s. We have exhibited some examples of compact \gtmfd s
$M^{7}$ where a single component of $\defstr$ has boundary points of both
orbifold and connected-sum type.  In fact, in these examples there are $7$
different kinds of connected-sum boundary points, corresponding to the
different directions in which $M$ can be pulled apart.  One can ask whether
$\defstr$ can be compactified in a natural way, so that different types of
boundary points lie on the same connected component.  Naively, one would expect
that a face of orbifold type and a face of connected-sum type would be joined
by a codimension $2$ edge, parametrised by classes of matching pairs of EAC
orbifolds.  Similarly, two different faces of connected-sum type boundary point
may meet at a codimension $2$ edge defined by quarters of $M$.  Defining the
boundary conditions for such quarters would be rather more complicated than the
definition for EAC manifolds.

Finally, one possible application of ``pulling-apart'' is to provide
a way to study Joyce's examples \emph{complex analytically}.
While we highlighted the cases where the connected components have holonomy
exactly $G_{2}$ for their novelty, the cases when the connected components
are reducible may give more information about the decomposed manifold.
When the connected components are of the form $S^{1} \times X_{\gi}^{6}$, where
$X_{\gi}$ are EAC Calabi-Yau $3$-folds, then $X_{\gi}$ can be considered
as the complement of an anti-canonical divisor in a compact complex
$3$-fold $\overline X_{\gi}$. This is very similar to the matching pairs of
reducible EAC \gtmfd s found in \cite{kovalev03}. In \cite{kovalev05} Kovalev
proposes a method to produce coassociative fibrations for compact \gtmfd s
obtained as the gluing of such pairs, by patching together fibrations of
each half by complex submanifolds of the $X_{\gi}$-factor.
This way one may hope to find coassociative fibrations of those of
Joyce's examples which can be pulled apart into reducible connected components.


\bibliographystyle{plain}
\bibliography{g2geom}

\end{document}
