\section{Definition of the Invariant}
\label{sec:DefNTheta}

In this section, we start with studying the metric dependence of the
quantity $\Inv$, with a view to showing the topological invariance of
$\NInv$. We begin with a few preliminary remarks on the
definition of $\Inv$.

\begin{defn}
A path $\{h_t\}_{t\in[0,1]}$ of metrics over $\Sigma$ is called {\em
$(U_0,U_1)$-allowable} if the metric $\met_0$ is $U_0$-allowable, and
the metric $\met_1$ is $U_1$-allowable. 
\end{defn}

Let $\ModSp(\met_t)$ denote the moduli space introduced in
Section~\ref{sec:Introduction}.

\begin{prop}
\label{prop:OurGenMet}
The moduli space $\ModSp(\met_t)$ can be naturally partitioned into
components $\ModSp(\met_t,\spinc)$ labeled by $\SpinC$ structures over
$Y$. Moreover, for a generic $(U_0,U_1)$-allowable path of metrics
$\met_t$ on $\Sigma$, the moduli space is a compact, oriented
$0$-manifold which does not contain any points with $s=t$.
\end{prop}

\begin{proof}
The orientation and partitioning into $\SpinC$ structures are
described in Section~\ref{Theta:sec:DefTheta} of~\cite{Theta}. When
$g>1$, the genericity statement follows from
Proposition~\ref{prop:GenericMetrics}, which is proved in
Section~\ref{sec:GenMet} of the present paper. Note that solutions
with $s=t$ correspond to points in $\Sym^{g-1}(\Sigma)$ which map to
$\Lag(U_0)\cap \Lag(U_1)$, which is a discrete set of points (since
$Y$ is a homology three-sphere); so these are excluded by dimension
counting.

When $g=1$, it is easy to see that the
moduli space is empty: in that case, the theta divisor consists of a
single, isolated spin structure, which does not bound.
\end{proof}

Armed with Proposition~\ref{prop:OurGenMet}, we can define
$\theta_{\met_t}(\spinc)$ to be the signed number of points in the component of
$\ModSp(\met_t,\spinc)$. 

\begin{defn}
Given a pair of
$(U_0,U_1)$-allowable paths $\met_t$ and $\met_t'$, a
$(U_0,U_1)$-allowable homotopy from $\met_t$ to $\met_t'$ is a smooth,
two-parameter family of metrics $$H\colon[0,1]\times[0,1]\rightarrow
\Met(\Sigma)$$ so that for each $t\in[0,1]$, $H(0,t)=\met_t$ and
$H(1,t)=\met_t'$, and for each $u\in[0,1]$ the path $t\mapsto H(u,t)$
is a $(U_0,U_1)$-allowable path.
\end{defn}

We will drop the handlebodies from the notation when they are clear
from the context.  Since the space of $U_i$-allowable metrics is
path-connected for $i=0,1$, see~\cite{Theta} (and the space of metrics
over $\Sigma$ is simply-connected), any two $(U_0,U_1)$-allowable
paths can be connected by a $(U_0,U_1)$-allowable homotopy.

Note that since $Y$ is a rational homology sphere, the $\SpinC$
structures naturally correspond to the intersection points of
$L_0=L(U_0)$ and $L_1=L(U_1)$, see also~\cite{Theta}.

\begin{prop}
\label{prop:ThetaChanges}
Let $\met_t$ and $\met_t'$ be a pair of generic $(U_0,U_1)$-allowable
paths. Then,
$$\theta_{\met_t}(\spinc)-\theta_{\met_t'}(\spinc)=
\#\{(D,u,t)\in\Sym^{g-1}(\Sigma)\times[0,1]\times[0,1]\big|
\Theta_{H(u,t)}(D)=p\},$$
where $H$ is any allowable homotopy from $\met_t$ to $\met_t'$, and
$p\in \Lag_0\cap \Lag_1$ is the point corresponding to the $\SpinC$
structure $\spinc$.
\end{prop}

\begin{remark}
Note that the set
$\{(D,u,t)\in\Sym^{g-1}(\Sigma)\times[0,1]\times[0,1]\big|
\Theta_{H(u,t)}(D)=p\}$ is not necessarily transversally cut out, as
one can easily see by considering spin structures (by Serre duality,
it is easy to see that if $p$ corresponds to a spin structure, then
the local multiplicities are always even). Thus, the intersection
number is to be interpreted by perturbing $p$ slightly.
\end{remark}

\begin{proof}
Let $H$ be a generic allowable homotopy connecting $\met_t$ and
$\met_t'$.  Consider the moduli space $$M(H)= \left\{(D,u,s,t)|s\leq
t, \Theta_{H_{u,s}}(D)\in \Lag(U_0),
\Theta_{H_{u,t}}(D)\in \Lag(U_1)\right\}.$$ According to the generic
metrics result, Proposition~\ref{prop:GenericMetrics}, this space is
an oriented one-manifold with boundary. Since $H$ is an allowable
homotopy, the only boundary components are $u=0$, $u=1$ and
$s=t$. Thus, counting boundaries, with sign and multiplicity, we get
the result as stated.
\end{proof}

The difference term appearing above has another natural interpretation
as a spectral flow on the cylinder $\R\times\Sigma$, inspired by work
of Yoshida (\cite{YoshidaII}, see also~\cite{KK}, \cite{CLMII}).  An
allowable path $\met_t$ and a scale factor $\scale$
naturally induces a metric on the cylinder
$\R\times\Sigma$ given by $(\scale dt)^2 + \met_t$,
where we extend $\met_t$ to all $t\in\R$ by requiring
$\met_t=\met_0$ (resp. $\met_1$) for $t\leq 0$ (resp. $\geq 1$). To
describe the relevant spectral flow, we introduce some 
terminology. 

\begin{defn}
\label{def:Reducible}
A connection $A$ on a spinor bundle $W$ over $Y$ is called {\em
reducible} if the trace of its curvature (or, equivalently, the
curvature induced on its determinant line bundle) vanishes. 
\end{defn}

On a rational homology sphere $Y$ equipped with a Riemannian metric,
each $\SpinC$ structure has a unique reducible connection (up to
gauge). 

\begin{prop}
\label{prop:SFCyl}
Fix a $\SpinC$ structure $\spinc\in \SpinC(Y)$.  Let $\met_t$ be an
allowable path for some Heegaard decomposition of a rational homology
sphere $Y$. Let $A$ be a reducible over $[0,1]\times\Sigma$ which
extends to a reducible over $Y$ for the $\SpinC$ structure $\spinc$.
Then, the Dirac operator coupled to $A$ acting on
$L^2(\R\times\Sigma)$ is Fredholm. If for each $t\in [0,1]$, the theta
divisor for $\met_t$ misses the point corresponding to $\spinc$, then
for all sufficiently large $\scale$, the Dirac operator for the metric
$(\scale dt)^2+\met_t$ and reducible belonging to $\spinc$ has no $L^2$
kernel over $\R\times \Sigma$.  Moreover, let $H$ be an allowable homotopy from
$\met_t$, $\met_t'$ over $\Sigma$. Then, there is some $\scale_0\geq
0$ such that for all sufficiently large $\scale,\scale'\geq \scale_0$,
the spectral flow between the two induced Dirac operators on $\Cyl$
coupled to reducible connections obtained by restricting the
reducibles in $\spinc\in\SpinC(Y)$ is given by
\begin{eqnarray*}
\lefteqn{\SF\left((\mu dt)^2+\met_t, (\mu'dt)^2 + \met_t'\right)} \\
&=& \#\left\{(D,u,t)\in\Sym^{g-1}(\Sigma)\times[0,1]\times[0,1]\big|
\Theta_{H(u,t)}(D)=p\right\},
\end{eqnarray*}
where $p$ is the point in $\Jac(\Sigma)$ corresponding to the $\SpinC$
structure $\spinc$.
\end{prop}

The above is a special case of the more general result
(Proposition~\ref{prop:SFCylBig}), which is proved in
Section~\ref{sec:ALimit}. 

Together, Propositions~\ref{prop:ThetaChanges} and \ref{prop:SFCyl}
show that $\theta_{\met_t}(\spinc)$ depends on the family of metrics
and $\SpinC$ connection used on the cylinder only through the spectral
flow of its Dirac operator. We call metric-dependent quantities which
depend on the metric through only the spectral flow of its associated
Dirac operator {\em chambered metric invariants}.  We use $\theta$,
together with another invariant  $\CorrU$ for handlebodies, to construct a
chambered invariant for certain metrics on $Y$.  Note that an allowable path
$\met_t$ and extensions $\Umet_0$ and $\Umet_1$ of the metrics
$\met_0$ and $\met_1$ over $U_0$ and $U_1$ respectively can be spliced
together naturally to form a metric $(\Umet_0)\#(\met_t)\#(\Umet_1)$
over $Y$.

In Section~\ref{sec:Handlebodies}, we construct a canonical chambered
invariant $\CorrU$ for handlebodies of arbitrary genus, which is uniquely
determined by an excision property, and a normalization condition
which guarantees that $\CorrU(\CDisk^3)=0$ for a non-negative
sectional curvature metric on the three-disk (see
Proposition~\ref{prop:CorrU} for a precise statement of the excision
property). Strictly speaking, this function depends on the metric and
a connection used on the spinor bundle over $U$, and an exact
perturbation of a reducible connection over $U$.  
Thus, the expression $\CorrU(\spinc|U_i,\Umet_i,a_i)$, denotes
the invariant evaluated on the metric $\Umet_i$ and the $\SpinC$
connection obtained by restricting the reducible in $\spinc$ to $U_i$,
and perturbing by $a_i$, where $a_i$ is a one-form which is compactly
supported in the interior of $U_i$. (This one-form is used to ensure
that the kernel of the Dirac operator on the handlebody has no kernel,
see Lemma~\ref{lemma:UDirac}.)

Splitting properties of spectral flow, then, show that the quantity
$\CorrU(\spinc|U_0,\Umet_0,a_0)+\theta_{\met_t}(\spinc)+\CorrU(\spinc|U_1,\Umet_1,a_1)$
depends on the metric $(\Umet_0)\#(\met_t)\#(\Umet_1)$ and the
connections used only through the spectral flow of the associated
Dirac operator, as described in the next proposition. To state the
proposition properly, we must analyze the choices made in ``splicing''
metrics to obtain a metric on $Y$.

Given metrics $\Umet_0$, $\Umet_1$ on the handlebodies $U_0$ and
$U_1$, and a path $\met_t$ on $\Sigma$ which interpolates between the
metrics on $\Sigma$ induced on the boundaries of $U_0$ and $U_1$, we
can splice to obtain a metric on $Y$. The spliced metric requires
three additional parameters: a scale $\scale$ to be used on the path
of metrics $\met_t$, and a pair of ``neck-length'' parameters which
specify lengths of cylinders to be spliced before and after the middle
neck.  More precisely, the metric $$(\Umet_0)\#_{T_0}\left((\scale dt)^2 +
\met_t\right)\#_{T_1} (\Umet_1)$$ will denote the metric on $Y$ which
is obtained by inserting cylinders $[0,T_i]\times \Sigma$ given the
product metric $dt^2 + \met_i$ for $i=0,1$ between the handlebodies
$U_0$ and $U_1$ and the cylinder $[0,1]\times\Sigma$ endowed with the
metric $(\scale dt)^2 + \met_t$.

\begin{prop}
\label{prop:ExtendToY}
Fix a $\SpinC$ structure $\spinc\in\SpinC(Y)$.
Let $\met_t$ be a generic $(U_0,U_1)$-allowable path. For a scale $\scale$ and
necklength parameters $T_0$ and $T_1$, let $\metY(\scale,T_0,T_1)$
denote the metric on $Y$ obtained by splicing 
$$\Umet_0\#_{T_0}\left(\left(\mu dt\right)^2 + \met_t\right) \#_{T_1} \Umet_1.
$$
Let $A(\mu,T_0,T_1)$ denote the corresponding reducible
connections. Then, for generic, compactly supported
one-forms $a_0$, $a_1$ in $U_0$ and $U_1$, the Dirac operator on $Y$
for the metric $\metY(\scale,T_0,T_1)$ and connection
$A(\mu,T_0,T_1)+a_0+a_1$ has no kernel, provided that $\mu$, $T_0$,
and $T_1$ are sufficiently large. Moreover, suppose that $H$ is an allowable
homotopy from $\met_t$ to another allowable path $\met_t'$ which
interpolates between metrics $\Umet_0'$ and $\Umet_1'$. Then, for
generic compactly-supported one-forms  $a_0$, $a_0'$ and $a_1$,
$a_1'$ on $U_0$ and $U_1$ respectively we have that for all 
sufficiently large $\scale$ and necklength parameters $T_0,T_1$ the
spectral flow of the Dirac operator $\Dirac(\scale,T_0,T_1)$
for the metric
$\metY(\scale,T_0,T_1)$ and connection $A(\mu,T_0,T_1)+a_0+a_1$
to the Dirac operator $\Dirac'(\scale,T_0,T_1)$ for the metric 
$\metY'(\scale,T_0,T_1)$ and connection $A'(\mu,T_0,T_1)+a_0'+a_1'$
is given by
\begin{eqnarray*}
\lefteqn{\SF(\Dirac(\scale,T_0,T_1),\Dirac'(\scale,T_0,T_1))} && \\
&=&
\left(\CorrU(\spinc|U_0,\Umet_0',a_0')+
\theta_{\met_t'}(\spinc)+\CorrU(\spinc|U_1,\Umet_1',a_1')\right) \\
&& -
\big(\CorrU(\spinc|U_0,\Umet_0,a_0)+\theta_{\met_t}(\spinc)+
\CorrU(\spinc|U_1,\Umet_1,a_1)\big).
\end{eqnarray*}
\end{prop}

\begin{proof}
The vanishing of the kernel follows from the vanishing of the kernel
over the three pieces: $U_0$, $[0,1]\times \Sigma$, and $U_1$
respectively. Over the handlebodies the kernel vanishes for generic
choices of $a_0$ and $a_1$ (this will be proved in
Lemma~\ref{lemma:UDirac}). Over the cylinder, it vanishes for generic
paths $\met_t$, according to Proposition~\ref{prop:SFCyl}.  Moreover,
the spectral flow statement follows from the splitting principle, the
chambered property of $\CorrU$, and the chambered property of
$\theta$, which in turn follows from
Proposition~\ref{prop:ThetaChanges} together with
Proposition~\ref{prop:SFCyl}.
\end{proof}

In the above proposition, the connection $A=A(\mu,T_0,T_1)+a_0+a_1$
has no kernel for generic $a_0$ and $a_1$. Thus, if we consider a
four-manifold $X$, equipped with a cylindrical-end metric $g_X$ and
$\SpinC$ structure $\spincX$ which bounds $\Umet_Y$ and $\spinc$
respectively, we can find a $\SpinC$ connection ${\widetilde A}$ which
$dt + A$ in a collar neighborhood of its boundary. According
to~\cite{APSI}, the Dirac operator coupled to ${\widetilde A}$ acting
on $L^2(X)$ (given a cylindrical end $Y\times [0,\infty)$, and endowed
with the natural extension of ${\widetilde A}$) is a Fredholm
operator, and indeed the quantity 
\begin{equation}
\label{eq:DefCorrY}
\Corr_{\metY}(\spinc,A)=
\ind_{X}(\Dirac_{\widetilde A})-
\left(\frac{c_1({\widetilde A})^2-\sgn(X)}{8}\right),
\end{equation}
where $\ind_X(\Dirac)$ denotes the complex index of the Dirac operator,
and $c_1({\widetilde A})$, of course, denotes the Chern-Weil
representative of the first Chern class. 

According to the above proposition, if we take the difference between
$\CorrU(\spinc|U_0,\Umet_0,a_0)+\theta_{\met_t}(\spinc)+
\CorrU(\spinc|U_1,\Umet_1,a_1)$ and $\Corr_{\metY}(\spinc,A)$, we get a quantity
which is independent of the extending metrics $\Umet_0$, $\Umet_1$ and
the path $\met_t$.  In fact, we get something which is independent of
the Heegaard decomposition as well, and hence a topological invariant
of $Y$.

\begin{theorem}
\label{thm:WellDefd}
Let $\met_t$ be a $(U_0,U_1)$-allowable path, and the $\metY$ be the
metric formed from $\Umet_0$, $\met_t$, and $\Umet_1$, where
$\Umet_i\in\Met(U_i)$ are metrics which bound $\met_0$ and $\met_1$
respectively.  Then, for generic $a_0$ and $a_1$, the quantity
$$\NInv(\spinc)=\CorrU(\spinc|U_0,\Umet_0,a_0)+
\theta_{\met_t}(\spinc)+
\CorrU(\spinc|U_1,\Umet_1,a_1)
-\Corr_{\metY}(\spinc,A)$$
(where $A=A(\scale,T_0,T_1)$ and $\metY=\metY(\mu,T_0,T_1)$ 
for sufficiently large $\scale$, $T_0$,
and $T_1$)
is a topological invariant of $Y$.
\end{theorem}

The independence of $\NInv$ of the Heegaard decomposition
relies on the corresponding result for $\theta$, which was
established in~\cite{Theta}.  To state the result, use a connected sum
for paths of metrics. For a fixed metric on the torus $S^1\times S^1$,
a metric $\met$ over $\Sigma$ (which is flat in a neighborhood of the
connected sum point $p\in\Sigma$), and a real number $T>0$, let
$\met(T)$ denote the metric on $\Sigma'=\Sigma\#(S^1\times S^1)$
obtained by a connected sum with neck-length $T$. Similarly, for a
one-parameter family $\met_t$ over $\Sigma$, we let $\met_t(T)$ denote
the one-parameter family of metrics on $\Sigma'$ obtained in this
manner. Then, we have the following:

\begin{prop}
\label{prop:StabInvarTheta}
Let $U_0\#_{\Sigma} U_1$ be a Heegaard decomposition of $Y$, and let 
$U_0'\#_{\Sigma'}U_1'$ be the ``stabilized'' Heegaard decomposition; i.e.
$U_0'=U_0\#(S^1\times \CDisk)$, 
$\Sigma'=\Sigma\#(S^1\times S^1)$, 
$U_1'=U_1\#(\CDisk\times S^1)$. 
For a path of metrics on $\met_t$ and a neck-length $T$, let 
$\met_t(T)$ denote the path of metrics on $\Sigma'$ obtained by
forming the connected sum of metrics.
Given a generic $(U_0,U_1)$-allowable
path $\met_t$, for all sufficiently large $T$, $\met_t(T)$ is a
generic $(U_0',U_1')$-allowable path, and the moduli spaces are
diffeomorphic. Thus, 
$$\theta_{\met(T)}(\spinc)=\theta_{\met_t(T)}(\spinc).$$
\end{prop}

\begin{proof}
The allowability of $\met_t(T)$ and diffeomorphism statement for 
$\theta$ were
proved in Proposition~\ref{Theta:prop:Stabilization} of~\cite{Theta};
note that the hypothesis that $b_1(Y)>0$ was not used in the proof of
this fact. One can arrange for $\met_t$ and $\met_t(T)$ to be generic
simultaneously by varying the family $\met_t$ in a region $U\subset
\Sigma$ which does not contain the connected sum point. This 
statement is proved in
Proposition~\ref{prop:GenericMetrics}.
\end{proof}

The proof of Theorem~\ref{thm:WellDefd} is not difficult, given the
excisive properties of spectral flow, and the above
proposition. However, spelling out the precise form of excision needed
involves some notation which we give in the
Section~\ref{sec:Handlebodies}, so we defer the proof to the end of that
section.
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