\section{Calculations when $b_1(Y)>1$}
\label{sec:CalcBig}

The aim of this section is to prove the following:

\begin{theorem}
\label{thm:CalcBig}
When $b_1(Y)>1$, then the polynomial ${\underline\theta}$ is equal up
to sign to the symmetrized Alexander polynomial of $Y$.
\end{theorem}

In the proof of this theorem, we will naturally meet certain tori in
the symmetric product.  Given a Heegaard decomposition of $Y$, let
$\{\alpha_i\}$, $\{\beta_i\}$ be complete sets of attaching circles
for the two handlebodies. Given any $i$ and $j$, we have tori in
$\Sym^{g-1}(\Sigma)$ $$\Torus{}_i(\alpha)=\alpha_1\times..\times
{\widehat
\alpha}_i\times...\times\alpha_g$$
(where the notation indicates omission of the $i^{th}$ factor) and
$$\Torus{}_j(\beta)=\beta_1\times..\times {\widehat
\beta}_j\times...\times\beta_g.$$
We will show that the invariant $\tInv$ can be extracted from certain
polynomials associated to these tori; these polynomials are defined as
follows.  Let $\OurSym\rightarrow \Sym^{g-1}(\Sigma)$ be the covering
space of $\Sigma$ (as in Section~\ref{sec:DefTheta}) corresponding to
the kernel of the composite map:
$$\pi_1(\Sym^{g-1}(\Sigma))\rightarrow H_1(\Sym^{g-1}(\Sigma);\Z)\cong
H_1(\Sigma;\Z)\rightarrow H_1(Y;\Z)\cong H.$$ (Recall that $H$ is by
definition $H^2(Y;\Z)$.)  Thus, $H$ acts freely on $\OurSym$.  Let
${\AbTa{i}}$, ${\AbTb{j}}$ be a pair of lifts of $\Ta{i}$ and
$\Tb{j}$. Note that these lifts are tori, and indeed they map
isomorphically to $\Ta{i}$ and $\Tb{j}$ respectively. The intersection
points of $\Ta{i}$ with $\Tb{j}$ correspond to the intersection points
of ${\AbTa{i}}$ with the various translates under $H$ of the
torus $\AbTb{j}$. Then, we define a polynomial (an element of $\Z[H]$)
associated to the lifts $\AbTa{i}$ and $\AbTb{j}$ by the formula
\begin{equation}
\label{eq:DefOfC}
C_{i,j}=\sum_{h\in H}
\#\left(\AbTa{i}\cap h\AbTb{j}\right) [h].
\end{equation}
(For the intersection numbers here, we use orientations for the
$\AbTa{i}$ and $\AbTb{j}$ induced from orientations of $\Ta{i}$ and
$\Tb{j}$; we return to a more careful discussion of signs in
Section~\ref{subsec:Refinements}.) Summing over the action of $\Tors$,
we get an induced polynomial ${\underline C}_{i,j}\in
\Z[\tH]$ (recall that $\tH=H/\Tors$). In Section~\ref{sec:Alex}, we
will show that the Alexander polynomial of $Y$ is the greatest common
divisor of the ${\underline C}_{i,j}$ for $i,j=1,...,g$. Different
lifts of the $\Ta{i}$ and $\Tb{j}$ give rise to translates of the
$C_{i,j}$ and ${\underline C}_{i,j}$ by elements in $H$.

The main ingredient in the proof of Theorem~\ref{thm:CalcBig} is a
perturbation of the invariants, which corresponds to moving the tori
$L_0$ and $L_1$. Let $\La{i}$ be the space of $B\in\Jac(\Sigma)$ with
$\Hol_{\alpha_k}B=0$ for all $k\neq i$, and similarly let $\Lb{j}$ be
the space of $B\in\Jac(\Sigma)$ with $\Hol_{\beta_k}B=0$ for all
$k\neq j$. We will move the tori $L_0$ and $L_1$ inside $\La{i}$ and
$\Lb{j}$, and obtain an expression of $\theta$ in terms of the 
intersection of the tori $\La{i}$ and $\Lb{j}$ with the theta divisor. 
An important point, then, is that we can concretely
understand these intersections, 
for favorable initial metrics. It is with the help of this
description, then, that we meet the polynomials described above.
But first, we describe how to calculate $\theta$ in terms of the lifts of
$\La{i}$ and $\Lb{j}$. To do this, we discuss the lifts in detail.

There is a lift ${\widetilde\delta}\colon \OurJac\longrightarrow
H^2(Y;\R)$ of the coboundary map $H^1(\Sigma;S^1)\rightarrow
H^2(Y;S^1)$, which is uniquely specified once we ask that
${\widetilde\delta}(\Lag_0(\spinc))=0$. With our conventions, then,
${\widetilde\delta}(\Lag_1(\spinc))={\underline\spinc}$. 

To $\alpha_i$, assign an element $\alpha_i^*\in H^1(\Sigma;\Z)$, as
follows. Let $\gamma_i$ be the core of the $i^{th}$ one-handle in
$U_0$ (i.e. this is the oriented curve which intersects only the
attaching disk associated to $\alpha_i$, which it intersects
positively in a single point), then $\alpha_i^*$ is the Poincar\'e
dual (in $\Sigma$) to a class in $H_1(\Sigma;\Z)$ whose image in
$H_1(U;\Z)$ is represented by $\gamma_i$.  (The class $\alpha_i^*$ is
not uniquely determined by this property, but the our constructions
involving $\alpha_i^*$ are independent of its choice.) Note that the
class $\mu_i=\delta\alpha_i^*\in H^2(Y;\Z)$, is Poincar\'e dual (in
$Y$) to the homology class represented by $\gamma_i$. The element
$\beta_i^*$ is defined in the analogous manner, only using $U_1$
instead of $U_0$. We let $\nu_j$ denote $\delta\beta_j^*$.  Choose $i$
and $j$ so that $\mu_i$ and $\nu_j$ are not torsion classes; we can
find such $i$ and $j$ since $H_1(U_0)$ and $H_1(U_1)$ both surject
onto $H_1(Y)$.

By multiplying $\alpha_i$ by $-1$ if necessary, we can assume that
${\underline\mu}_i$ and ${\underline\nu}_j$ in $H^2(Y;\Z)/\Tors$ are
not negative multiples of each other. (In keeping with the conventions
introduced in Section~\ref{sec:Introduction}, we underline objects
when viewing them modulo the action of torsion.) We define subsets
$\Lambda_0^+(\spinc),\Lambda_1^+(\spinc)\subset \OurJac$ which
correspond to all translates of (small perturbations of) $\Lag_0$ and
$\Lag_1$ in the directions determined by $\alpha_i^*$ and $\beta_j^*$;
more precisely, 
\begin{eqnarray*}
\Lambda_0^+(\spinc)=\Lag_0(\spinc)+\eta_0+ \R^+{\alpha_i^*}
&{\text{and}}&
\Lambda_1^+(\spinc)=\Lag_1(\spinc)+\eta_1- \R^+{\beta_j^*}.
\end{eqnarray*}
Under the map $\OurJac\rightarrow \Jac$, the spaces
$\Lambda_0^+(\spinc)$ and $\Lambda_1^+(\spinc)$ project to $\La{i}$
and $\Lb{j}$ respectively.

In the case where $b_1(Y)=2$, we make use of special allowable metrics:

\begin{defn}
Fix an $1/2 > \epsilon>0$, and let $U$ be a handlebody which bounds
$\Sigma$.  A metric $\met$ on $\Sigma$ is said to be {\em strongly
allowable for $\epsilon$} if it is product-like in a neighborhood of
$g$ attaching circles $\{\gamma_1,...,\gamma_g\}$, and any point in
the theta divisor for $\Sigma$ must have holonomy around some
attaching circle $\gamma_i$ within $\OneHalf\epsilon$ of $\OneHalf$.
\end{defn}

Given any $\epsilon>0$, there exist metrics which are 
strongly allowable for $\epsilon$ thanks to Lemma~\ref{lemma:MissTheta}.

\begin{prop}
\label{prop:Intersection}
Let $\spinc$ be any $\SpinC$ structure on $Y$, and fix
$\Lambda^+_i(\spinc)$ for $i=0,1$ as above -- using classes
${\underline\mu}_i$ and ${\underline\nu}_j$ which are not negative
multiples of one another. There is an $\epsilon>0$ with the property
that for any metrics $\met_0$ and $\met_1$ which are $U_0$ and
$U_1$-allowable respectively, where $\met_0$ is also
$\epsilon$-strongly $U_0$-allowable then, we can find $\eta_0$,
$\eta_1$ sufficently small, with $$\theta(\spinc)=
\#\left(\OurTheta^{-1}_{\met_0}\left(\Lambda_0^+(\spinc)+\eta_0\right)
	\cap \OurTheta^{-1}_{\met_1}\left(\Lambda_1^+(\spinc)+\eta_1\right)\right).
$$
\end{prop}

\begin{proof}
The proof will rely on the fact that $\OurDelta\circ \OurTheta_{\met_t}$
has bounded variation along any one-parameter family of
metrics. Specifically, let $\met_t$ be a one-parameter family of
metrics, and fix a norm on $H^2(Y;\R)$. Then, there is a constant $K$
with the property that for any $D\in\OurSym$, $s,t\in [0,1]$,
\begin{equation}
\label{eq:BoundedVariation}
|\OurDelta\circ\OurTheta_{\met_s}(D)-\OurDelta\circ\OurTheta_{\met_t}(D)|<
K.
\end{equation} This follows immediately from the compactness of
$\Sym^{g-1}(\Sigma)$, together with the fact that
$\OurDelta\circ\OurTheta$ is an $H^2(Y;\Z)$-equivariant map.

Note that $\theta(\spinc)$ is calculated by the number of points
(counted with signs) in the zero-dimensional submanifold of
$\OurSym\times[0,1]\times[0,1]$ $$\{(D,s,t)\big| s\leq t,
\OurTheta_{\met_s}(D)\in
\Lag_0(s)+\eta_0, \OurTheta_{\met_t}(D)\in\Lag_1(\spinc)+\eta_1\},$$ 
a space we denote by $M(\spinc)$. We construct a cobordism between
this space and the points in the intersection stated in the lemma, as
follows.  We can assume without loss of generality that $\met_t$ is
constant between $[0,1/4]$ and $[3/4,1]$. Moreover, let $\psi$ be a
non-decreasing smooth function on $[0,1]$ which is monotone increasing
in the range $[0,1/4]$, with $\psi(0)=0$ and $\psi|_{[1/4,1]}\equiv
1$. Consider the subspace of $\OurSym\times[0,1]\times[0,1]$ (which
agrees with $M(\spinc)$ when $u_1=u_2=0$): $$M_{u_1,u_2}(\spinc)=
\left\{
(D,s,t)\Bigg|
\begin{array}l
s\leq t, \\
\OurTheta_{\met_s}(D)\in
\Lag_0(\spinc)+\eta_0+u_1\psi(s) \alpha_i^*, \\
\OurTheta_{\met_t}(D)\in\Lag_1(\spinc)+\eta_1-u_2\psi(1-t) \beta_j^*
\end{array}
\right\}.$$

We argue that for all sufficiently large $u$,
\begin{equation}
\label{eq:PerturbedPoints}
M_{u,u}=\OurTheta^{-1}_{\met_0}\left(\Lambda_0^+(\spinc)+\eta_0\right)\cap
\OurTheta^{-1}_{\met_1}\left(\Lambda_1^+(\spinc)+\eta_1\right).
\end{equation}
Since ${\underline\mu}_i$ and ${\underline\nu}_j$ are not negative
multiples of one another, we see that that as $u\goesto\infty$, the
distance between the point ${\widetilde\delta}(L_0(\spinc)+\eta_0+u
\alpha_i^*)\in H^2(Y;\R)$ and the ray
${\widetilde\delta}(L_1(\spinc)+\eta_1-\R^+\beta_j^*)$ goes to
infinity, and similarly the distance between the point
${\widetilde\delta}(L_1(\spinc)+\eta_1-u\beta_j)$ and the ray
${\widetilde\delta}(L_0(\spinc)+\eta_0+\R^+\alpha_i^*)$ goes to
infinity. Fix $u$ large enough that both distances are larger than the
constant $K$ from Inequality~\eqref{eq:BoundedVariation}. This
condition ensures that all points $(D,s,t)\in M_{u,u}$ have $s\leq
1/4$ and $t\geq 3/4$. Monotonicity of $\psi$ over $[0,1/4]$, and the
choice of $u$ then also ensures that the identification
\eqref{eq:PerturbedPoints} holds.

Thus, Proposition~\ref{prop:Intersection} is established once we
construct a smooth cobordism between $M_{0,0}$ and $M_{u,u}$.
Consider the spaces obtained by connecting $M_{0,0}$ to $M_{u,u}$ by
first allowing $u_1$ to go from $0$ to $u$ (to connect $M_{0,0}$ to
$M_{u,0}$) and then allowing $u_2$ to go from $0$ to $u$ (to connect
$M_{u,0}$ to $M_{u,u}$). Since $\met_0$ and $\met_1$ are allowable
metrics, the $s=0$ and $t=1$ boundaries are excluded in this
one-parameter family for all small $\eta_0$ and $\eta_1$.  Thus, we
get a cobordism between $M_{0,0}$ and $M_{u,u}$, provided that the
$M_{u_1,u_2}$ do not hit the $s=t$ boundary, which is guaranteed if
$$\big(\La{i}+\eta_0\big)\cap
\big(\Lb{j}+\eta_1\big)=\emptyset. $$ Taking $\OurDelta$ of both
spaces, we get a pair of lines in $H^1(Y;\R)$, which generically miss
each other when $b_1(Y)>2$.

The case where $b_1(Y)=2$ requires a slightly closer investigation.
In the first part of the cobordism, where we allow $u_1$ to vary in
$M_{u_1,0}$, there are still no $s=t$ boundaries, as we can arrange
for $$(\La{i}+\eta_0)\cap (\Lag_1+\eta_1)=\emptyset$$ (since, applying
$\OurDelta$, we have a point and a line in a two-space).  Now, as
$u_2$ varies in the $M_{u,u_2}$, it is easy to see that the only
possible $s=t$ boundaries lie in the range where $s\leq 1/4$, by our
hypothesis on $\psi$, and hence they must lie in the set
$$\Theta_{\met_0}^{-1}\big((\La{i}+\eta_0)\cap (\Lb{j}+\eta_1)\big),$$
since $\met_t$ is constant for $t\leq 1/4$.  Now, consider the
intersection point $p$ of the induced rays $\OurDelta(\La{i}+\eta_0)$
and $\OurDelta(\Lb{j}+\eta_1)$.  Note that as $\met_0$ is stretched
out normal to the attaching disks, the image under $\OurDelta$
of the intersection
intersection $$(\La{i}+t\alpha_i^*)\cap
\Theta_{\met_0}(\Sym^{g-1}(\Sigma))$$
converges to a discrete subset of the ray
$\R^+{\underline\mu}_i\subset H^2(Y;\R)$, consisting of points
separated by some distance $\delta>0$ (which depends on the $\mu_i$
and $\nu_j$). If $p$ misses this discrete set, then if $\met_0$ is
sufficiently stretched out, then all sufficiently small $\eta_0$ and
$\eta_1$ have the property that
$$\Theta_{\met_0}^{-1}\big((\La{i}+\eta_0)\cap
(\Lb{j}+\eta_1)\big)=\emptyset.$$
If, on the other hand, $p$ lies on the discrete set, then, given any
sufficiently small
$0<\gamma$, if $\met_0$ is
sufficiently stretched out, then all sufficiently small $\eta_0$ and
$\eta_1$ have the property that
$$\Theta_{\met_0}^{-1}\big((\La{i}+\eta_0+\gamma \alpha_i^*)\cap
(\Lb{j}+\eta_1)\big)=\emptyset.$$
Thus, in both cases, we have obtained the requisite cobordism.
\end{proof}

Moreover, we can describe the intersection appearing in
Proposition~\ref{prop:Intersection}, in terms of the tori $\Ta{i}$ and
$\Tb{j}$ described in the beginning of the section.  But first, we
state a relevant lemma, whose proof fits naturally into the framework
of Section~\ref{sec:MoveTheta}.

\begin{lemma}
\label{lemma:StretchEverything}
Let $\Sigma$ be a surface, realized as a connected sum of $g$ tori as
in Corollary~\ref{cor:HolonomyConstraints}, and let
$\{\alpha_1,...,\alpha_g\}$ a complete set of attaching circles for
the handlebody $U$ which bounds $\Sigma$, each of which is disjoint
from the separating curves for the connected sum decomposition of
$\Sigma$. Fix a metric $h$ which is product-like near the $\alpha_i$
and the separating curves. Then, there is a $T_0$ so that any metric
which is obtained from $h$ by stretching each of the $2g-1$ curves at
least by $T_0$ is $U$-allowable.
\end{lemma}

\begin{proof}
Take a weak limit of connections which lie in $L(U)$, as all the
$2g-1$ curves are stretched. Under this limit, the surface degenerates
into a collection of genus zero surfaces (with cylindrical ends),
whose ends correspond to attaching circles for $U$ or separating
curves for $\Sigma$. Thus, the weak limit of connections in $L(U)$
induces a connection over these genus zero surfaces, whose holonomies
around all its bounding circles is zero.  But none of these support
harmonic spinors according to Proposition~\ref{prop:LTwoCohom},
proving the lemma.
\end{proof}

\begin{prop}
\label{prop:IdentTori}
There is a $U_0$-allowable metric $h$, for which
$\Theta_{h}^{-1}(\La{i}+\eta)$ is isotopic to $\Ta{i}$ for all
generic, small $\eta$.
\end{prop}

\begin{proof}
Fix a metric $\met$ on $\Sigma$ as in
Lemma~\ref{lemma:StretchEverything}.  Let $\met(T)$ denote the
metric which is stretched by $T_0$ along the $\alpha_i$ and $T$ along
the separating curves. The lemma guarantees
that for all $T>T_0$, $\met(T)$ is allowable. Now, for all
sufficiently large $T$, Corollary~\ref{cor:HolonomyConstraints} gives
the isotopy of $\Ta{i}$ with the subset $\Theta_{h}^{-1}(\La{i})$,
where $h=\met(T)$. 
\end{proof}

Putting together Propositions~\ref{prop:Intersection} and
\ref{prop:IdentTori}, we obtain the following topological
description of $\theta$.

\begin{prop}
\label{prop:CalcTheta}
For some $\SpinC$ structure $\spinc\in\SpinC(Y)$, we have that
$$C_{i,j}[\spinc]=(1-\mu_i)(1-\nu_j)\cm\theta.$$
\end{prop}

\begin{proof}
Let $\met_k$ be $U_k$-allowable metrics for $k=0,1$.  Given $i,j$, we
construct a natural element ${\widetilde C}_{i,j}\in\Z[\SpinC(Y)]$
closely related to the $C_{i,j}$ defined in the beginning of this
section. The ${\widetilde C}_{i,j}$ will be a translate of the
following analogue of the $C_{i,j}$, which is assigned to a pair of
lifts ${\widetilde A}_i$ and ${\widetilde B}_j$ (to $\OurSym$) of the
manifolds $A_i=\Theta^{-1}_{\met_0}(\La{i}+\eta_0)$ and
$B_j=\Theta^{-1}_{\met_1}(\Lb{j}+\eta_1)$: $$ X_{i,j}= \sum_{h\in H}
\# \left({\widetilde B}_j \cap h {\widetilde A}_i\right) [h].
$$ To do define the ${\widetilde C}_{i,j}$, we must assign a $\SpinC$
structure to each intersection point of $A_i$ with $B_j$. 
To this end,
we assume that $\met_0$ and $\met_1$ are strongly allowable for some
$\epsilon>0$.  If $p\in A_i\cap B_j$, then let ${\widetilde p}$ be a
lift of $p$. There is a pair of lifts ${\widetilde A}_i$ and
${\widetilde B}_j$ of $A_i$ and $B_j$ which meet in ${\widetilde
p}$. 
Note that there are unique lifts $\Lag_0$ and $\Lag_1$
whose image under $\OurDelta$ lie
in an $\epsilon$ neighborhood of ${\widetilde p}-\OneHalf{\underline\mu}_i$
and ${\widetilde p}+\OneHalf{\underline\nu}_j$ respectively.
Let $G(p)$ denote the $\SpinC$ structure which corresponds
to the difference between these two lifts (i.e. if
${\widetilde\Lag}_0$ and ${\widetilde\Lag_1}$ are the two lifts, then
the pair $L_0(G(p))$ and $L_1(G(p))$ are translates of
${\widetilde\Lag}_0$ and ${\widetilde\Lag_1}$ by a single cohomology
class in $H^2(Y;\Z)$).  By summing over all intersection points which
correspond to a given $\SpinC$ structure (with signs), we obtain the
element ${\widetilde C}_{i,j}\in\Z[\SpinC(Y)]$, which is clearly the
translate by some $\SpinC$ structure of the polynomial $X_{i,j}\in
\Z[H]$ defined above.

It follows from Proposition~\ref{prop:Intersection} that
$\theta(\spinc)$ is given by adding up the intersection number of
certain lifts of $A_i$ and
$B_j$. Moreover, the intersection point $p$ will
contribute for each $k,\ell\geq 0$ in the $\SpinC$ structure
$$G(p)+k\mu_i+\ell\nu_j$$ (as those are the $\SpinC$ structures for
which $p$ lies on the corresponding rays).  This proves that
\begin{equation}
\label{eq:CalculateTheta}
\theta={\widetilde C}_{i,j}\left(\sum_{k=0}^{\infty}\mu_i^k\right)
\left(\sum_{\ell=0}^{\infty}\nu_j^\ell\right).
\end{equation}

Finally, the proposition is proved once we establish that the
${\widetilde C}_{i,j}$ is a translate of $C_{i,j}$, as defined in
Equation~\eqref{eq:DefOfC}.  To see this, recall that
Proposition~\ref{prop:IdentTori} guarantees that the spaces $A_i$ and
$B_j$ are isotopic, for suitable choices of allowable metrics $\met_0$
and $\met_1$, to $\Ta{i}$ and $\Tb{j}$ respectively.  Now the
polynomial $X_{i,j}$, which is clearly a translate of ${\widetilde
C}_{i,j}$, depends on the submanifolds $A_i$ and $B_j$ only up to
isotopy.  Thus, the proposition follows.
\end{proof}

In particular, we have the following:

\begin{cor}
If ${\underline\mu}_i$ and ${\underline\nu}_j$ are not negative
multiples of one another, then $${\underline
C}_{i,j}=(1-{\underline\mu}_i)(1-{\underline\nu}_j){\underline\theta}.$$
\end{cor}

Theorem~\ref{thm:CalcBig} follows from this corollary. First, after
handle-slides, we can arrange that all the ${\underline\mu}_i$ and
${\underline\nu}_j$ are non-zero, so that ${\underline\theta}$ divides
each ${\underline C}_{i,j}$ and hence the Alexander polynomial.
Furthermore, after additional handleslides we can arrange that
${\underline\mu}_1={\underline\nu}_1$,
${\underline\mu}_2={\underline\nu}_2$, and ${\underline\mu}_1$ and
${\underline\mu}_2$ are linearly independent in $H^2(Y;\R)$. This
shows that the greatest common divisor of ${\underline C}_{1,1}$ and
${\underline C}_{2,2}$ is ${\underline\theta}$, so the latter agrees
with the Alexander polynomial.
{"config": "arxiv", "file": "math0006192/calcbig.tex"}