define the perturbation space of the Seiberg-Witten equaions on $\mathcal{X}$.
This is crucial for the transversality result in Section 16, cf. Theorem 16.5;

  * •

prove that the linearized operator on $\mathcal{X}$ is Fredholm.

They are addressed in Subsection 14.1 and 14.2 respectively.

####  14.1. Perturbations

Given a morphism
$\mathbb{X}:(\mathbb{Y}_{1},\widehat{\mathfrak{s}}_{1})\to(\mathbb{Y}_{2},\widehat{\mathfrak{s}}_{2})$
in the strict $spin^{c}\ $cobordism category $\operatorname{SCob}_{s}$, the
perturbation $\mathfrak{q}_{i}\in\mathcal{P}(Y_{i})$ encoded in the definition
of $(\mathbb{Y}_{i},\widehat{\mathfrak{s}}_{i})$ is admissible by (P8). Take a
critial point

$\mathfrak{a}_{i}\in\operatorname{Crit}(\hbox
to0.0pt{$-$\hss}\mathcal{L}_{\omega_{i},\widehat{Y}_{i}})\subset\mathcal{C}_{k}(\widehat{Y}_{i},\widehat{\mathfrak{s}}_{i}),$

for each $i=1,2$. Pick a smooth configuration $\gamma$ on $\mathcal{X}$ such
that

(14.1)
$\left\\{\begin{array}[]{rl}\check{\gamma}(t)&\equiv\mathfrak{a}_{1}\text{ if
}t<-1/2;\\\ \check{\gamma}(t)&\equiv\mathfrak{a}_{2}\text{ if }t>1/2\\\
\gamma(t)&\text{is in the temporal gauge when }|t|>1/2,\\\
\gamma\big{|}_{\widehat{X}}&\in\mathcal{C}_{k}(\widehat{X},\widehat{\mathfrak{s}}).\end{array}\right.$

Now consider the configuration space on $\mathcal{X}$:

$\mathcal{C}_{k}(\mathfrak{a}_{1},\mathcal{X},\mathfrak{a}_{2})\colonequals\\{(A,\Phi)=\gamma_{0}+(a,\phi):(a,\phi)\in
L^{2}_{k}(\mathcal{X},iT^{*}\mathcal{X}\oplus S^{+})\\}.$

and the gauge group

$\mathcal{G}_{k+1}(\mathcal{X})=\\{u:\mathcal{X}\to S^{1}:u-1\in
L^{2}_{k+1}(\mathcal{X},\mathbb{C})\\}.$

The linearized action of $\mathcal{G}_{k+1}(\mathcal{X})$ at
$\gamma=(A,\Phi)\in\mathcal{C}_{k}(\mathfrak{a}_{1},\mathcal{X},\mathfrak{a}_{2})$
is given by:

$\displaystyle\operatorname{\mathbf{d}}_{\gamma}:L^{2}_{k+1}(\mathcal{X},i\mathbb{R})$
$\displaystyle\to
T_{\gamma}\mathcal{C}(\mathfrak{a}_{1},\mathcal{X},\mathfrak{a}_{2})$
$\displaystyle f(t)$ $\displaystyle\mapsto(-df,f\Phi)$

whose $L^{2}$-formal adjoint is

$\displaystyle\operatorname{\mathbf{d}}_{\gamma}^{*}:T_{\gamma}\mathcal{C}(\mathfrak{a}_{1},\mathcal{X},\mathfrak{a}_{2})$
$\displaystyle\to L^{2}_{k-1}(\mathcal{X},i\mathbb{R})$ $\displaystyle(\delta
a,\delta\phi)$
$\displaystyle\mapsto-d^{*}a+i\operatorname{Re}\langle\delta\phi,i\Phi\rangle.$

Let us now specify the class of perturbations involved in the Seiberg-Witten
equations. Choose a cut-off function $\beta:\mathbb{R}_{t}\to\mathbb{R}$ with
$\beta(t)\equiv 1$ if $|t|>3$ and $\beta(t)\equiv 0$ if $|t|<2$. Pick another
cut-off function $\beta_{0}:\mathbb{R}_{t}\to\mathbb{R}$ supported on
$[1,2]_{t}\subset\mathbb{R}_{t}$, equal to $1$ when $t\in[5/4,7/4]$. Now
consider the perturbed Seiberg-Witten equation:

(14.2) $\displaystyle\mathfrak{F}_{\mathcal{X},\mathfrak{p}}(\gamma)$
$\displaystyle=0,\
\gamma\in\mathcal{C}_{k}(\mathfrak{a}_{1},\mathcal{X},\mathfrak{a}_{2}),$
$\displaystyle\mathfrak{F}_{\mathcal{X},\mathfrak{p}}(\gamma)$
$\displaystyle\colonequals\mathfrak{F}_{\mathcal{X}}(\gamma)+\beta(t)[\widehat{\mathfrak{q}}_{1}(\gamma)+\widehat{\mathfrak{q}}_{2}(\gamma)]+\beta_{0}(t)(\widehat{\mathfrak{q}}_{3}(\gamma))+(\rho_{4}(\omega_{3}^{+}),0),$

where $\mathfrak{F}_{\mathcal{X}}$ is the unperturbed Seiberg-Witten map
defined by the formula (3.7). Here $\mathfrak{p}$ denotes the quadruple

$\mathfrak{p}\colonequals(\mathfrak{q}_{1},\mathfrak{q}_{2},\mathfrak{q}_{3},\omega_{3})\in\mathcal{P}(Y_{1})\times\mathcal{P}(Y_{2})\times\mathcal{P}(Y_{2})\times\Omega^{2}_{c}([1,2]\times
Y_{2},i\mathbb{R}).$

where $\mathfrak{q}_{3}\in\mathcal{P}(Y_{2})$ is a tame perturbation supported
on $Y_{2}$ and $\omega_{3}$ is an imaginary-valued exact 2-form compactly
supported on $[1,2]\times Y_{2}$. The effect of $\omega_{3}$ is to deform
$\omega_{X}$ into $\omega_{X}-\omega_{3}$, so the first equation of (3.7) is
changed into

$\frac{1}{2}\rho_{4}(F_{A^{t}}^{+}-2\omega^{+}_{X})-(\Phi\Phi^{*})_{0}=-\rho_{4}(\omega_{3}^{+}),$

modulo perturbations from $\mathfrak{q}_{i}$’s. In practice, it suffices to
consider $\omega_{3}$ in the special form:

(14.3) $\omega_{3}=d_{\mathcal{X}}(\beta_{0}(t)f_{3}dt)=-\beta_{0}(t)dt\wedge
d_{Y_{2}}f_{3}.$

for a compactly supported smooth function $f_{3}:[1,2]_{t}\times Y_{2}\to
i\mathbb{R}$.

Within the space of all compactly supported smooth functions on
$[1,2]_{t}\times Y_{2}$, we choose a countable subset that is dense in
$C^{\infty}$-topology and form a Banach space as in Theorem 8.17:

$\mathcal{P}_{\operatorname{Form}}.$

The space $\mathcal{P}_{\operatorname{Form}}$ is dense in
$C^{\infty}_{c}([1,2]_{t}\times Y_{2},i\mathbb{R})$, and we define
$\omega_{3}$ by the formula (14.3) with
$f_{3}\in\mathcal{P}_{\operatorname{Form}}$. In all, the quadruple
$\mathfrak{p}$ takes value in a Banach space

$\mathfrak{p}=(\mathfrak{q}_{1},\mathfrak{q}_{2},\mathfrak{q}_{3},\omega_{3})\in\mathcal{P}(Y_{1})\times\mathcal{P}(Y_{2})\times\mathcal{P}(Y_{2})\times\mathcal{P}_{\operatorname{Form}}.$

Here $\mathfrak{q}_{1}$ and $\mathfrak{q}_{2}$ are encoded in the cylindrical
ends of $\mathcal{X}$; only the last two terms

$(\mathfrak{q}_{3},\omega_{3})$

give rise to the actual perturbation in (14.2), allowing us to achieve
transversality in Section 16. Note that

$\beta_{0}(t)\widehat{\mathfrak{q}}_{3}(\gamma)\text{ and
}(\rho_{4}(\omega_{3}^{+}),0)$

are both supported in the compact region $[1,2]_{t}\times Y_{2}$. Finally, we
form the moduli space
$\mathcal{M}_{k}(\mathfrak{a}_{1},\mathcal{X},\mathfrak{a}_{2})$ by taking the
quotient space:

(14.4)
$\mathcal{M}_{k}(\mathfrak{a}_{1},\mathcal{X},\mathfrak{a}_{2})\colonequals\\{\mathfrak{F}_{\mathcal{X},\mathfrak{p}}(\gamma)=0:\gamma\in\mathcal{C}_{k}(\mathfrak{a}_{1},\mathcal{X},\mathfrak{a}_{2})\\}/\mathcal{G}_{k+1}(\mathcal{X}),$

which is in fact independent of the subscript $k$, due to the exponential
decay of the local energy functional, cf. Theorem 9.3.

####  14.2. Linearized Operators

Similar to the case for $\widehat{Z}=\mathbb{R}_{t}\times\widehat{Y}$, the
linearization of $\mathfrak{F}_{\mathcal{X},\mathfrak{p}}$ together with
$\operatorname{\mathbf{d}}_{\gamma}^{*}$ forms a Fredholm operator. In
particular, the cokernel is finite dimensional.

######  Proposition 14.1.

For any $i=1,2$, let $\mathfrak{a}_{i}$ be a smooth non-generate critical
point of $\hbox to0.0pt{$-$\hss}\mathcal{L}_{\omega_{i}}$ in
$\mathcal{C}_{k}(\widehat{Y}_{i},\widehat{\mathfrak{s}}_{i})$. Then for any
$\gamma\in\mathcal{C}_{k}(\mathfrak{a}_{1},\mathcal{X},\mathfrak{a}_{2})$, the
operator

$(\operatorname{\mathbf{d}}_{\gamma}^{*},\mathcal{D}_{\gamma}\mathfrak{F}_{\mathcal{X},\mathfrak{p}}):L^{2}_{k}(\mathcal{X},iT^{*}\mathcal{X}\oplus
S^{+})\to L^{2}_{k-1}(\mathcal{X},i\mathbb{R}\oplus
i\Lambda^{+}\mathcal{X}\oplus S^{-})$

is Fredholm.

######  Definition 14.2.

The moduli space
$\mathcal{M}_{k}(\mathfrak{a}_{1},\mathcal{X},\mathfrak{a}_{2})$ is called
regular, if the operator
$(\operatorname{\mathbf{d}}_{\gamma}^{*},\mathcal{D}_{\gamma}\mathfrak{F}_{\mathcal{X},\mathfrak{p}})$
is surjective at any solution
$[\gamma]\in\mathcal{M}_{k}(\mathfrak{a}_{1},\mathcal{X},\mathfrak{a}_{2})$, ∎

###### Proof of Proposition 14.1.

It suffices to deal with the case for the reference configuration
$\gamma=\gamma_{0}$ and when $(\mathfrak{q}_{3},\omega_{3})=0$. As
$\mathfrak{a}_{i}$ is non-degenerate, the operator on the infinite cylinder

$D_{i}\colonequals\frac{d}{dt}+\widehat{\operatorname{Hess}}_{\mathfrak{q}_{i},\mathfrak{a}_{i}}:L^{2}_{k}(\mathbb{R}_{t}\times\widehat{Y}_{i},i\mathbb{R}\oplus
iT^{*}\widehat{Y}_{i}\oplus S)\to
L^{2}_{k-1}(\mathbb{R}_{t}\times\widehat{Y}_{i},i\mathbb{R}\oplus
iT^{*}\widehat{Y}_{i}\oplus S)$

is invertible for $i=1,2$. Denote the inverse by $Q_{i}$. Unlike Theorem 10.2,
the cut-off functions involved in the parametrix patching argument are more
sophisticated, as we explain now. There are three of them:

$\beta_{1},\ \beta_{2}\text{ and }\beta_{X}\text{ with
}\beta_{1}+\beta_{2}+\beta_{X}\equiv 1\text{ and }\beta_{X}\text{ compactly
supported}$

Over the region $\\{s\leq 2\\}\subset\mathcal{X}$, choose a partition of unity
$\\{\beta_{1}^{\prime},\beta_{2}^{\prime},\beta_{X}\\}$ subordinate to the
open cover $U_{1}\cup U_{2}\cup U_{X}$:

\begin{overpic}[scale={.17}]{F2.png} \put(46.0,15.0){$X$}
\put(-2.0,15.0){$(-\infty,-1]\times\widehat{Y}_{1}$}
\put(72.0,15.0){$[1,\infty)\times\widehat{Y}_{2}$} \put(16.0,5.0){$t=$}
\put(26.0,5.0){\small$-2$} \put(36.0,5.0){\small$-1$}
\put(47.0,5.0){\small$0$} \put(57.0,5.0){\small$1$} \put(66.0,5.0){\small$2$}
\put(8.0,46.0){\small$s=2$} \put(15.0,36.0){\small$0$}
\put(12.0,26.0){\small$-2$}
\put(60.0,23.0){\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}$U_{X}$}
\put(21.0,39.0){\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}$U_{1}$}
\put(71.0,39.0){\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}$U_{2}$}
\end{overpic} Figure 1. An open cover of $\\{s\leq 2\\}$

Over the region $\\{s\geq 2\\}$, $\beta_{X}\equiv 0$ and
$\beta_{i}(s,t)=\beta_{i}^{T}(t),i=1,2$ where
$\\{\beta_{1}^{T},\beta_{2}^{T}\\}$ is a partition of unity on the real line
$\mathbb{R}_{t}$ subordinate to the cover

$\mathbb{R}_{t}=(-\infty,T]\cup[-T,\infty),$

such that $|d\beta_{i}^{T}|\leq 4/T$. The value of $\beta_{i}$ in the
transition area $\\{1\leq s\leq 2\\}$ is filled in by interpolation. To be
more precise, pick a partition of unity $\\{\alpha^{L},\alpha^{U}\\}$ on
$\mathbb{R}_{s}$ such that $\alpha^{U}(s)\equiv 1$ when $s\geq 2$ and
$\alpha^{U}(s)\equiv 0$ when $s\leq 1$. Set

$\beta_{i}=\alpha^{L}(s)\beta_{i}^{\prime}+\alpha^{U}(s)\beta_{i}^{T}(t),\
i=1,2.$

Finally, we take

$Q=\tilde{\beta}_{1}Q_{1}\beta_{1}+\tilde{\beta}_{2}Q_{2}\beta_{2}+\tilde{\beta}_{X}Q_{X}\beta_{X},$

with $\tilde{\beta}_{i}$ constructed in a similar manner. Here we require that
$\tilde{\beta}_{i}\equiv 1$ on $\text{supp}\ \beta_{i}$ so that
$\tilde{\beta}_{i}\beta_{i}=\beta_{i}$. The same holds for
$(\tilde{\beta}_{X},\beta_{X})$; and also $\text{supp}\ \tilde{\beta}_{X}$ is
compact.

The parametrix $Q_{X}$ is given by a local patching argument as usual. By
taking $T\gg 0$, one verifies that $Q$ is indeed a parametrix for the operator
$(\operatorname{\mathbf{d}}_{\gamma}^{*},\mathcal{D}_{\gamma}\mathfrak{F}_{\mathcal{X},\mathfrak{p}})$.
∎

##  Part V Transversality

The primary goal of this part is to prove the key transversality result:
Theorem 16.1, which states that admissible perturbations on
$(\widehat{Y},\widehat{\mathfrak{s}})$, in the sense of Definition 13.3, exist
and are in fact generic. Because the perturbation space $\mathcal{P}(Y)$ that
we consider are supported on the truncated 3-manifold $Y=\\{s\leq 0\\}$, only
a weak separating property is satisfied, cf. Theorem 8.20. As a result, a
stronger unique continuation property is required in order to achieve
transversality.

Section 15 is devoted to the proof of unique continuation properties, which
uses the Carleman estimates from [Kim95]. In Section 16, we prove Theorem 16.1
as well as its analogue for a general morphism
$\mathbb{X}:(\mathbb{Y}_{1},\widehat{\mathfrak{s}}_{1})\to(\mathbb{Y}_{2},\widehat{\mathfrak{s}}_{2})$
in the $\operatorname{SCob}_{s}$, cf. Theorem 16.5.

###  15\. Unique Continuation

####  15.1. Statements

In this section, we prove the unique continuation properties of the perturbed
Seiberg-Witten equations (13.1), which are crucial for the proof of Theorem
16.1. The main results are listed as follows:

  * •

the non-linear version: Theorem 15.1;

  * •

the linearized version: Theorem 15.2; and

  * •

the irreducibility of spinors: Theorem 15.3.

These theorems are summarized in the first subsection, while the rest of
section is devoted to their proofs. Let us start with the non-linear version
of unique continuation:

######  Theorem 15.1.

Let $I=(t_{1},t_{2})_{t}$ be an open finite interval. Consider a tame
perturbation $\mathfrak{q}\in\mathcal{P}$ supported on the truncated
3-manifold $Y=\\{s\leq 0\\}\subset\widehat{Y}$ and the perturbed Seiberg-
Witten equations on $\widehat{Z}\colonequals I\times\widehat{Y}$:

(15.1)
$0=\mathfrak{F}_{\widehat{Z},\mathfrak{q}}(\gamma)\colonequals\mathfrak{F}_{\widehat{Z}}(\gamma)+\widehat{\mathfrak{q}}(\gamma).$

If two solutions $\gamma_{1},\gamma_{2}$ are gauge equivalent on the slice
$\\{t_{0}\\}\times Y$ at some $t_{0}\in I$, i.e there exists a gauge
transformation $u\in\mathcal{G}(\widehat{Y})$ such that

$u(\gamma_{1}|_{\\{t_{0}\\}\times\widehat{Y}})=\gamma_{2}|_{\\{t_{0}\\}\times\widehat{Y}}\text{
on }Y,$

then $\gamma_{1}$ and $\gamma_{2}$ are gauge equivalent over the whole
manifold $\widehat{Z}$.

The analogous result for closed 3-manifolds is [KM07, Proposition 7.2.1]. The
main difference here is that $\gamma_{1}$ and $\gamma_{2}$ are not assumed to
be gauge equivalent on the whole time slice $\\{t_{0}\\}\times\widehat{Y}$;
thus, the proof of [KM07, Proposition 7.2.1] does not apply directly here.

Theorem 15.1 will follow from the strong unique continuation of the Seiberg-
Witten equations if $\mathfrak{q}=0$. The problem arises from the tame
perturbation $\mathfrak{q}$, which gives rise to non-local operators. We will
provide a toy model in the next subsection to clarify this point, cf. Remark
15.5. It is essential here that the region $\\{t_{0}\\}\times Y$ over which
$\gamma_{1}$ and $\gamma_{2}$ agree contains the support of $\mathfrak{q}$.

Before we proceed any further, let us state the linearized version of Theorem
15.1 and the version that concerns the irreducibility of spinors.

######  Theorem 15.2 (The Linearized Version).

Let $I=(t_{1},t_{2})_{t}\subset\mathbb{R}_{t}$ be an open interval. Consider a
tame perturbation $\mathfrak{q}\in\mathcal{P}$ supported on the truncated
3-manifold $Y=\\{s\leq 0\\}\subset\widehat{Y}$ and a smooth solution $\gamma$
to the perturbed Seiberg-Witten equation (15.1) on the 4-manifold
$\widehat{Z}=I\times(\widehat{Y},\widehat{\mathfrak{s}})$. Suppose a smooth
tangent vector at $\gamma$

$V(t)=(\delta c(t),\delta b(t),\delta\psi(t))\in
L^{2}_{k}(\widehat{Z},iT^{*}\widehat{Z}\oplus S)$

lies in the kernel of the linearized Seiberg-Witten map:

(15.2) $0=\mathcal{D}_{\gamma}\mathfrak{F}_{\widehat{Z},\mathfrak{q}}(V),$

or equivalently, it solves the equation (13.2). If $V$ is generated by the
linearized gauge action on $\\{t_{0}\\}\times Y$ at some $t_{0}\in I$, i.e.
there exists a smooth function $\xi\in L^{2}_{k+1/2}(\widehat{Y},i\mathbb{R})$
such that

$(\delta
b(0),\delta\psi(0))=\operatorname{\mathbf{d}}_{\check{\gamma}(t)}\xi\text{ on
}\\{t_{0}\\}\times Y.$

then $V$ is generated by the linearized gauge action on the whole manifold
$\widehat{Z}$, i.e. there exists a smooth function $\xi^{\prime}\in
L^{2}_{k+1}(\widehat{Z},i\mathbb{R})$ such that

$V=\operatorname{\mathbf{d}}_{\gamma}\xi^{\prime}\text{ on }\widehat{Z}.$

######  Theorem 15.3 (Irreducibiliy of Spinors).

Let $I=(t_{1},t_{2})_{t}\subset\mathbb{R}_{t}$ be an open interval. For any
tame perturbation $\mathfrak{q}\in\mathcal{P}$ and a solution
$\gamma=(A,\Phi)$ to the perturbed Seiberg-Witten equations (15.1) on the
4-manifold $\widehat{Z}=I\times\widehat{Y}$, if the spinor

$\Phi\equiv 0\text{ on }\\{t_{0}\\}\times Y,$

for some $t_{0}\in I$, then $\Phi\equiv 0$ on $\widehat{Z}$.

The proofs of Theorem 15.1-15.3 will not be used elsewhere in this paper. They
will dominate the rest of the section.

####  15.2. A Motivating Problem

To better explain the ideas and point out the difference from the standard
theory [KM07, Section 7], let us first discuss a motivating problem that
concerns the $\bar{\partial}$-operator on the complex plane. Let

$f:\mathbb{C}_{z}\to\mathbb{C}$

be a holomorphic function and $z=t+is$ be the complex coordinate of the
domain. It is well-known that if $f$ vanishes along the interval
$\\{0\\}\times[0,1]_{s}$, then $f\equiv 0$ over $\mathbb{C}_{z}$.

We investigate a class of perturbations of the $\bar{\partial}-$operator. The
equation $\bar{\partial}f=0$ can be formally cast into an evolution equation:

$\partial_{t}f=-D(f)$

where $D(f)=i\partial_{s}f$ is a self-adjoint operator on
$L^{2}(\mathbb{R}_{s},\mathbb{C})$ (although we do not assume $f(t)\in
L^{2}(\mathbb{R}_{s},\mathbb{C})$ for any time slice
$\\{t\\}\times\mathbb{R}_{s}$). Consider a smooth function
$K_{1}:\mathbb{R}_{s}\times\mathbb{R}_{s}\to\mathbb{C}$ with

$\text{supp}\ K_{1}\subset[0,1]_{s}\times[0,1]_{s}$

and form the convolution operator

$\displaystyle K:C^{\infty}(\mathbb{R}_{s},\mathbb{C})$ $\displaystyle\to
C^{\infty}(\mathbb{R}_{s},\mathbb{C})$ $\displaystyle f$ $\displaystyle\mapsto
K(f)(s)=\int_{\mathbb{R}}K_{1}(s,s^{\prime})f(s^{\prime})ds^{\prime}.$

Then $D_{K}\colonequals D+K$ is a compact perturbation of $D$, not necessarily
self-adjoint anymore. More generally, let $V:\mathbb{C}_{z}\to\mathbb{C}$ be
any smooth function and consider the equation

(15.3) $\partial_{t}f=-D_{K}(f)-V\cdot f\text{ on
}\mathbb{C}=\mathbb{R}_{t}\times\mathbb{R}_{s}.$

The potential $V$ can be viewed as a time-dependent perturbation of $D_{K}$.

######  Proposition 15.4.

Suppose $f\in C^{\infty}(\mathbb{C}_{z},\mathbb{C})$ is a solution to the
perturbed $\bar{\partial}$-equation (15.3) and $f(z)=0$ for any
$z\in\\{0\\}\times[0,1]_{s}$, then $f\equiv 0$ on $\mathbb{C}_{z}$.

######  Remark 15.5.

If we only assume $f\equiv 0$ on $\\{0\\}\times[\epsilon,1]_{s}$ for some
small $\epsilon>0$, then for some kernel $K_{1}$ and potential $V$, the
conclusion fails. Indeed, set $f(t,s)\equiv g(s)$ and $V\equiv 0$. Let $g$ be
a cut-off function such that

$g(s)\equiv 0,\forall s\geq\epsilon\text{ and }g(s)\equiv 1,\forall
s<\epsilon/2.$

Then one can find $K_{1}$ with $K_{1}*g=-D(g)=-i\partial_{s}g$, so $g\in\ker
D_{K}$. ∎

The problem here is that the convolution operator $K$ is not local: even if a
function $g:[0,1]_{s}\to\mathbb{C}$ is supported on a small interval
$[0,\epsilon]\subset[0,1]_{s}$, $K(g)=K_{1}*g$ might be non-vanishing on a
much larger region. This is the analogue of the tame perturbation
$\mathfrak{q}$ in the Seiberg-Witten equations.

The proofs of Theorem 15.1-15.3 are modeled on that of Proposition 15.4, which
involves Carleman estimates, as we discuss in the next subsection.

####  15.3. Carleman Estimates

There are two classical ways to prove a strong unique continuation property
like Proposition 15.4. The first follows Agmon and Nirenberg [AN67] and relies
on a differential inequality. This is the approach adopted in the book [KM07,
Section 7]. In this paper, we follow the second strategy and base our works on
Carleman estimates [Car39]. The primary result that we consult is [Kim95,
Theorem 1].

Let us first state a result in an abstract Hilbert space.

######  Proposition 15.6.

Let $H$ be a Hilbert space and $L_{i}:H\to H,\ i=1,2$ be (unbounded) self-
adjoint operators on $H$ satisfying the relation

(15.4) $(L_{1}+rL_{2}+\alpha)^{2}-rL_{2}\geq 0$

for any $r>0$ and $\alpha>\alpha_{0}(H,L_{1},L_{2})$; or equivalently,

(15.5) $\|(L_{1}+rL_{2}+\alpha)v\|_{H}^{2}-\operatorname{Re}\langle
v,(rL_{2})v\rangle\geq 0\ \forall v\in D(L_{1})\cap D(L_{2}).$

Here, $\alpha_{0}>0$ is a fixed large number depending only on $H$, $L_{1}$
and $L_{2}$.

Suppose $w:[0,r_{0}]_{r}\to D(L_{1})\cap D(L_{2})$ is a smooth function such
that

  * •

for a constant $C_{0}>0$, the following estimate holds for any
$r\in(0,r_{0}]$:

(15.6) $\|(\partial_{r}+\frac{1}{r}\cdot L_{1}+L_{2})w(r)\|_{H}\leq
C_{0}\|w(r)\|_{H};$

  * •

$w(r)$ vanishes at the origin to the infinite order, i.e
$(\partial_{r}^{n}w)(0)=0$ for any $n\geq 0$. In practice, we will only need
the property that

(15.7) $\|w(r)\|_{H},\|\partial_{r}w(r)\|_{H}=\mathcal{O}(r^{n})\text{ as
}r\to 0,$

for any $n\geq 1$.

Then $w\equiv 0$.

With loss of generality, we assume $r_{0}=1$ and let $x\colonequals\ln
r\in(-\infty,0]$. Then the inequality (15.6) becomes

(15.8) $\displaystyle g(x)$
$\displaystyle\colonequals(\partial_{x}+L_{1}+e^{x}L_{2})w(x),$
$\displaystyle\|g(x)\|_{H}$ $\displaystyle\leq C_{0}e^{x}\|w(x)\|_{H},\
\forall x\in(-\infty,0].$

The key ingredient is the Carleman estimate. We follow the idea from [AB80].
For any $\epsilon\in(0,1)$, consider the weight function
$\varphi:(-\infty,0]\to\mathbb{R}_{+}$ implicitly determined by the relation
$-\varphi(x)+\exp(-\epsilon\varphi(x))=x$, so $\varphi(x)\sim-x$ and

(15.9) $\displaystyle\partial_{x}\varphi(x)$
$\displaystyle=-\frac{1}{1+\epsilon
e^{-\epsilon\varphi(x)}}\in(-1,-\frac{1}{2}),$ (15.10)
$\displaystyle\partial_{x}^{2}\varphi(x)$
$\displaystyle=\frac{\epsilon^{2}e^{-\epsilon\varphi(x)}}{(1+\epsilon
e^{-\epsilon\varphi(x)})^{3}}\geq C_{1}\epsilon^{2}\cdot e^{2\epsilon x},$

for a constant $C_{1}>0$. In what follows, we will always treat
$\epsilon\in(0,1)$ as a fixed constant.

######  Proposition 15.7 (Carleman Estimates, [Kim95] Theorem 1).

Under the assumptions of Proposition 15.6, for any $\epsilon\in(0,1)$, there
is a constant $C(\epsilon)>0$ such that for any $\tau>2\alpha_{0}$ and $u\in
C_{c}^{\infty}((-\infty,0),D(L_{1})\cap D(L_{2}))$, we have

$\tau\int_{(-\infty,0)}\|e^{\tau\varphi(x)+\epsilon x}u(x)\|_{H}^{2}dx\leq
C(\epsilon)\int_{(-\infty,0)}\|e^{\tau\varphi(x)}(\partial_{x}+L_{1}+e^{x}L_{2})u(x)\|_{H}^{2}dx.$

This estimate is uniform in $\tau$.

###### Proof of Proposition 15.6.

Fix some $x_{0}<0$. To apply Carleman estimates, choose a cut-off function
$\chi:(-\infty,0]_{x}\to[0,1]$ such that $\chi(x)\equiv 1$ when $x<x_{0}$ and
$\chi(0)=0$. Set $u(x)=\chi(x)w(x)$. The function $u(x)$ is not compactly
supported on $(-\infty,0)$, but its decay is faster than any exponential
function as $x\to-\infty$, by (15.7). In this case, Proposition 15.7 still
applies, cf. Remark 15.8; so

$\displaystyle\frac{\tau}{2C(\epsilon)}\int_{(-\infty,x_{0}]}$
$\displaystyle\|e^{\tau\varphi(x)+\epsilon
x}w(x)\|_{H}^{2}dx\leq\frac{\tau}{2C(\epsilon)}\int_{(-\infty,0]}\|e^{\tau\varphi(x)+\epsilon
x}u(x)\|_{H}^{2}dx,$
$\displaystyle\leq\frac{1}{2}\int_{(-\infty,0]}\|e^{\tau\varphi(x)}(\partial_{x}+L_{1}+e^{x}L_{2})u(x)\|_{H}^{2}dx,$
$\displaystyle\leq\int_{(-\infty,0]}\|e^{\tau\varphi(x)}g(x)\|_{H}^{2}dx+\int_{[x_{0},0]}\|e^{\tau\varphi(x)}[\partial_{x},\chi(x)]w(x)\|_{H}^{2}dx,$
$\displaystyle(\text{by }\eqref{E20.16})$ $\displaystyle\leq
C_{0}\int_{(-\infty,0]}\|e^{\tau\varphi(x)+x}w(x)\|_{H}^{2}dx+C_{2}e^{\tau\varphi(x_{0})}\int_{[x_{0},0]}\|w\|_{H}^{2}dx,$

where $C_{2}=\|\partial_{x}\chi\|_{\infty}^{2}$. The upshot is that this
inequality holds for any $\tau\gg 1$, so when $\tau>4C_{0}C(\epsilon)$, we use
the rearrangement argument to derive that

$\frac{\tau}{2}\int_{(-\infty,x_{0}]}\|e^{\tau\varphi(x)+\epsilon
x}w(x)\|_{H}^{2}dx\leq
2C(\epsilon)C_{2}e^{\tau\varphi(x_{0})}\int_{[x_{0},0]}\|w\|_{H}^{2}dx.$

Let $\tau\to\infty$. We conclude that $w(x)\equiv 0$ when $x<x_{0}$. Since
$x_{0}<0$ is arbitrary, $w\equiv 0$ on $(-\infty,0]$. ∎

To complete the proof of Proposition 15.6, it remains to prove Carleman
estimates.

###### Proof of Proposition 15.7.

It is essentially the same argument as [Kim95, Theorem 1]. We record the proof
here because a slight modification will be made in our actual applications.
Set $v(x)\colonequals e^{\tau\varphi(x)}u(x)$, then

$e^{\tau\varphi(x)}(\partial_{x}+L_{1}+e^{x}L_{2})e^{-\tau\varphi(x)}v(x)=(\partial_{x}+L_{1}+e^{x}L_{2}+\tau(-\partial_{x}\varphi(x)))v(x).$

Define $L(x)\colonequals L_{1}+e^{x}L_{2}+\tau(-\partial_{x}\varphi(x))$ and
compute

$\displaystyle\int_{(-\infty,0]}\|(\partial_{x}+L(x))v(x)\|_{H}^{2}dx$
$\displaystyle=\int_{(-\infty,0]}\|\partial_{x}v(x)\|_{H}^{2}+\|L(x)v(x)\|_{H}^{2}$
$\displaystyle\qquad+\int_{(-\infty,0]}2\operatorname{Re}\langle\partial_{x}v(x),L(x)v(x)\rangle_{H}dx.$

Using the fact that $L(x):H\to H$ is a self-adjoint operator, we integrate by
parts:

(15.11)
$\displaystyle\int_{(-\infty,0]}2\operatorname{Re}\langle\partial_{x}v(x),L(x)v(x)\rangle_{H}=-\int_{(-\infty,0]}\operatorname{Re}\langle
v(x),(\partial_{x}L(x))v(x)\rangle$ $\displaystyle=$
$\displaystyle\int_{(-\infty,0]}\operatorname{Re}\langle
v(x),e^{x}(-L_{2})v(x)\rangle+\tau\int_{(-\infty,0]}\langle
v(x),(\partial_{x}^{2}\varphi(x))v(x)\rangle$ $\displaystyle(\text{by
}\eqref{E20.4})\geq$ $\displaystyle\int_{(-\infty,0]}\operatorname{Re}\langle
v(x),e^{x}(-L_{2})v(x)\rangle+C_{1}\epsilon^{2}\tau\int_{(-\infty,0]}\|e^{\epsilon
x}v(x)\|_{H}^{2}.$

Set $\alpha=\tau(-\partial_{x}\varphi)$. If $\tau>2\alpha_{0}$, then by
(15.9), $\alpha=\tau(-\partial_{x}\varphi)>\tau/2>\alpha_{0}.$ Now we use the
relation (15.4) to conclude that

$\int_{(-\infty,0]}\|(\partial_{x}+L(x))v(x)\|_{H}^{2}dx\geq
C_{1}\epsilon^{2}\tau\int_{(-\infty,0]}\|e^{\epsilon x}v(x)\|_{H}^{2}$

for any $\tau>2\alpha_{0}$ and $\epsilon\in(0,1)$. ∎

######  Remark 15.8.

When $u(x):(-\infty,0)\to D(L_{1})\cap D(L_{2})$ is not compactly supported
and yet $u(0)=0$, we have to verify that the boundary term in (15.11)
vanishes:

$\lim_{x\to-\infty}\operatorname{Re}\langle v(x),L(x)v(x)\rangle.$

Then one may assume that $\|u(x)\|_{H},\ \|\partial_{x}u(x)\|_{H}$ and
$\|(L_{1}+e^{x}L_{2})u(x)\|_{H}$ decay faster than any exponential functions
as $x\to-\infty$. In Proposition 15.6, this is guaranteed by (15.8) and
(15.7). ∎

####  15.4. Applications

In this subsection, we give a few examples of $(H,L_{1},L_{2})$ for which the
assumption (15.5) is fulfilled and derive Proposition 15.4 from the abstract
Proposition 15.6. We will work out the Seiberg-Witten equations in the next
subsection.

######  Lemma 15.9.

If self-adjoint operators $L_{1},L_{2}:H\to H$ anti-commute, i.e.

$\\{L_{1},L_{2}\\}\colonequals L_{1}L_{2}+L_{2}L_{1}=0,$

then the condition (15.5) holds.

###### Proof.

We rewrite the left hand side of (15.5) as

$\|(L_{1}+(1-\frac{1}{2\alpha})rL_{2}+\alpha)v\|_{H}^{2}+(1-(1-\frac{1}{2\alpha})^{2})\|(rL_{2})v\|_{H}^{2}+\frac{\operatorname{Re}\langle
L_{1}v,(rL_{2})v\rangle}{\alpha}\geq 0\text{ if }\alpha>\frac{1}{4}.$

The last term vanishes because $\\{L_{1},L_{2}\\}=0.$ ∎

######  Example 15.10.

The first example is the Dirac operator on $\mathbb{C}_{z}\times\Sigma$ where
$\Sigma=\partial Y$ is a union of 2-tori endowed with a flat metric. We choose
a $spin^{c}\ $connection $A$ on $\mathbb{C}_{z}$ such that

$A=\frac{d}{dt}+\frac{d}{ds}+\check{B}$

for a fixed $spin^{c}\ $connection $\check{B}$ on the surface $\Sigma$.

Using the polar coordinate $(r,\theta)$ on the complex plane, the Dirac
operator $D_{A}^{+}$ can be written as

$D_{A}^{+}=\rho_{4}(dr)(\partial_{r}+\rho_{3}(rd\theta)\cdot(\frac{1}{r}\partial_{\theta}+D_{\check{B}}^{\Sigma}))$

where $D_{\check{B}}^{\Sigma}$ is the Dirac operator associated to $\check{B}$
on the surface. Unlike $\rho_{4}(rd\theta)$,

$\rho_{3}(rd\theta)=\rho_{4}(dr)^{-1}\cdot\rho_{4}(rd\theta)=-\rho_{4}(dr\wedge
rd\theta)$

is a constant bundle map. Proposition 15.6 applies to the operator
$\rho_{4}^{-1}(dr)\cdot D_{A}^{+}$ with

$L_{1}^{\mathbb{D}}=\rho_{3}(rd\theta)\cdot\partial_{\theta},\
L_{2}^{\mathbb{D}}=\rho_{3}(rd\theta)D_{\check{B}}^{\Sigma}.$

and $H=L^{2}(S^{1}\times\Sigma,S^{+})$. Indeed, by Lemma 15.9,
$\\{L_{1}^{\mathbb{D}},L_{2}^{\mathbb{D}}\\}=0$. ∎

######  Example 15.11.

The second example concerns the self-dual operator

$\displaystyle\Omega^{1}(X,i\mathbb{R})$
$\displaystyle\to\Omega^{+}(X,i\mathbb{R}),$ $\displaystyle b$
$\displaystyle\mapsto d^{+}b,$

on the 4-manifold $X=\mathbb{C}_{z}\times\Sigma$. Using the polar coordinate
at the origin $0\in\mathbb{C}_{z}$, we regard $b$ as an 1-form on

$X^{\prime}=[0,r_{0})_{r}\times S^{1}\times\Sigma.$

Suppose $b$ does not contain the $dr$-component and write

$b(r)=b_{1}(r)(rd\theta)+b_{2}(r)$

with $b_{1}(r)\in H_{1}\colonequals L^{2}(S^{1}\times\Sigma,\mathbb{R})$ and
$b_{2}(r)\in H_{2}\colonequals L^{2}(S^{1}\times\Sigma,T^{*}\Sigma)$. As the
metric on $X^{\prime}$ is given by

$dr^{2}+(rd\theta)^{2}+g_{\Sigma},$

the equation $d^{+}b=0$ is equivalent to that

$\partial_{r}\begin{pmatrix}b_{1}(r)\\\
b_{2}(r)\end{pmatrix}+\bigg{[}\frac{1}{r}\begin{pmatrix}1&0\\\
0&*_{\Sigma}\partial_{\theta}\end{pmatrix}+\begin{pmatrix}0&*_{\Sigma}d_{\Sigma}\\\
-*_{\Sigma}d_{\Sigma}&0\end{pmatrix}\bigg{]}\begin{pmatrix}b_{1}(r)\\\
b_{2}(r)\end{pmatrix}=0.$

To apply Proposition 15.6, set $H=H_{1}\oplus H_{2}$ and

$L_{1}^{\mathbb{S}}=\begin{pmatrix}1&0\\\ 0&L_{3}\end{pmatrix},\
L_{2}^{\mathbb{S}}=\begin{pmatrix}0&L_{4}\\\ L_{4}^{*}&0\end{pmatrix},$

with $L_{3}\colonequals*_{\Sigma}\partial_{\theta}:H_{2}\to H_{2}$ and
$L_{4}\colonequals*_{\Sigma}d_{\Sigma}:H_{2}\to H_{1}$. To verify the
condition (15.5), we calculate for each $v=(b_{1},b_{2})\in H$ that

$\displaystyle\|(L_{1}^{\mathbb{S}}+rL_{2}^{\mathbb{S}}+\alpha)v\|_{H}^{2}-\langle
v,rL_{2}^{\mathbb{S}}v\rangle$ $\displaystyle=\|\alpha
b_{1}+rL_{4}b_{2}\|_{H_{1}}^{2}+\|rL_{4}^{*}b_{1}+(L_{3}+\alpha)b_{2}\|_{H_{2}}^{2}$
$\displaystyle\qquad+(2\alpha+1)\|b_{1}\|_{H_{1}}^{2}\geq 0\text{ if
}\alpha>-\frac{1}{2}.$

In this case, Lemma 15.9 is not applicable because the anti-commutator
$\\{L_{1}^{\mathbb{S}},L_{2}^{\mathbb{S}}\\}\neq 0$. ∎

In the proof of Proposition 15.4 below, we will work with operator
$L_{1},L_{2}$ that are not self-adjoint on $H$. Nevertheless, the abstract
Proposition 15.6 still applies, since we can verify the first step of (15.11)
directly: this is the only place the self-adjointness was used.

###### Proof of Proposition 15.4.

Let $I=[0,1]_{s}$. For any $r\geq 0$, consider the contour
$\Gamma_{r}=\Gamma_{r}^{(1)}+\Gamma_{r}^{(2)}+\Gamma_{r}^{(3)}+\Gamma_{r}^{(4)}$
with

$\displaystyle\Gamma_{r}^{(1)}$ $\displaystyle=\\{r\\}\times I,$
$\displaystyle\Gamma_{r}^{(2)}=$
$\displaystyle\\{i+re^{i\theta}:0\leq\theta\leq\pi\\},$
$\displaystyle\Gamma_{r}^{(3)}$ $\displaystyle=\\{-r\\}\times I,$
$\displaystyle\Gamma_{r}^{(4)}=$
$\displaystyle\\{re^{i\theta}:\pi\leq\theta\leq 2\pi\\},$

and define

$\displaystyle v_{1}(r)$
$\displaystyle=f|_{\Gamma_{r}^{(1)}\coprod\Gamma_{r}^{(3)}}\in
H_{1}\colonequals L^{2}(I\coprod(-I),\mathbb{C}),$ $\displaystyle v_{2}(r)$
$\displaystyle=f|_{\Gamma_{r}^{(2)}\coprod\Gamma_{r}^{(4)}}\in
H_{2}\colonequals
L^{2}([0,\pi]_{\theta}\coprod[\pi,2\pi]_{\theta},\mathbb{C}).$

where $(-I)$ stands for the orientation reversal of $I$. Finally, set

$w(r)=(w_{1}(r),w_{2}(r))\colonequals(v_{1}(r),\sqrt{r}v_{2}(r))\in
H\colonequals H_{1}\oplus H_{2}.$

Our assumptions imply that the function $w:[0,1)\to H$ vanishes to the
infinite order at the origin. To apply Proposition 15.6, we look for the
differential equation that governs $w(r)$. As the function $f$ solves the
perturbed $\bar{\partial}$-equation, we have

(15.12) $\partial_{r}w(r)+\big{(}\frac{1}{r}L_{1}+L_{2}\big{)}w(r)=h(r)$

with

$L_{1}=(0,i\partial_{\theta}-\frac{1}{2})\text{ and
}L_{2}=(i\partial_{s},0)\text{ on }H=H_{1}\oplus H_{2}.$

The error term $h(r)$ in (15.12) is determined by the convolution operator $K$
and the smooth potential $V$, so the assumption (15.6) is satisfied in our
case.

Neither $L_{1}$ nor $L_{2}$ is a self-adjoint operator on $H$, but we still
have

(15.13)
$\operatorname{Re}\langle(\frac{1}{r}L_{1}+L_{2})w(r),\partial_{r}w(r)\rangle=\operatorname{Re}\langle
w(r),(\frac{1}{r}L_{1}+L_{2})\partial_{r}w(r)\rangle$

which justifies the equality (15.11) in the proof of Proposition 15.7. Indeed,

$\displaystyle\langle(\frac{1}{r}L_{1}+L_{2})w(r),\partial_{r}w(r)\rangle-\langle
w(r),(\frac{1}{r}L_{1}+L_{2})\partial_{r}w(r)\rangle=\frac{i}{2r}\int_{[0,\pi]\coprod[\pi,2\pi]}\partial_{\theta}|v_{2}(r,\theta)|^{2}d\theta$

is purely imaginary. As the relation

$2\operatorname{Re}\langle L_{1}v,L_{2}v\rangle=0,\forall v\in D(L_{1})\cap
D(L_{2})$

still holds in our case, the proof of Lemma 15.9 remains valid. Now we use
Lemma 15.9 and Proposition 15.6 to complete the proof. ∎

####  15.5. The Seiberg-Witten Equations

Having discussed some toy problems, we are now ready to prove the strong
unique continuation property for the perturbed Seiberg-Witten equations, by
combining Example 15.10 and 15.11.

###### Proof of Theorem 15.1.

With loss of generality, assume $I=[-1,1]$ and $t_{0}=0$. It suffices to show
that $\gamma_{1}$ and $\gamma_{2}$ are gauge equivalent in an open
neighborhood of $\\{0\\}\times Y$, then one may use induction to extend this
neighborhood to the whole space $\widehat{Z}=I\times\widehat{Y}$.

To imitate the proof of Proposition 15.4, consider the closed 3-manifold
$\mathcal{Y}_{r}=\mathcal{Y}_{r}^{(1)}\cup\mathcal{Y}_{r}^{(2)}\cup\mathcal{Y}_{r}^{(3)}$
where

$\displaystyle\mathcal{Y}_{r}^{(1)}$ $\displaystyle\colonequals\\{r\\}\times
Y,$ $\displaystyle\mathcal{Y}_{r}^{(2)}$
$\displaystyle=\\{re^{i\theta}:0\leq\theta\leq\pi\\}\times\Sigma,$
$\displaystyle\mathcal{Y}_{r}^{(3)}$
$\displaystyle\colonequals(-\\{-r\\}\times Y),$ $\displaystyle\forall r$
$\displaystyle\in[0,1].$

Here $\mathcal{Y}_{r}^{(3)}$ is the orientation reversal of $\\{-r\\}\times
Y$. Let $B_{0}$ be the reference $spin^{c}\ $connection on $\widehat{Y}$, so
$B_{0}$ agrees with the $\mathbb{R}_{s}$-invariant connection

$\frac{d}{ds}+\check{B}$

on the cylindrical end $[-1,\infty)_{s}\times\Sigma$. Set
$\gamma_{0}=(B_{0},0)$.

Extend the gauge transformation $u$ constantly in the time direction and
replace $\gamma_{1}$ by $u(\gamma_{1})$. Construct gauge transformations
$u_{i},\ i=1,2$ such that $u_{i}\equiv\operatorname{Id}$ on
$\\{0\\}\times\widehat{Y}$ and $\gamma_{i}^{\prime}\colonequals
u_{i}(\gamma_{i})$ is in the temporal gauge (the $dt$-component vanishes).
Consider the difference

$\delta_{i}(t)\colonequals\gamma_{i}^{\prime}|_{\\{t\\}\times\widehat{Y}}-\gamma_{0}\in
C^{\infty}(\widehat{Y},iT^{*}\widehat{Y}\oplus S)$

Formally, $\delta_{i}$ is subject to an evolution equation:

$\partial_{t}\delta_{i}(t)+L_{2}^{Y}\delta_{i}(t)+\delta_{i}(t)\\#\delta_{i}(t)+\mathfrak{q}(\delta_{i}(t)+\gamma_{0})=c.$

where $\\#$ is a symmetric bilinear form that involves only point-wise
multiplications. Here $c$ is a constant error term determined by $\gamma_{0}$
and

$L_{2}^{Y}=\begin{pmatrix}*_{3}d_{\widehat{Y}}&0\\\ 0&D_{B_{0}}\end{pmatrix}.$

Now take the difference $\delta(t)\colonequals\delta_{2}(t)-\delta_{1}(t)$.
Over the space $[-1,1]_{t}\times Y$, we have

(15.14) $\partial_{t}\delta(t)+L^{Y}_{2}(\delta(t))=h_{1}(t)\in
C^{\infty}(Y,iT^{*}Y\oplus S)$

and $\|h_{1}(t)\|_{L^{2}(Y)}\leq C\|\delta(t)\|_{L^{2}(Y)}$ for a uniform
constant $C>0$. Moreover,

$\partial_{t}^{n}\delta(0)\equiv 0\text{ on }Y\text{ for any }n\geq 0.$

When $n=0$, this follows from the assumption that $\gamma_{1}=\gamma_{2}$ on
$\\{0\\}\times Y$. When $n\geq 1$, this is a consequence of the equation
(15.14) and its higher time derivatives. As a result, all derivatives of
$\delta$ vanish on $\\{0\\}\times Y$.

Set $H_{1}=L^{2}(Y,iT^{*}Y\oplus S)$ and define

$v_{1}(r)=(\delta(r)|_{Y},\delta(-r)|_{Y})\in H_{1}\oplus H_{1}.$

Then $\partial_{r}^{n}v_{1}(0)=0$ for any $n\geq 0$.

To deal with the middle part $\mathcal{Y}_{r}^{(2)}$, consider the polar
coordinate at $0\in\mathbb{C}_{z}$ and restrict $\delta$ to a section of

$iT^{*}X^{\prime}\oplus S\to
X^{\prime}\colonequals[0,1]_{r}\times[0,\pi]_{\theta}\times\Sigma\subset\mathbb{R}_{t}\times\\{s\geq
0\\}\times\Sigma.$

The section $\delta$ is not necessarily in the radial temporal gauge: the
$dr$-component of $\delta$ only vanishes when $\theta=0,\pi$. One has to
construct gauge transformations $u^{\prime}_{i}:X^{\prime}\to S^{1}$ on
$X^{\prime}$ such that
$u_{i}^{\prime}|_{\\{0\\}\times[0,\pi]_{\theta}\times\Sigma}\equiv\operatorname{Id}$
and $u_{i}^{\prime}(\gamma_{i}^{\prime})$ is the radial temporal gauge. Then
we define

$v_{2}(r)=u_{2}^{\prime}(\gamma_{2}^{\prime})(r)-u_{1}^{\prime}(\gamma_{1}^{\prime})(r)\in
H_{2}\colonequals L^{2}([0,\pi]_{\theta}\times\Sigma,i\mathbb{R}\oplus
iT^{*}\Sigma\oplus S).$

Then the path $v_{2}(r)$ is subject to the equation

$\partial_{r}v_{2}(r)+\bigg{(}\frac{1}{r}\begin{pmatrix}L_{1}^{\mathbb{S}}&0\\\
0&L_{1}^{\mathbb{D}}\end{pmatrix}+\begin{pmatrix}L_{2}^{\mathbb{S}}&0\\\
0&L_{2}^{\mathbb{D}}\end{pmatrix}\bigg{)}v_{2}(r)=h_{2}(r)\in H_{2}.$

and $\|h_{2}(r)\|_{H_{2}}\leq C\|v_{2}(r)\|_{H_{2}}$ for a constant $C>0$. The
Seiberg-Witten equations are not perturbed on $X^{\prime}$, so the error term
$h_{2}(r)$ involves only point-wise multiplications with $v_{2}(r)$. Operators
$L_{i}^{\mathbb{S}}$ and $L_{i}^{\mathbb{D}},i=1,2$ are defined as in Example
15.10 and 15.11.

As all derivatives of $\delta$ vanish on $(0,0)\times\Sigma$,
$\partial_{r}^{n}v_{2}(0)=0$ for any $n\geq 0$.

Finally, let $H=(H_{1}\oplus H_{1})\oplus H_{2}$ and define

$w(r)=(w_{1}(r),w_{2}(r))\colonequals(v_{1}(r),\sqrt{r}v_{2}(r))\in H.$

Now the path $w:[0,1)_{r}\to H$ is subject to the equation

(15.15)
$\partial_{r}w(r)+(\frac{1}{r}L_{1}+L_{2})w(r)=(h_{1}(r),-h_{1}(-r),\sqrt{r}h_{2}(r)).$

with

$L_{1}=(0,0,\begin{pmatrix}L_{1}^{\mathbb{S}}&0\\\
0&L_{1}^{\mathbb{D}}\end{pmatrix}-\frac{1}{2}),L_{2}=(L_{2}^{Y},-L_{2}^{Y},\begin{pmatrix}L_{2}^{\mathbb{S}}&0\\\
0&L_{2}^{\mathbb{D}}\end{pmatrix}).$

To apply Proposition 15.6, we have to verify:

  * •

the positivity condition (15.5);

  * •

the symmetry condition (15.13); note that neither $L_{1}$ nor $L_{2}$ is self-
adjoint.

At this point, we have reduced the problem to some formal properties of
$L_{1}$, $L_{2}$ and $w(r)$. We will treat the form component and the spinor
component of (15.15) separately. The verification of (15.5) and (15.13) will
dominate the rest of the proof.

Step 1\. The Form Component and the Self-Dual operators. In this case, the
positivity condition (15.5) follows from the same argument as in Example 15.11
and Proposition 15.4. It can be checked separately on each of
$\mathcal{Y}_{r}^{(i)},\ 1\leq i\leq 3$. As for (15.13), we focus on the
common boundary of $\mathcal{Y}_{r}^{(1)}$ and $\mathcal{Y}_{r}^{(2)}$.
Suppose the form components of $v_{1}(r)$ and $v_{2}(r)$ are given
respectively by

$\displaystyle v_{1}|_{\\{r\\}\times(-1,0]_{s}\times\Sigma}$
$\displaystyle\rightsquigarrow a_{1}ds+a_{2},$ $\displaystyle a_{1}(r)\in
C^{\infty}((-1,0]_{s}\times\Sigma,i\mathbb{R}),$ $\displaystyle a_{2}(r)\in
C^{\infty}((-1,0]_{s}\times\Sigma,iT^{*}\Sigma),$ $\displaystyle v_{2}$
$\displaystyle\rightsquigarrow b_{1}(rd\theta)+b_{2},$ $\displaystyle
b_{1}(r)\in C^{\infty}([0,\pi]_{\theta}\times\Sigma,i\mathbb{R}),$
$\displaystyle b_{2}(r)\in
C^{\infty}([0,\pi]_{\theta}\times\Sigma,iT^{*}\Sigma).$

Near the boundary of $\mathcal{Y}_{r}^{(1)}$, we have

$(*_{3}d_{Y})\begin{pmatrix}a_{1}(r)\\\
a_{2}(r)\end{pmatrix}=\begin{pmatrix}0&*_{\Sigma}d_{\Sigma}\\\
-*_{\Sigma}d_{\Sigma}&*_{\Sigma}\partial_{s}\end{pmatrix}\begin{pmatrix}a_{1}(r)\\\
a_{2}(r)\end{pmatrix}.$

Then we calculate (the operator $L_{2}^{\mathbb{S}}$ is ignored here as it is
always self-adjoint):

$\displaystyle\langle(*_{3}d_{Y})v_{1},(\partial_{r}v_{1})\rangle_{\\{r\\}\times
Y}-\langle v_{1},(*_{3}d_{Y})(\partial_{r}v_{1})\rangle_{\\{r\\}\times Y}$
$\displaystyle=\langle*_{\Sigma}a_{2}(r,0),(\partial_{r}a_{2})(r,0)\rangle_{(r,0)\times\Sigma}.$
$\displaystyle\langle\frac{1}{r}L_{1}^{\mathbb{S}}w_{2},(\partial_{r}w_{2})\rangle_{H_{2}}-\langle
w_{2},\frac{1}{r}L_{1}^{\mathbb{S}}(\partial_{r}w_{2})\rangle_{H_{2}}$
$\displaystyle=-\langle*_{\Sigma}b_{2}(r,0),(\partial_{r}b_{2})(r,0)\rangle_{(r,0)\times\Sigma}$
$\displaystyle\qquad+\frac{1}{r}\underbrace{\int_{[0,\pi]_{\theta}}\partial_{\theta}\langle*_{\Sigma}b_{2},b_{2}\rangle_{\Sigma}}_{=0}+\cdots.$

It remains to verify that $a_{2}(r,0)=b_{2}(r,0)$ on
$\mathcal{Y}_{r}^{(1)}\cap\mathcal{Y}_{r}^{(2)}$. Suppose the restriction of
the form component of $\delta$ on
$X^{\prime}=[0,1]_{r}\times[0,\pi]_{\theta}\times\Sigma$ is
$fdr+c_{1}(rd\theta)+c_{2}$ with

$\displaystyle f(r),c_{1}(r)\in
C^{\infty}([0,\pi]_{\theta}\times\Sigma,i\mathbb{R}),\ c_{2}(r)\in
C^{\infty}([0,\pi]_{\theta}\times\Sigma,iT^{*}\Sigma).$

It is clear that $a_{2}$ and $c_{2}$ agree along the common boundary of
$\mathcal{Y}_{r}^{(1)}$ and $\mathcal{Y}_{r}^{(2)}$. Moreover,

$f(r,\theta)\equiv 0\text{ if }\theta=0\text{ or }\pi.$

To put $\delta$ into radial temporal gauge, we applied further gauge
transformations, so $(b_{1},b_{2})$ is related to $\delta$ by the formulae:

$\displaystyle b_{1}(r)$
$\displaystyle=c_{1}(r)-\frac{1}{r}\int_{0}^{r}(\partial_{\theta}f)(r^{\prime})dr^{\prime},$
$\displaystyle b_{2}(r)$
$\displaystyle=c_{2}(r)-\int_{0}^{r}(d_{\Sigma}f)(r^{\prime})dr^{\prime}.$

As a result, $a_{2}(r,s)|_{s=0}=b_{2}(r,\theta)|_{\theta=0}$. This equality
does not a priori hold for $a_{1}$ and $b_{1}$, but it is not needed in the
proof.

Step 2\. The Spinor Component and the Dirac operators. The proof of (15.13)
proceeds in the same way as in Step 1\. We focus on the positivity condition
(15.5). Suppose the spinor components of $v_{i}(r),\ 1\leq i\leq 3$ are given
respectively by

$\displaystyle v_{1}|_{\\{r\\}\times Y}$
$\displaystyle\rightsquigarrow\Phi_{1}(r)\in C^{\infty}(Y,S),$ $\displaystyle
v_{2}$ $\displaystyle\rightsquigarrow\Phi_{2}(r)\in
C^{\infty}([0,\pi]_{\theta}\times\Sigma,S),$ $\displaystyle
v_{3}|_{\\{r\\}\times Y}$ $\displaystyle\rightsquigarrow\Phi_{3}(r)\in
C^{\infty}(Y,S).$

We focus on sections

$(\Phi_{1}(r),\Phi_{3}(r),\sqrt{r}\Phi_{2}(r))\in L^{2}(Y,S)\oplus
L^{2}(Y,S)\oplus L^{2}([0,\pi]_{\theta}\times\Sigma,S)$

and operators:

$\frac{1}{r}\begin{pmatrix}0&0&0\\\ 0&0&0\\\
0&0&L_{1}^{\mathbb{D}}-\frac{1}{2}\end{pmatrix}+\begin{pmatrix}D_{B_{0}}&0&0\\\
0&-D_{B_{0}}&0\\\ 0&0&L_{2}^{\mathbb{D}}\end{pmatrix}.$

Unlike Example 15.10, $L^{\mathbb{D}}_{1}$ is not self-adjoint in this case.
In general,

$2\operatorname{Re}\langle
L^{\mathbb{D}}_{1}v,L^{\mathbb{D}}_{2}v\rangle_{L^{2}([0,\pi]_{\theta}\times\Sigma)}=\int_{\\{\theta\\}\times\Sigma}\langle
v,D_{\check{B}}^{\Sigma}v\rangle\ \bigg{|}_{\theta=0}^{\theta=\pi}\neq 0,\
v\in L^{2}([0,\pi]_{\theta},S).$

Let $v=\sqrt{r}\Phi_{2}(r)$ and follow the proof of Lemma 15.9:

$\displaystyle\|(L_{1}^{\mathbb{D}}+rL_{2}^{\mathbb{D}}+(\alpha-\frac{1}{2}))\sqrt{r}\Phi_{2}\|^{2}_{L^{2}([0,\pi]_{\theta}\times\Sigma)}-\operatorname{Re}\langle\sqrt{r}\Phi_{2},(rL_{2}^{\mathbb{D}})\sqrt{r}\Phi_{2}\rangle$
$\displaystyle\geq$
$\displaystyle\frac{2r^{2}}{2\alpha-1}\operatorname{Re}\langle
L_{1}^{\mathbb{D}}\Phi_{2},L_{2}^{\mathbb{D}}\Phi_{2}\rangle_{L^{2}([0,\pi]_{\theta}\times\Sigma)}$
$\displaystyle=$
$\displaystyle\frac{r^{2}}{2\alpha-1}\bigg{(}\int_{\\{\pi\\}\times\Sigma}\langle\Phi_{2},D_{\check{B}}^{\Sigma}\Phi_{2}\rangle-\int_{\\{0\\}\times\Sigma}\langle\Phi_{2},D_{\check{B}}^{\Sigma}\Phi_{2}\rangle\bigg{)}.$

Just as in Step 1, sections $\Phi_{1}$ and $\Phi_{2}$ have the same boundary
value along $\mathcal{Y}_{r}^{(1)}\cap\mathcal{Y}_{r}^{(2)}$:

$\Phi_{1}(r,s)\big{|}_{s=0}=\Phi_{2}(r,\theta)\big{|}_{\theta=0}.$

Therefore, it remains to verify the inequality:

$\|(rD_{B_{0}}+\alpha)\Phi_{1}\|^{2}_{L^{2}(Y)}-\operatorname{Re}\langle\Phi_{1},(rD_{B_{0}})\Phi_{1}\rangle\geq\frac{r^{2}}{2\alpha-1}\int_{\\{0\\}\times\Sigma}\langle\Phi_{1},D_{\check{B}}^{\Sigma}\Phi_{1}\rangle.$

The left hand side can be rewritten as

$(1-\frac{1}{2\alpha-1})\|(rD_{B_{0}}+\frac{(2\alpha-1)^{2}}{4\alpha-4})\Phi_{1}\|^{2}_{L^{2}(Y)}+\frac{r^{2}}{2\alpha-1}\|D_{B_{0}}\Phi_{1}\|^{2}_{L^{2}(Y)}+\frac{4\alpha^{2}-6\alpha+1}{8\alpha-8}\|\Phi_{1}\|^{2}_{L^{2}(Y)}.$

Using the Weitzenböck formula [KM07, (4,15)], the last two terms are bounded
below by

$\displaystyle\frac{r^{2}}{2\alpha-1}\bigg{(}\|\nabla_{B_{0}}\Phi_{1}\|^{2}_{L^{2}(Y)}+\int_{\\{0\\}\times\Sigma}\langle\Phi_{1},D_{\check{B}}^{\Sigma}\Phi_{1}\rangle+\int_{Y}\frac{s}{4}|\Phi_{1}|^{2}+\langle\Phi_{1},\frac{1}{2}\rho_{3}(F_{B_{0}^{t}})\Phi_{1}\rangle\bigg{)}$
$\displaystyle\qquad+\frac{\alpha-1}{2}\|\Phi_{1}\|^{2}_{L^{2}(Y)}\geq\frac{r^{2}}{2\alpha-1}\int_{\\{0\\}\times\Sigma}\langle\Phi_{1},D_{\check{B}}^{\Sigma}\Phi_{1}\rangle+\frac{\alpha-\alpha_{0}}{2}\|\Phi_{1}\|^{2}_{L^{2}(Y)}.$

Then we take $\alpha>\alpha_{0}\colonequals
e^{100}\max\\{\|s\|_{\infty},\|F_{B_{0}^{t}}\|_{\infty},1\\}$.

The common boundary $\mathcal{Y}_{r}^{(2)}\cap\mathcal{Y}_{r}^{(3)}$ is dealt
with similarly. Hence, the positivity condition (15.5) holds when
$\alpha>\alpha_{0}$. Now we use Proposition 15.6 and 15.7 to complete the
proof. ∎

####  15.6. Irreducibility of Spinors

We accomplish the proof of Theorem 15.3 in this subsection, following the idea
above. The spinor part of the equation (15.1) is cast into the form

$\frac{d}{dt}\Psi(t)+D_{B(t)}\Psi(t)+\mathfrak{q}^{1}(B(t),\Psi(t))=0.$

where $\mathfrak{q}^{1}$ is the spinor part of the perturbation
$\mathfrak{q}=(\mathfrak{q}^{0},\mathfrak{q}^{1})$. As
$\mathfrak{q}^{1}(B(t),0)\equiv 0$, we have

$\displaystyle\|\mathfrak{q}^{1}(B(t),\Psi(t))\|_{2}$
$\displaystyle=\|\mathfrak{q}^{1}(B(t),\Psi(t))-\mathfrak{q}^{1}(B(t),0)\|_{2}$
$\displaystyle=\int_{0}^{1}\|\mathcal{D}_{(B(t),r\Psi(t))}\mathfrak{q}^{1}(\Psi(t))\|_{2}dr\leq
C\|\Psi(t)\|_{L^{2}(Y)},$

for a constant $C>0$ and any $t\in[t_{0}-\epsilon,t_{0}+\epsilon]$. Now the
proof of Theorem 15.1 can proceed with no difficulty.

####  15.7. The Linearized Version

In this subsection, we accomplish the proof of Theorem 15.2. To some extent,
it suffices to “linearize” each step of the proof of Theorem 15.1. Again,
assume $I=[-1,1]_{t}$ and $t_{0}=0$.

$\xi^{(1)}(t)=\xi-\int_{0}^{t}\delta c(t^{\prime})dt^{\prime}\in
C^{\infty}(\widehat{Z},i\mathbb{R}),$

and set $V_{1}=V-\operatorname{\mathbf{d}}_{\gamma}\xi^{(1)}$. This new
section $V_{1}$ is smooth, and

$\displaystyle V_{1}(t)$ $\displaystyle=(0,\delta
b_{1}(t),\delta\psi_{1}(t))\in L^{2}_{k}(Z,iT^{*}Z\oplus S),$ $\displaystyle
V_{1}(0)$ $\displaystyle=0\text{ on }\\{0\\}\times Y.$

As $\gamma$ solves the non-linear equation (15.1),
$\textbf{d}_{\gamma}f^{(1)}$ is a solution to the linear equation (15.2), and
so is $V_{1}$. The equation (13.2) is formally an evolutionary equation on
$I\times Y$:

(15.16) $\frac{d}{dt}\begin{pmatrix}\delta b_{1}(t)\\\
\delta\psi_{1}(t)\end{pmatrix}+\begin{pmatrix}*_{3}d_{Y}&0\\\
0&D_{B_{0}}\end{pmatrix}\begin{pmatrix}\delta b(t)\\\
\delta\psi(t)\end{pmatrix}=\eta(t)\begin{pmatrix}\delta b(t)\\\
\delta\psi(t)\end{pmatrix},t\in\mathbb{R}.$

where $\eta(t):L^{2}(Y)\to L^{2}(Y)$ is a family of bounded linear operators
determined by $\check{\gamma}(t)$.

To borrow the proof of Theorem 15.2, we focus on $\mathcal{Y}_{r}^{(2)}$.
Using polar coordinates, we write

$V_{1}(r)=(\delta c_{1}^{\prime}(r),\delta
b_{1}^{\prime}(r),\delta\psi_{1}^{\prime}(r))\in
C^{\infty}(X^{\prime},i\mathbb{R}\oplus
iT^{*}([0,\pi]_{\theta}\times\Sigma)\oplus S),$

on
$X^{\prime}=[0,1]_{r}\times[0,\pi]_{\theta}\times\Sigma\subset\mathbb{H}^{2}_{+}\times\Sigma$.
To put $V_{1}(r)$ into radial temporal gauge, consider the function

$f^{(2)}(r)=-\int_{0}^{r}\delta c_{1}^{\prime}(r^{\prime})dr^{\prime}\text{ on
}X^{\prime}.$

Then $f^{(2)}(r,\theta)\equiv 0$ when $\theta=0,\pi$, and the section
$V_{1}-\textbf{d}_{\gamma}f^{(2)}$ solves the linear equation (15.2) on $X$.
The proof of Theorem 15.1 is now applicable. We conclude that

(15.17) $\displaystyle V_{1}(t)$ $\displaystyle\equiv 0\text{ on }I\times Y$
$\displaystyle V_{1}-\textbf{d}_{\gamma}f^{(2)}$ $\displaystyle\equiv 0\text{
on }X^{\prime}.$

We extend $f^{(2)}$ by zero over the product $I\times Y$. One might worry that
$f^{(2)}$ does not form a smooth function on the union

$(I\times Y)\bigcup X^{\prime},$

as we pointed out in Step 1 in the proof of Theorem 15.2. However, once the
unique continuation property is established, the smoothness of $f^{(2)}$
follows from (15.17) and the smoothness of $V_{1}$. As a result,

$V_{1}=\textbf{d}_{\gamma}f^{(2)}\text{ on }(I\times Y)\bigcup X^{\prime}.$

By induction, we can extend the region where this equality holds. This
completes the proof of Theorem 15.2.

###  16\. Transversality

With all machinery developed so far, we are ready to prove the transversality
result on the cylinder $\mathbb{R}_{t}\times\widehat{Y}$ in this section. Here
is the main result:

######  Theorem 16.1.

For any relative $spin^{c}\ $manifold $(\widehat{Y},\widehat{\mathfrak{s}})$
satisfying constraints in the strict cobordism category
$\operatorname{Cob}_{s}$, one can find an admissible perturbation
$\mathfrak{q}\in\mathcal{P}(\widehat{Y},\widehat{\mathfrak{s}})$, in the sense
of Definition 13.3. Here $\mathcal{P}(\widehat{Y},\widehat{\mathfrak{s}})$ is
the Banach space of tame perturbations constructed Subsection 8.5.

Pick an admissible perturbation $\mathfrak{q}(\widehat{\mathfrak{s}})$ for
each relative $spin^{c}\ $structure $\widehat{\mathfrak{s}}$ on $Y$. By
putting them altogether, we obtain an object
$\mathbb{Y}=(Y,\psi,g_{Y},\omega,\mathfrak{q})$ in the category
$\operatorname{Cob}_{s}$: the property (P8) is fulfilled. In this case, the
moduli spaces $\mathcal{M}_{[\gamma]}(\mathfrak{a},\mathfrak{b})$ defined in
Section 13 will become a smooth manifold, and the Floer homology of
$(\mathbb{Y},\widehat{\mathfrak{s}})$ will be defined in Part VI.

Theorem 16.1 is a formal consequence of the unique continuation properties,
Theorem 15.1-15.3 and the separating properties of cylinder functions, Theorem
8.20. The transversality result for a general morphism
$\mathbb{X}:(\mathbb{Y},\widehat{\mathfrak{s}}_{1})\to(\mathbb{Y}_{2},\widehat{\mathfrak{s}}_{2})$
in the category $\operatorname{SCob}_{s}$ is proved in Subsection 16.3, cf.
Theorem 16.5.

####  16.1. Transversality for the 3-Dimensional Equations

Consider the Banach space of perturbations $\mathcal{P}$ and a tame
perturbation $\mathfrak{q}=\operatorname{grad}f\in\mathcal{P}$. We start with
the first condition (E1) in Definition 13.3 which concerns the 3-dimensional
equation

$\operatorname{grad}\hbox
to0.0pt{$-$\hss}\mathcal{L}_{\omega}(\mathfrak{a})=0,$

Recall from Definition 12.2 that a critical point
$\mathfrak{a}\in\mathcal{C}_{k}(\widehat{Y},\widehat{\mathfrak{s}})$ of $\hbox
to0.0pt{$-$\hss}\mathcal{L}_{\omega}=\mathcal{L}_{\omega}+f$ is called non-
degenerate if the extended Hessian at $\mathfrak{a}$

$\widehat{\operatorname{Hess}}_{\mathfrak{q},\mathfrak{a}}$

is invertible. In fact, this is a generic condition for a perturbation
$\mathfrak{q}\in\mathcal{P}$.

######  Theorem 16.2 (cf. [KM07] Theorem 12.1.12).

There is a residue (and in particular non-empty) subset of $\mathcal{P}$ such
that for every $\mathfrak{q}$ in this subset, any critical point
$\mathfrak{a}\in\operatorname{Crit}(\hbox
to0.0pt{$-$\hss}\mathcal{L}_{\omega})$ is non-degenerate. For such a
perturbation, $\operatorname{Crit}(\hbox
to0.0pt{$-$\hss}\mathcal{L}_{\omega})$ comprises a finite collection of gauge
orbits.

###### Proof.

The proof follows the same argument as in [KM07, Section 12.5] with one slight
modification, as we explain now. Suppose for some $\mathfrak{q}\in\mathcal{P}$
and $\mathfrak{a}\in\operatorname{Crit}{\hbox
to0.0pt{$-$\hss}\mathcal{L}_{\omega}}$, the tangent vector $v=(0,\delta
b,\delta\psi)\neq 0$ lies in the kernel of
$\widehat{\operatorname{Hess}}_{\mathfrak{q},\mathfrak{a}}$:

(16.1) $(0,\delta
b,\delta\psi)\in\ker\widehat{\operatorname{Hess}}_{\mathfrak{q},\mathfrak{a}}.$

We have to show that $v$ is separated by a cylinder function. To apply
Proposition 8.6. we verify that $v$ is not generated by the infinitesimal
gauge action on $Y$. Suppose on the contrary that

(16.2) $(\delta
b,\delta\psi)=\operatorname{\mathbf{d}}_{\mathfrak{a}}\xi\text{ on }Y$

for some $\xi\in L^{2}_{k+1}(\widehat{Y},i\mathbb{R})$, then by the unique
continuation property of tangent vectors, Theorem 15.2, for a possibly
different function $\xi^{\prime}\in L^{2}_{k+1}(\widehat{Y},\mathbb{R})$, the
equation (16.2) holds on $\widehat{Y}$:

$(\delta b,\delta\psi)=\operatorname{\mathbf{d}}_{\mathfrak{a}}\xi^{\prime}.$

By (16.1), $\operatorname{\mathbf{d}}_{\mathfrak{a}}^{*}(\delta
b,\delta\psi)=0$ , so $(\delta b,\delta\psi)$ is $L^{2}$-orthogonal to the
subspace $\mathcal{J}_{k,\mathfrak{a}}\subset\mathcal{T}_{k,\mathfrak{a}}$.
This implies that $v=0$, which a contradiction. Alternatively, we may apply
the linearized version of [KM07, Theorem 7.2.1] on the 4-manifold

$S^{1}\times\widehat{Y},$

which possesses a cylindrical end $S^{1}\times[0,\infty)_{s}\times\Sigma$. Now
we use Proposition 8.6 to find a cylinder function
$f\in\operatorname{Cylin}(Y)$ supported on $Y\subset\widehat{Y}$ such that

$df(v)\neq 0.$

The rest of the proof then follows [KM07, Section 12.5]. ∎

####  16.2. Transversality on Cylinders

Suppose a tame perturbation $\mathfrak{q}_{1}=\operatorname{grad}f_{1}$ in the
residue subset of Theorem 16.2 has been chosen. Then the critical set of
$\hbox to0.0pt{$-$\hss}\mathcal{L}_{\omega}^{1}\colonequals\hbox
to0.0pt{$-$\hss}\mathcal{L}_{\omega}+f_{1}$ consists of a finite collection of
gauge orbits; let their representatives be

$\mathfrak{a}_{i},\ 1\leq i\leq r.$

We wish to find a closed Banach subspace $\mathcal{P}^{\prime}$ of
$\mathcal{P}$ such that for any generic
$\mathfrak{q}_{2}\in\mathcal{P}^{\prime}$ with $\|\mathcal{P}\|\ll 1$, the sum

$\mathfrak{q}=\mathfrak{q}_{1}+\mathfrak{q}_{2}$

is an admissible perturbation. The Banach subspace $\mathcal{P}^{\prime}$ that
we consider is

(16.3)
$\mathcal{P}^{\prime}\colonequals\\{\mathfrak{q}_{2}\in\mathcal{P}:\mathfrak{q}_{2}(\mathfrak{a}_{i})=0,\mathcal{D}_{\mathfrak{a}_{i}}^{1}\mathfrak{q}_{2}=0,\forall
i=1,\cdots,r\\},$

so the perturbation $\mathfrak{q}_{2}$ vanishes to the first order at each
representative $\mathfrak{a}_{i}$. The subspace $\mathcal{P}^{\prime}$ is
clearly closed inside $\mathcal{P}$. Let us first verify the property (E1) for
$\mathfrak{q}=\mathfrak{q}_{1}+\mathfrak{q}_{2}$.

######  Lemma 16.3 ([KM07] Lemma 15.1.2).

There exists some $\eta>0$ such that for any
$\mathfrak{q}_{2}=\operatorname{grad}_{2}f_{2}\in\mathcal{P}^{\prime}$ with
$\|\mathfrak{q}_{2}\|_{\mathcal{P}}<\eta$, the critical set of $\hbox
to0.0pt{$-$\hss}\mathcal{L}_{\omega}\colonequals\mathcal{L}_{\omega}+(f_{1}+f_{2})$
agrees with that of $\hbox
to0.0pt{$-$\hss}\mathcal{L}_{\omega}^{1}=\mathcal{L}_{\omega}+f_{1}$. As a
result, the first condition (E1) of Definition 13.3 continues to hold for the
sum $\mathfrak{q}=\mathfrak{q}_{1}+\mathfrak{q}_{2}$.

In particular, for any $\mathfrak{q}_{2}\in\mathcal{P}^{\prime}$, the critical
points of $\hbox to0.0pt{$-$\hss}\mathcal{L}_{\omega}$ in the quotient
configuration space $\mathcal{B}_{k-1/2}(\widehat{Y},\widehat{\mathfrak{s}})$
are still given by $[\mathfrak{a}_{i}],1\leq i\leq r$ and

$\mathcal{D}_{\mathfrak{a}_{i}}\operatorname{grad}(\mathcal{L}_{\omega}+f_{1})=\mathcal{D}_{\mathfrak{a}_{i}}\operatorname{grad}(\mathcal{L}_{\omega}+f_{1}+f_{2}),\
1\leq i\leq r.$

So each $\mathfrak{a}_{i}$ is still non-degenerate in the sense of Definition
12.2. Here $[\mathfrak{a}_{i}]$ is the image of $\mathfrak{a}_{i}$ in
$\mathcal{B}_{k-1/2}(\widehat{Y},\widehat{\mathfrak{s}})$.

###### Proof of Lemma.

Suppose on the contrary that there is a sequence of tame perturbations
$\mathfrak{q}_{2}^{(j)}\in\mathcal{P}^{\prime}$ and a sequence of
configurations
$\beta_{i}\in\mathcal{C}_{k-1/2}(\widehat{Y},\widehat{\mathfrak{s}})$ such
that

$\|\mathfrak{q}_{j}\|_{\mathcal{P}}\to 0,\ (\operatorname{grad}\hbox
to0.0pt{$-$\hss}\mathcal{L}_{\omega}^{1}+\mathfrak{q}_{2}^{(j)})(\beta_{j})=0$

and each $\beta_{j}$ is not gauge equivalent to any of $\mathfrak{a}_{i},1\leq
i\leq r$. By Proposition 9.7, a subsequence of $\\{\beta_{j}\\}$ converges to
some $\mathfrak{a}_{i}$ up to gauge. Fix $0<\epsilon\ll 1$ and let
$\mathcal{O}_{i}(\epsilon)$ be the $\epsilon$-neighborhood of
$\mathfrak{a}_{i}$ in
$\mathcal{C}_{k+1/2}(\widehat{Y},\widehat{\mathfrak{s}})$. When $j\gg 1$, each
$\beta_{j}\in\mathcal{O}_{i}(\epsilon)$, and one may use gauge transformations
to put $\beta_{j}$ into the Coulomb gauge slice at $\mathfrak{a}_{i}$, i.e.

$\operatorname{\mathbf{d}}_{\mathfrak{a}_{i}}^{*}(\beta-\mathfrak{a}_{i})=0.$

Then

(16.4) $\operatorname{grad}\hbox
to0.0pt{$-$\hss}\mathcal{L}_{\omega}^{1}(\beta_{j})-\operatorname{grad}\hbox
to0.0pt{$-$\hss}\mathcal{L}_{\omega}^{1}(\mathfrak{a}_{i})=-(\mathfrak{q}_{2}^{(j)}(\beta_{j})-\mathfrak{q}_{2}^{(j)}(\mathfrak{a}_{j})).$

As $\mathfrak{a}_{i}$ is non-degenerate as a critical point of $\hbox
to0.0pt{$-$\hss}\mathcal{L}_{\omega}^{1}$, the $L^{2}_{k-1/2}$-norm of the
left hand side is bounded below by

$c\|\beta_{j}-\mathfrak{a}_{i}\|_{L^{2}_{k+1/2,\mathfrak{a}_{i}}}$

for some $c>0$. On the other hand, as $\mathfrak{q}_{2}^{(j)}\to 0$ in
$\mathcal{P}$, the $C^{2}$-norm of $\mathfrak{q}$ over the bounded
neighborhood $\mathcal{O}_{i}(\epsilon)$ converges to zero, by Corollary 8.19:

$\sup_{\gamma\in\mathcal{O}_{i}(\epsilon)}\|\mathcal{D}^{2}_{\gamma}\mathfrak{q}_{2}^{(j)}\|\to
0\text{ as }j\to\infty.$

As a result, the $L^{2}_{k-1/2}$-norm of the right hand side of (16.4) is
bounded above by

$\|\beta_{j}-\mathfrak{a}_{i}\|_{L^{2}_{k-1/2,\mathfrak{a}_{i}}}^{2}\leq\|\beta_{j}-\mathfrak{a}_{i}\|_{L^{2}_{k+1/2,\mathfrak{a}_{i}}}^{2}\leq\epsilon\|\beta_{j}-\mathfrak{a}_{i}\|_{L^{2}_{k+1/2,\mathfrak{a}_{i}}},$

when $j\gg 1$, which yields a contradiction if $\epsilon<c$. ∎

Theorem 16.1 now follows from the strong unique continuation property Theorem
15.1-15.3 together with Lemma 13.5. The proof is modeled on [KM07, Section
15]. In what follows, we will only point out the necessary changes to be made.

###### Proof of Theorem 16.1.

Let $\mathfrak{a},\mathfrak{b}\in\operatorname{Crit}(\hbox
to0.0pt{$-$\hss}\mathcal{L}_{\omega})$ be critical points of $\hbox
to0.0pt{$-$\hss}\mathcal{L}_{\omega}$ and
$\widehat{Z}=\mathbb{R}_{t}\times\widehat{Y}$ be the infinite cylinder.
Following the scheme of [KM07, Section 15] and notations from Subsection 13.1,
it suffice to show for any $\mathfrak{q}_{2}\in\mathcal{P}^{\prime}$ and any
solution $\gamma\in\mathcal{C}_{k}(\mathfrak{a},\mathfrak{b})$ to the
perturbed equation

$0=\mathfrak{F}_{\widehat{Z},\mathfrak{q}}\colonequals\mathfrak{F}_{\widehat{Z}}+\widehat{\mathfrak{q}},$

the operator

(16.5) $\displaystyle\mathcal{P}^{\prime}\times
L^{2}_{k}(\widehat{Z},i\mathbb{R}\oplus iT^{*}\widehat{Y}\oplus S)\to
L^{2}_{k-1}(\widehat{Z},i\mathbb{R}\oplus iT^{*}\widehat{Y}\oplus S)$
$\displaystyle(\delta\mathfrak{q},V)\mapsto\delta\widehat{\mathfrak{q}}(\gamma)+(\operatorname{\mathbf{d}}_{\gamma}^{*},\mathcal{D}_{\gamma}\mathfrak{F}_{\widehat{Z},\mathfrak{q}})(V)$

is surjective. The section $\delta\widehat{\mathfrak{q}}(\gamma)$ lies in
$L^{2}_{k-1}$ as the underlying path
$\check{\gamma}:\mathbb{R}\to\mathcal{C}_{k-1/2}(\widehat{Y},\widehat{\mathfrak{s}})$
decay exponentially to either $\mathfrak{a}$ or $\mathfrak{b}$ as
$t\to\pm\infty$ and $\delta\mathfrak{q}$ vanishes at $\mathfrak{a}$ and
$\mathfrak{b}$ to the first order.

Suppose first that $\delta\widehat{\mathfrak{q}}=0$ in (16.5), then (16.5)
becomes a Fredholm operator by Proposition 13.1, and its cokernel is finite
dimensional. It remains to show that for any section

$U=(\delta c^{\prime}(t),\delta b^{\prime}(t),\delta\psi^{\prime}(t))\in
L^{2}(\widehat{Z},i\mathbb{R}\oplus iT^{*}\widehat{Y}\oplus S)$

that is $L^{2}$-orthogonal to the image of
$(\operatorname{\mathbf{d}}_{\gamma}^{*},\mathcal{D}_{\gamma}\mathfrak{F}_{\widehat{Z},\mathfrak{q}})$,
there exist some $\delta\widehat{\mathfrak{q}}\in\mathcal{P}^{\prime}$ such
that

(16.6)
$\langle\delta\widehat{\mathfrak{q}}(\gamma(t)),U\rangle_{L^{2}(\mathbb{R}\times\widehat{Y})}\neq
0.$

We first explain how to achieve (16.6) for a generalized cylinder function
$f:\mathcal{C}_{k-1/2}(\widehat{Y},\widehat{\mathfrak{s}})\to\mathbb{R}$:

(16.7)
$\int_{t\in\mathbb{R}_{t}}\langle\operatorname{grad}f(\check{\gamma}),U(t)\rangle_{L^{2}(\widehat{Y})}\neq
0.$

By the unique continuation properties, Theorem 15.1, 15.2 and 15.3, the
underlying path
$\check{\gamma}:\mathbb{R}\to\mathcal{C}_{k-1/2}(\widehat{Y},\widehat{\mathfrak{s}})$
satisfies the following properties

  * •

for any $t_{1}\neq t_{2}\in\mathbb{R}_{t}$, $\check{\gamma}(t_{1})$ and
$\check{\gamma}(t_{2})$ are not gauge equivalent over $Y$;

  * •

for any $t\in\mathbb{R}_{t}$, $\check{\gamma}(t)$ is not gauge equivalent to
$\mathfrak{a}_{i}$ on $Y$ for any $1\leq i\leq r$; moreover,
$\check{\gamma}(t)$ is irreducible on $Y$;

  * •

for any $t\in\mathbb{R}_{t}$, its derivative $\partial_{t}\check{\gamma}(t)$
is not generated by the infinitesimal gauge action over $Y$.

As for the section $U$ in the cokernel, by Lemma 13.5, we have

  * •

$\delta c^{\prime}(t)\equiv 0$;

  * •

for some $t_{0}\in\mathbb{R}_{t}$, $U(t_{0})=(0,\delta
b^{\prime}(t_{0}),\delta\psi^{\prime}(t_{0}))$ are not generated by the
infinitesimal gauge action over $Y$.

Take a large constant $T>0$ such that $t_{0}\in[-T,T]$. To apply Theorem 8.20,
let the compact subset $K$ be the image of

$\\{\mathfrak{a}_{i}:1\leq i\leq
r\\}\bigcup\\{\check{\gamma}(t):t\in[-T,T]\\}$

in the quotient configuration space
$\mathcal{B}^{*}_{k-1/2}(\widehat{Y},\widehat{\mathfrak{s}})$. Then we can
find a finite collection of cylinder functions $\\{f_{j},\ 1\leq j\leq l\\}$
defined using embeddings $\iota_{j}:S^{1}\times D^{2}\hookrightarrow Y$ such
that the map

$\Xi^{\prime}_{*}=(f_{1},\cdots,f_{l}):\mathcal{B}_{k-1/2}(\widehat{Y},\widehat{\mathfrak{s}})\to\mathbb{R}^{l}$

gives an embedding of $K$ and $\Xi^{\prime}(U(t_{0}))\neq 0$. Choose a smooth
function

$g^{\prime}:\mathbb{R}^{l}\to\mathbb{R}$

supported in a small neighborhood $\overline{\Omega}$ of
$\Xi^{\prime}([\check{\gamma}(t_{0})])$ with the following additional
properties

  * •

$\Xi^{\prime}([\mathfrak{a}_{i}])\not\in\Omega,\ \forall 1\leq i\leq r;$

  * •

$(\Xi^{\prime}\circ\check{\gamma})^{-1}(\overline{\Omega})$ is a small
connected interval $[t_{0}-\epsilon_{1},t_{0}+\epsilon_{2}]$ around $t_{0}$;
to achieve this, we take $T\gg 1$;

  * •

lastly, the integral

(16.8) $\int_{\mathbb{R}_{t}}dg^{\prime}(\Xi^{\prime}_{*}(U(t)))dt\neq 0.$

The last property would be impossible if for some constant
$\alpha\in\mathbb{R}$,
$\Xi_{*}^{\prime}(U(t))=\alpha\Xi_{*}^{\prime}(\partial_{t}\check{\gamma}_{t})$
for any $t\in[t_{0}-\epsilon,t_{0}+\epsilon]$. However, this cannot hold for
the whole real line; otherwise one may draw a contradiction from equations
(13.2) and (13.8). Then we can achieve (16.8) by taking a different time slice
$t_{0}\in\mathbb{R}_{t}$ and possibly a different $\Xi^{\prime}$.

As a result, the inequality (16.7) is achieved for the composition:

$f\colonequals
g^{\prime}\circ\Xi^{\prime}:\mathcal{B}_{k-1/2}(\widehat{Y},\widehat{\mathfrak{s}})\to\mathbb{R},$

Note that $f\equiv 0$ in some $L^{2}_{k-1/2}$-neighborhood of
$\\{[\mathfrak{a}_{i}]:1\leq i\leq r\\}$, so $\operatorname{grad}f$ satisfies
the constraints in (16.3). By the density of the Banach space $\mathcal{P}$,
we can approximate $\operatorname{grad}f$ by an element
$\delta\widehat{\mathfrak{q}}$ in $\mathcal{P}^{\prime}$ and the inequality
(16.6) holds for this approximation.

The rest of the proof follows the same line of argument as in [KM07,
Proposition 15.1.3]. ∎

####  16.3. Transversality on 4-Manifolds in General

Recall the set up from Section 14. For a morphism
$\mathbb{X}:(\mathbb{Y}_{1},\widehat{\mathfrak{s}}_{1})\to(\mathbb{Y}_{2},\widehat{\mathfrak{s}}_{2})$,
the Seiberg-Witten equations $\mathfrak{F}_{\mathcal{X},\mathfrak{p}}=0$ on
the complete Riemannian 4-manifold $\mathcal{X}$ is perturbed by a quadruple

$\mathfrak{p}=(\mathfrak{q}_{1},\mathfrak{q}_{2},\mathfrak{q}_{3},\omega_{3}).$

While $(\mathfrak{q}_{1},\mathfrak{q}_{2})$ are encoded in the objects
$\mathbb{Y}_{1}$ and $\mathbb{Y}_{2}$, the pair

$(\mathfrak{q}_{3},\omega_{3})\in\mathcal{P}(Y_{3})\times\mathcal{P}_{\operatorname{Form}}$

is the actual perturbation that allows us to achieve transversality.

######  Definition 16.4.

The quadruple $\mathfrak{p}$ is said to be admissible if

  * •

each $\mathfrak{q}_{i}\in\mathcal{P}(Y_{i}),i=1,2$ is admissible in the sense
of Definition 13.3;

  * •

for any $spin^{c}\ $cobordism
$(\widehat{X},\widehat{\mathfrak{s}}_{X}):(\widehat{Y}_{1},\widehat{\mathfrak{s}}_{1})\to(\widehat{Y}_{2},\widehat{\mathfrak{s}}_{2})$
(with a prescribed planar metric $g_{X}$), the moduli space
$\mathcal{M}_{k}(\mathfrak{a}_{1},\mathcal{X},\mathfrak{a}_{2})$ is regular in
the sense of Definition 14.2. Here
$\mathfrak{a}_{i}\in\operatorname{Crit}(\hbox
to0.0pt{$-$\hss}\mathcal{L}_{\omega_{i},\widehat{Y}_{i}})$ is a critical point
of the perturbed Chern-Simons-Dirac functional $\hbox
to0.0pt{$-$\hss}\mathcal{L}_{\omega_{i},\widehat{Y}_{i}}$ on
$\widehat{Y}_{i}$, $i=1,2$. ∎

######  Theorem 16.5.

Under above assumptions, for any fixed admissible perturbations
$(\mathfrak{q}_{1},\mathfrak{q}_{2})$ on $\widehat{Y}_{1}$ and
$\widehat{Y}_{2}$ respectively, there is a residue subset of
$\mathcal{P}(Y_{2})\times\mathcal{P}_{\operatorname{Form}}$ such that for
every pair $(\mathfrak{q}_{3},\omega_{3})$ in this subset, the quadruple
$\mathfrak{p}$ is admissible.

###### Proof.

Following the proof of Theorem 16.1, it suffices to verify that the operator

(16.9)
$\displaystyle\mathcal{P}(Y_{2})\times\mathcal{P}_{\operatorname{Form}}\times
L^{2}_{k}(\mathcal{X},iT^{*}\mathcal{X}\oplus S^{+})$ $\displaystyle\to
L^{2}_{k-1}(\mathcal{X},i\mathbb{R}\oplus i\mathfrak{su}(S^{+})\oplus S-)$
$\displaystyle(\delta\mathfrak{q}_{3},\delta\omega_{3},V)$
$\displaystyle\mapsto(\operatorname{\mathbf{d}}_{\gamma}^{*},\mathcal{D}_{\gamma}\mathfrak{F}_{\mathcal{X},\mathfrak{p}})V+\beta_{0}(t)\delta\widehat{\mathfrak{q}}_{3}(\gamma)+\rho_{4}(\delta\omega_{3}^{+}),$

is surjective, for any solution
$\gamma\in\mathcal{C}_{k}(\mathfrak{a}_{1},\mathcal{X},\mathfrak{a}_{2})$ to
the perturbed equation $\mathfrak{F}_{\mathcal{X},\mathfrak{p}}=0$. We begin
with $(\delta\mathfrak{q}_{3},\delta\omega_{3})=0$, then (16.9) becomes a
Fredholm operator by Proposition 14.1. Suppose $U\in
L^{2}(\mathcal{X},i\mathbb{R}\oplus i\Lambda^{+}\mathcal{X}\oplus S^{-})$ is
$L^{2}$-orthogonal to the image of
$(\operatorname{\mathbf{d}}_{\gamma}^{*},\mathcal{D}_{\gamma}\mathfrak{F}_{\mathcal{X},\mathfrak{p}})$.
it remains to find $(\delta\widehat{\mathfrak{q}}_{3},\delta\omega_{3})$ such
that

(16.10) $\langle
U,\beta_{0}(t)\delta\widehat{\mathfrak{q}}_{3}(\gamma)+\rho_{4}(\delta\omega_{3}^{+})\rangle_{L^{2}}\neq
0.$

Let $I=[1,2]_{t}$ and write

$U=(\delta\xi,\delta\omega,\delta\phi)\text{ with }\delta\xi\in
L^{2}(\mathcal{X},i\mathbb{R}).$

The same argument as in the proof of Lemma 13.5 implies that $\delta\xi\equiv
0$. The inner product (16.10) is supported on the compact submanifold

$\widehat{Z}\colonequals I\times\widehat{Y}_{2},$

over which the formal adjoint of
$(\operatorname{\mathbf{d}}_{\gamma}^{*},\mathcal{D}_{\gamma}\mathfrak{F}_{\mathcal{X},\mathfrak{p}})$
is cast into the form (13.6). If instead we write

$U(t)=(0,\delta b(t),\delta\psi(t))\in L^{2}_{1}(\widehat{Z},i\mathbb{R}\oplus
iT^{*}\widehat{Y}_{2}\oplus S)\text{ on }I\times\widehat{Y}_{2},$

then we are back to the cylindrical case. Here we have used the bundle map

$(\rho_{3},\rho_{4}(dt))$

to identify $iT^{*}\widehat{Y}_{2}\oplus S$ with $i\mathfrak{su}(S^{+})\oplus
S^{-}$ over $\widehat{Z}$.

However, Lemma 13.5 does not apply directly here, so we argue as follows. If
there exists some $t_{0}\in\text{supp}\beta_{0}\subset[1,2]$ such that
$U(t_{0})$ is separated by some cylinder function $f$, then we set
$\delta\omega_{3}=0$ and proceed as in the proof of Theorem 15.1.

If not, then by the proof of Lemma 13.5, for any $t\in[5/4,7/4]$, there exists
some function $\xi(t)\in L^{2}_{1}(\widehat{Y},i\mathbb{R})$ such that

$(\delta
b(t),\delta\psi(t))=\operatorname{\mathbf{d}}_{\check{\gamma}(t)}\xi(t)\text{
on }\\{t\\}\times Y_{2}.$

Moreover,

(16.11) $\frac{d}{dt}d_{Y_{2}}\xi(t)\equiv 0\text{ and
}\Delta_{Y_{2}}\xi(t)+\xi(t)|\Psi(t)|^{2}=0\text{ on }[5/4,7/4]\times Y_{2}.$

Recall that $\delta\omega_{3}=-\beta_{0}(t)dt\wedge d_{Y_{2}}f_{3}$ for a
compactly supported function $f_{3}:I\times Y_{2}\to i\mathbb{R}$, so

$\rho_{4}(\delta\omega_{3}^{+})=\rho_{3}(d_{Y_{2}}(\beta_{0}(t)f_{3})).$

If $U$ is orthogonal to $\rho_{4}(\delta\omega_{3}^{+})$ for any
$\delta\omega_{3}\in\mathcal{P}_{\operatorname{Form}}$, then
$\Delta_{Y_{2}}\xi(t)\equiv 0$. By (16.11), $U(t)\equiv 0$ on $[5/4,7/4]\times
Y_{2}$. By unique continuation, $U\equiv 0$ on the whole manifold
$\mathcal{X}$. ∎

##  Part VI Floer Homology

Let $(\mathbb{Y},\widehat{\mathfrak{s}})\in\operatorname{SCob}_{s}$ be an
object in the strict $spin^{c}\ $cobordism category, as defined in Section 3.
The underlying 3-manifold $Y$ of $\mathbb{Y}$ is compact, connected and
oriented, whose boundary is identified with a disjoint union of 2-tori
$\Sigma$ by the diffeomorphism $\psi:\partial Y\to\Sigma$. The quintuple
$\mathbb{Y}=(Y,\psi,g_{Y},\omega,\\{\mathfrak{q}\\})$ also dictates a
cylindrical metric $g_{Y}$ and a closed 2-form
$\omega\in\Omega^{2}(Y,i\mathbb{R})$.
$\widehat{\mathfrak{s}}\in\operatorname{Spin^{c}_{R}}(Y)$ is a relative
$spin^{c}\ $structure of the 3-manifold $Y$.

The primary goal of this part is to define the functor

$\operatorname{\it
HM}_{*}:\operatorname{SCob}_{s}\to\mathcal{R}\text{-}\mathrm{Mod}$

which assigns the monopole Floer homology $\operatorname{\it
HM}_{*}(\mathbb{Y},\widehat{\mathfrak{s}})$ for each object
$(\mathbb{Y},\widehat{\mathfrak{s}})\in\operatorname{SCob}_{s}$, generalizing
the construction of Kronheimer-Mrowka for closed 3-manifolds.

So far we have addressed two fundamental problems in order to define the
functor $\operatorname{\it HM}_{*}$:

  * •

the compactness issue; see Theorem 6.1 for the unperturbed equations and
Theorem 9.5 for the perturbed ones;

  * •

the transversality issue; see Theorem 16.1 for the case of cylinders and
Theorem 16.5 for morphisms in $\operatorname{SCob}_{s}$.

Although the proof of the gluing theorem is omitted in this paper, it follows
from the standard procedure in [KM07, Section 17-19], as noted in Subsection
13.4.

Now the construction of monopole Floer homology becomes straightforward by
following the standard argument. Part VI is organized as follows. In Section
17, we explain the basic construction using $\mathbb{F}_{2}$-coefficient.
Section 18 is devoted to the canonical grading as well as the canonical mod
$2$ grading of $\operatorname{\it HM}_{*}(\mathbb{Y},\widehat{\mathfrak{s}})$.

In Section 19, we address the orientation issue, which allows us to define the
monopole Floer homology $\operatorname{\it
HM}_{*}(\mathbb{Y},\widehat{\mathfrak{s}})$ using $\mathbb{Z}$-coefficient.
The key ingredient is the notion of relative orientations, which compare the
orientations of two Fredholm operators using the excision principle, cf.
Theorem 19.2 and Definition A.2. The proof is postponed to Appendix A.

###  17\. The Basic Construction: $\mathbb{F}_{2}$-coefficient

In this section, we define the monopole Floer homology $\operatorname{\it
HM}_{*}(\mathbb{Y},\widehat{\mathfrak{s}})$ for each object
$(\mathbb{Y},\widehat{\mathfrak{s}})\in\operatorname{SCob}_{s}$ using
$\mathbb{F}_{2}$-coefficient. For the most general case, we have to use a
Novikov ring $\mathcal{R}_{2}$. To work with the field $\mathbb{F}_{2}$ of two
elements, we will pass to a subcategory of $\operatorname{SCob}_{s}$ in which
case a monotonicity condition is required.

####  17.1. Novikov Rings

Let us first explain the construction of $\operatorname{\it
HM}_{*}(\mathbb{Y},\widehat{\mathfrak{s}})$ using a Novikov ring

$\mathcal{R}_{2}=\\{\sum_{n_{i}}a_{i}q^{n_{i}}:\ a_{i}\in\mathbb{F}_{2},\
n_{i}\in\mathbb{R},\ \lim_{i}n_{i}=-\infty\\},$

which is a complete topological group. Each element of $\mathcal{R}_{2}$ is a
Laurent series in a formal variable $q$ with possibly infinitely many terms in
negative degrees. For any object
$(\mathbb{Y},\widehat{\mathfrak{s}})\in\operatorname{SCob}_{s}$, the
perturbation $\mathfrak{q}=\operatorname{grad}f$ encoded in the quintuple
$\mathbb{Y}$ is admissible in the sense of Definition 13.3. Let
$\mathfrak{C}(\mathbb{Y},\widehat{\mathfrak{s}})$ be the set of critical
points of $\hbox to0.0pt{$-$\hss}\mathcal{L}_{\omega}=\mathcal{L}_{\omega}+f$
in the quotient configuration space
$\mathcal{B}_{k}(\widehat{Y},\widehat{\mathfrak{s}})$, then
$\mathfrak{C}(\mathbb{Y},\widehat{\mathfrak{s}})$ is a finite set by Theorem
16.2. Then the chain group $C_{*}(\mathbb{Y},\widehat{\mathfrak{s}})$ is
freely generated by $\mathfrak{C}(\mathbb{Y},\widehat{\mathfrak{s}})$ over
$\mathcal{R}_{2}$:

$C_{*}(\mathbb{Y},\widehat{\mathfrak{s}})=\bigoplus_{[\mathfrak{a}]\in\mathfrak{C}(\widehat{Y},\widehat{\mathfrak{s}})}\mathcal{R}_{2}\cdot[\mathfrak{a}].$

with differential $\partial$ defined as

(17.1)
$\partial[\mathfrak{a}]=\sum_{\begin{subarray}{c}z\in\pi_{1}(\mathcal{B}_{k}(\widehat{Y},\widehat{\mathfrak{s}});[\mathfrak{a}],[\mathfrak{b}])\\\
\dim\widecheck{\mathcal{M}}_{z}([\mathfrak{a}],[\mathfrak{b}])=0\end{subarray}}[\mathfrak{b}]\cdot\\#\widecheck{\mathcal{M}}_{z}([\mathfrak{a}],[\mathfrak{b}])\cdot
q^{-\mathcal{E}_{top}^{\mathfrak{q}}([\mathfrak{a}],[\mathfrak{b}];z)}.$

The unparameterized moduli space
$\widecheck{\mathcal{M}}_{z}([\mathfrak{a}],[\mathfrak{b}])\colonequals\mathcal{M}_{z}([\mathfrak{a}],[\mathfrak{b}])/\mathbb{R}_{t}$
is defined as in (13.5). The topological energy
$\mathcal{E}_{top}^{\mathfrak{q}}([\mathfrak{a}],[\mathfrak{b}];z)$ for a
homotopy class of paths
$z\in\pi_{1}(\mathcal{B}_{k}(\widehat{Y},\widehat{\mathfrak{s}});[\mathfrak{a}],[\mathfrak{b}])$
equals twice the drop of $\hbox to0.0pt{$-$\hss}\mathcal{L}_{\omega}$ along
$\gamma$

$2(\hbox to0.0pt{$-$\hss}\mathcal{L}_{\omega}(\mathfrak{a})-\hbox
to0.0pt{$-$\hss}\mathcal{L}_{\omega}(\mathfrak{b}))$

if $\gamma:[0,1]\to\mathcal{C}_{k}(\widehat{Y},\widehat{\mathfrak{s}})$ is a
lift of $z$ with $\gamma(0)=\mathfrak{a}$ and $\gamma(1)=\mathfrak{b}$. This
expression is suggested by Proposition 9.1. To ensure the sum in (21.1) is
convergent in $\mathcal{R}_{2}$, we need a finiteness result:

######  Lemma 17.1.

For any $C>0$, there are only finitely many homotopy classes of paths
$z\in\pi_{1}(\mathcal{B}_{k}(\widehat{Y},\widehat{\mathfrak{s}});[\mathfrak{a}],[\mathfrak{b}])$
such that
$\mathcal{E}_{top}^{\mathfrak{q}}([\mathfrak{a}],[\mathfrak{b}],z)<C$ and
$\widecheck{\mathcal{M}}_{z}([\mathfrak{a}],[\mathfrak{b}])$ is non-empty.
Moreover, each $\widecheck{\mathcal{M}}_{z}([\mathfrak{a}],[\mathfrak{b}])$ is
compact if its dimension equals zero.

To show $\partial^{2}=0$, we follow the standard argument and look at the
compactification of moduli spaces
$\widecheck{\mathcal{M}}_{z}([\mathfrak{a}],[\mathfrak{b}])$ when $\dim=1$.
Readers are referred to [KM07, Section 22] for the details. The monopole Floer
homology of $(\mathbb{Y},\widehat{\mathfrak{s}})$ is then defined as the
homology of the chain complex
$(C_{*}(\widehat{Y},\widehat{\mathfrak{s}}),\partial)$:

$\operatorname{\it HM}_{*}(\mathbb{Y},\widehat{\mathfrak{s}})\colonequals
H_{*}((C_{*}(\widehat{Y},\widehat{\mathfrak{s}}),\partial)).$

To make $\operatorname{\it HM}_{*}$ into a functor:

$\operatorname{\it
HM}_{*}:\operatorname{SCob}_{s}\to\mathcal{R}_{2}\text{-}\mathrm{Mod},$

we assign for each morphism $\mathbb{X}:\mathbb{Y}_{1}\to\mathbb{Y}_{2}$ a
chain map:

$m(\mathbb{X};g_{X},\mathfrak{p}):(C_{*}(\mathbb{Y}_{1},\widehat{\mathfrak{s}}_{1}),\partial_{1})\to(C_{*}(\mathbb{Y}_{2},\widehat{\mathfrak{s}}_{2}),\partial_{2})$

which relies on a planar metric $g_{X}$ of the strict cobordism $X:Y_{1}\to
Y_{2}$ and a quadruple

$\mathfrak{p}=(\mathfrak{q}_{1},\mathfrak{q}_{2},\mathfrak{q}_{3},\omega_{3})\in\mathcal{P}(Y_{1})\times\mathcal{P}(Y_{2})\times\mathcal{P}(Y_{2})\times\mathcal{P}_{\operatorname{Form}}.$

Here $\mathfrak{p}$ is required to be admissible in the sense of Definition
16.4. While $(\mathfrak{q}_{1},\mathfrak{q}_{2})$ are encoded in the objects
$(\mathbb{Y}_{1},\mathbb{Y}_{2})$, $(\mathfrak{q}_{3},\omega_{3})$ are the
actual perturbations to the Seiberg-Witten equations on the complete
Riemannian 4-manifold $\mathcal{X}$. Now define

(17.2)
$m(\mathbb{X};g_{X},\mathfrak{q})[\mathfrak{a}_{1}]=\sum_{\begin{subarray}{c}\widehat{\mathfrak{s}}_{X}\in\operatorname{Spin^{c}}(X;\widehat{\mathfrak{s}}_{1},\widehat{\mathfrak{s}}_{2})\\\
\dim\mathcal{M}(\mathfrak{a}_{1},\widehat{\mathfrak{s}}_{X},\mathfrak{a}_{2})=0\end{subarray}}[\mathfrak{a}_{2}]\cdot\\#\mathcal{M}(\mathfrak{a}_{1},\widehat{\mathfrak{s}}_{X},\mathfrak{a}_{2})\cdot
q^{-\mathcal{E}_{top}^{\mathfrak{p}}(\mathfrak{a}_{1},\widehat{\mathfrak{s}}_{X},\mathfrak{a}_{2})},$

where $\mathfrak{a}_{i}$ is a lift of
$[\mathfrak{a}_{i}]\in\mathfrak{C}(\mathbb{Y}_{i})$ in
$\mathcal{C}_{k}(\widehat{Y},\widehat{\mathfrak{s}})$ for $i=1,2$. The moduli
space
$\mathcal{M}(\mathfrak{a}_{1},\widehat{\mathfrak{s}}_{X},\mathfrak{a}_{2})$ is
defined as in (14.4) with the admissible quadruple $\mathfrak{q}$ as
perturbations. The topological energy is given by the formula

(17.3)
$\mathcal{E}_{top}^{\mathfrak{p}}(\mathfrak{a}_{1},\widehat{\mathfrak{s}}_{X},\mathfrak{a}_{2})\colonequals
2\hbox to0.0pt{$-$\hss}\mathcal{L}_{\omega_{1}}(\mathfrak{a}_{1})-2\hbox
to0.0pt{$-$\hss}\mathcal{L}_{\omega_{2}}(\mathfrak{a}_{2})+C(A_{0},\omega_{X})$

where $A_{0}$ is a background $spin^{c}\ $connection on $\widehat{X}$ such
that the restriction $A_{0}|_{\widehat{Y}_{i}}$ is the reference connection on
$\widehat{Y}_{i}$ that defines the Chern-Simons-Dirac functional $\hbox
to0.0pt{$-$\hss}\mathcal{L}_{\omega_{i}}$ for $i=1,2$. The constant
$C(A_{0},\omega_{X})$ is given concretely by

(17.4) $C(A_{0},\omega_{X})=\frac{1}{4}\int_{\widehat{X}}F_{A^{t}_{0}}\wedge
F_{A^{t}_{0}}-\int_{\widehat{X}}F_{A^{t}_{0}}\wedge\omega_{X},$

as suggested by (5.2). To make sense of the expression (17.2), we need another
finiteness result:

######  Lemma 17.2.

For any $C>0$, any pair of critical points
$([\mathfrak{a}_{1}],[\mathfrak{a}_{2}])\in\mathfrak{C}(\mathbb{Y}_{1})\times\mathfrak{C}(\mathbb{Y}_{2})$
and any admissible quadruple $\mathfrak{p}$, there are only finitely many
relative $spin^{c}\ $cobordisms
$\widehat{\mathfrak{s}}_{X}\in\operatorname{Spin^{c}_{R}}(X;\widehat{\mathfrak{s}}_{1},\widehat{\mathfrak{s}}_{2})$
such that
$\mathcal{E}_{top}^{\mathfrak{p}}(\mathfrak{a}_{1},\widehat{\mathfrak{s}}_{X},\mathfrak{a}_{2})<C$
and
$\mathcal{M}(\mathfrak{a}_{1},\widehat{\mathfrak{s}}_{X},\mathfrak{a}_{2})$ is
non-empty. Moreover, each moduli space
$\mathcal{M}(\mathfrak{a}_{1},\widehat{\mathfrak{s}}_{X},\mathfrak{a}_{2})$ is
compact if its dimension equals zero.

Lemma 17.1 and Lemma 17.2 follow from the Compactness Theorem 9.5 and its
analogue for a general cobordism. Readers are referred to [KM07, Corollary
31.2.5] for more details; their proofs are omitted here. By analyzing the
moduli space
$\mathcal{M}(\mathfrak{a}_{1},\widehat{\mathfrak{s}}_{X},\mathfrak{a}_{2})$
with $\dim=1$, we conclude that $m(\mathbb{X};g_{X},\mathfrak{q})$ is a chain
map by the standard argument. The chain maps induced from different auxiliary
data $(g_{X},\mathfrak{q})$ are all chain homotopic to each other, so the
resulting maps on the homology are independent of $(g_{X},\mathfrak{q})$

$\operatorname{\it
HM}(\mathbb{X})\colonequals[m(\mathbb{X};g_{X},\mathfrak{p})]:\operatorname{\it
HM}_{*}(\mathbb{Y}_{1},\widehat{\mathfrak{s}}_{1})\to\operatorname{\it
HM}_{*}(\mathbb{Y}_{2},\widehat{\mathfrak{s}}_{2}),$

To show that $\operatorname{\it HM}$ defined this way is a functor and
satisfies the composition law in Theorem 1.6, we follow [KM07, Section 26].

####  17.2. Monotonicity

To define the monopole Floer homology using $\mathbb{F}_{2}$-coefficient, it
is necessary to pass to a subcategory of $\operatorname{SCob}_{s}$, as we
explain in this subsection.

######  Definition 17.3.

An object
$(\mathbb{Y},\widehat{\mathfrak{s}})=(Y,\psi,g_{Y},\omega,\mathfrak{q},\widehat{\mathfrak{s}})\in\operatorname{SCob}_{s}$
is called monotone if the period class $[\omega]\in H^{2}(Y;i\mathbb{R})$ is
proportional to the image of $c_{1}(\widehat{\mathfrak{s}})$ in
$\operatorname{Im}(H^{2}(Y,\partial Y;\mathbb{Z})\to H^{2}(Y;\mathbb{R})$:

$[\frac{\omega}{\pi i}]=\alpha\cdot c_{1}(\mathfrak{s})\in
H^{2}(Y;\mathbb{R})\text{ for some }\alpha\in\mathbb{R}.$

In addition, $(\widehat{Y},\widehat{\mathfrak{s}})$ is called

  * •

positively monotone if $\alpha<1$;

  * •

balanced if $\alpha=1$;

  * •

negatively monotone if $\alpha>1$.∎

In light of Lemma 3.8, under the monotonicity assumption, we have

$\hbox to0.0pt{$-$\hss}\mathcal{L}_{\omega}(u\cdot\gamma)-\hbox
to0.0pt{$-$\hss}\mathcal{L}_{\omega}(\gamma)=2(1-\alpha)\pi^{2}[u]\cup
c_{1}(\mathfrak{s}),$

for any $\gamma\in\mathcal{C}_{k}(\widehat{Y},\widehat{\mathfrak{s}})$ and
$u\in\mathcal{G}_{k+1}(\widehat{Y})$. In particular, $\hbox
to0.0pt{$-$\hss}\mathcal{L}_{\omega}$ becomes a real valued functional if
$(\mathbb{Y},\widehat{\mathfrak{s}})$ is balanced. One necessary condition of
monotonicity is that $\mu=0$. The construction described below will work in
general for any monotone objects, but let us focus on the special case when
the period class $[\omega]=0\in H^{2}(Y;i\mathbb{R})$ and the form
$\overline{\omega}$ defined in (P6) vanishes, for the sake of simplicity; so

$\omega=\omega_{\lambda}=\chi_{1}(s)ds\wedge\lambda.$

In this case, $(\widehat{Y},\widehat{\mathfrak{s}})$ is always positively
monotone, since $\alpha=0$.

Under this assumption, the chain group
$C_{*}(\mathbb{Y},\widehat{\mathfrak{s}};\mathbb{F}_{2})$ is a finite
dimensional $\mathbb{F}_{2}$-vector space:

$C_{*}(\mathbb{Y},\widehat{\mathfrak{s}};\mathbb{F}_{2})\colonequals\bigoplus_{[\mathfrak{a}]\in\mathfrak{C}(\widehat{Y},\widehat{\mathfrak{s}})}\mathbb{F}_{2}\cdot[\mathfrak{a}].$

with differential defined by

(17.5)
$\partial[\mathfrak{a}]=\sum_{\begin{subarray}{c}z\in\pi_{1}(\mathcal{B}_{k}(\widehat{Y},\widehat{\mathfrak{s}});[\mathfrak{a}],[\mathfrak{b}])\\\
\dim\widecheck{\mathcal{M}}_{z}([\mathfrak{a}],[\mathfrak{b}])=0\end{subarray}}[\mathfrak{b}]\cdot\\#\widecheck{\mathcal{M}}_{z}([\mathfrak{a}],[\mathfrak{b}])$

In light of Lemma 17.1, to make sense of this expression, we need an upper
bound on the topological energy
$\mathcal{E}_{top}^{\mathfrak{q}}([\mathfrak{a}],[\mathfrak{b}];z)$:

######  Lemma 17.4.

For any
$[\mathfrak{a}],[\mathfrak{b}]\in\mathfrak{C}(\mathbb{Y},\widehat{\mathfrak{s}})$,
there exists a constant $C>0$ such that

$\mathcal{E}_{top}^{\mathfrak{q}}([\mathfrak{a}],[\mathfrak{b}];z)<C,$

for any homotopy classes of paths
$z\in\pi_{1}(\mathcal{B}_{k}(\widehat{Y},\widehat{\mathfrak{s}}),[\mathfrak{a}],[\mathfrak{b}])$
with $\dim\widecheck{\mathcal{M}}_{z}([\mathfrak{a}],[\mathfrak{b}])=0$.

As for a morphism
$\mathbb{X}:(\mathbb{Y}_{1},\widehat{\mathfrak{s}}_{1})\to(\mathbb{Y}_{2},\widehat{\mathfrak{s}}_{2})$
with $\omega_{1}=\omega_{2}=\omega_{\lambda}$, $\overline{\omega}_{X}$ is a
compactly supported 2-form (see • ‣ (Q6)) on $X$. We require that the class
defined in (Q7) vanishes: $[\omega_{X}]_{cpt}=0\in H^{2}(X,\partial
X;\mathbb{Z})$. This time the chain map $m(\mathbb{X};g_{X},\mathfrak{q})$ is
defined as

$\displaystyle
m(\mathbb{X};g_{X},\mathfrak{q}):C_{*}(\mathbb{Y}_{1},\widehat{\mathfrak{s}}_{1};\mathbb{F}_{2})$
$\displaystyle\to
C_{*}(\mathbb{Y}_{2},\widehat{\mathfrak{s}}_{2};\mathbb{F}_{2})$
$\displaystyle[\mathfrak{a}_{1}]$
$\displaystyle\mapsto\sum_{\begin{subarray}{c}\widehat{\mathfrak{s}}_{X}\in\operatorname{Spin^{c}}(X;\widehat{\mathfrak{s}}_{1},\widehat{\mathfrak{s}}_{2})\\\
\dim\mathcal{M}(\mathfrak{a}_{1},\widehat{\mathfrak{s}}_{X},\mathfrak{a}_{2})=0\end{subarray}}[\mathfrak{a}_{2}]\cdot\\#\mathcal{M}(\mathfrak{a}_{1},\widehat{\mathfrak{s}}_{X},\mathfrak{a}_{2}).$

Again, we need a upper bound on
$\mathcal{E}_{top}(\mathfrak{a}_{1},\widehat{\mathfrak{s}}_{X},\mathfrak{a}_{2})$
to ensure the sum in the expression above is finite:

######  Lemma 17.5.

Under above assumptions, for any pair of critical points
$([\mathfrak{a}_{1}],[\mathfrak{a}_{2}])\in\mathfrak{C}(\mathbb{Y}_{1},\widehat{\mathfrak{s}}_{1})\times\mathfrak{C}(\mathbb{Y}_{2},\widehat{\mathfrak{s}}_{2})$,
any planar metric $g_{X}$ and any admissible quadruple $\mathfrak{p}$, there
is a constant $C>0$ such that

$\mathcal{E}_{top}^{\mathfrak{p}}(\mathfrak{a}_{1},\widehat{\mathfrak{s}}_{X},\mathfrak{a}_{2})<C$

which holds for any
$\widehat{\mathfrak{s}}_{X}\in\operatorname{Spin^{c}_{R}}(X;\widehat{\mathfrak{s}}_{1},\widehat{\mathfrak{s}}_{2})$
with
$\dim\mathcal{M}(\mathfrak{a}_{1},\widehat{\mathfrak{s}}_{X},\mathfrak{a}_{2})=0$.

Lemma 17.4 and 17.5 follow directly from a general statement relating the
dimension with the topological energy $\mathcal{E}_{top}$. In Proposition 17.6
below, we will think of a homotopy class of paths as a relative $spin^{c}\
$cobordism, following the ideas in Subsection 3.5.

######  Proposition 17.6.

Under above assumptions, for any relative $spin^{c}\ $cobordism
$\widehat{\mathfrak{s}}_{X},\widehat{\mathfrak{s}}_{X}^{\prime}\in\operatorname{Spin^{c}_{R}}(X;\widehat{\mathfrak{s}}_{1},\widehat{\mathfrak{s}}_{2})$,
we have

$\displaystyle\mathcal{E}_{top}(\mathfrak{a}_{1},\widehat{\mathfrak{s}}_{X}^{\prime},\mathfrak{a}_{2})-\mathcal{E}_{top}(\mathfrak{a}_{1},\widehat{\mathfrak{s}}_{X},\mathfrak{a}_{2})=-4\pi^{2}\big{(}\dim\mathcal{M}(\mathfrak{a}_{1},\widehat{\mathfrak{s}}_{X}^{\prime},\mathfrak{a}_{2})-\dim\mathcal{M}(\mathfrak{a}_{1},\widehat{\mathfrak{s}}_{X},\mathfrak{a}_{2})\big{)}$

In particular, the topological energy
$\mathcal{E}_{top}^{\mathfrak{q}}(\mathfrak{a}_{1},\widehat{\mathfrak{s}}_{X},\mathfrak{a}_{2})$
is independent of the choice of
$\widehat{\mathfrak{s}}_{X}\in\operatorname{Spin^{c}_{R}}(X;\widehat{\mathfrak{s}}_{1},\widehat{\mathfrak{s}}_{2})$
if
$\dim\mathcal{M}(\mathfrak{a}_{1},\widehat{\mathfrak{s}}_{X},\mathfrak{a}_{2})=0$.

###### Proof.

Suppose $\widehat{\mathfrak{s}}_{X}^{\prime}=\widehat{\mathfrak{s}}_{X}\otimes
L$ for a relative complex line bundle in the class $[L]\in H^{2}(X,\partial
X;\mathbb{Z})$. In terms of (17.3) and (17.4), we compute the difference of
the topological energy

$\displaystyle\mathcal{E}_{top}(\mathfrak{a}_{1},\widehat{\mathfrak{s}}_{X}^{\prime},\mathfrak{a}_{2})-\mathcal{E}_{top}(\mathfrak{a}_{1},\widehat{\mathfrak{s}}_{X},\mathfrak{a}_{2})$
$\displaystyle=C(A_{0}(\widehat{\mathfrak{s}}_{X}^{\prime}),\omega_{X})-C(A_{0}(\widehat{\mathfrak{s}}_{X}),\omega_{X})$
$\displaystyle=-2\pi^{2}[L]\cup(c_{1}(\widehat{\mathfrak{s}}_{X})+c_{1}(\widehat{\mathfrak{s}}_{X}^{\prime}))[X,\partial
X]$
$\displaystyle=-4\pi^{2}[L]\cup(c_{1}(\widehat{\mathfrak{s}}_{X})+[L])[X,\partial
X].$

where $c_{1}(\widehat{\mathfrak{s}}_{X})$ and $\
c_{1}(\widehat{\mathfrak{s}}_{X}^{\prime})$ are understood as elements in
$H^{2}(X,[-1,1]\times\Sigma;\mathbb{Z})$. On the other hand, pick an arbitrary
non-vanishing section $\Phi_{0}$ of

$S^{+}|_{\partial X}\to\partial X.$

Any relative $spin^{c}\ $structure
$\widehat{\mathfrak{s}}_{X}\in\operatorname{Spin^{c}_{R}}(X;\widehat{\mathfrak{s}}_{1},\widehat{\mathfrak{s}}_{2})$
dictates an identification of $\widehat{\mathfrak{s}}_{X}|_{\partial X}$ with
a standard $spin^{c}\ $structure on the boundary $\partial X$, so it makes
sense to define the relative Euler number
$e(\widehat{\mathfrak{s}}_{X};\Phi_{0})[X,\partial X]$ for any non-vanishing
section $\Phi_{0}$ of the spin bundle $S^{+}\to\partial X$. In particular,

$\big{(}e(\widehat{\mathfrak{s}}_{X}^{\prime};\Phi_{0})-e(\widehat{\mathfrak{s}}_{X};\Phi_{0})\big{)}[X,\partial
X]=[L]\cup(c_{1}(\widehat{\mathfrak{s}}_{X})+[L])[X,\partial X].$

In Proposition 18.6 below, we will associate a homotopy class of non-vanishing
sections $[\Phi_{0}(\mathfrak{a}_{1},\mathfrak{a}_{2})]$ to any pair
$(\mathfrak{a}_{1},\mathfrak{a}_{2})$ such that

(17.6)
$e(\widehat{\mathfrak{s}}_{X};\Phi_{0}(\mathfrak{a}_{1},\mathfrak{a}_{2}))[X,\partial
X]=\dim\mathcal{M}(\mathfrak{a}_{1},\widehat{\mathfrak{s}}_{X},\mathfrak{a}_{2})$

for any
$\widehat{\mathfrak{s}}_{X}\in\operatorname{Spin^{c}_{R}}(X;\widehat{\mathfrak{s}}_{1},\widehat{\mathfrak{s}}_{2})$.
In fact, (17.6) follows from the Index Axiom (A-I) of the canonical grading of
$\operatorname{\it HM}_{*}(\mathbb{Y},\widehat{\mathfrak{s}})$. Another
approach is to show

$\big{(}e(\widehat{\mathfrak{s}}_{X}^{\prime};\Phi_{0})-e(\widehat{\mathfrak{s}}_{X};\Phi_{0})\big{)}[X,\partial
X]=\dim\mathcal{M}(\mathfrak{a}_{1},\widehat{\mathfrak{s}}_{X}^{\prime},\mathfrak{a}_{2})-\dim\mathcal{M}(\mathfrak{a}_{1},\widehat{\mathfrak{s}}_{X},\mathfrak{a}_{2})$

for any non-vanishing section $\Phi_{0}$ directly using the excision
principle. This completes the proof of Proposition 17.6 ∎

Finally, one has to verify that $m(\mathbb{X};g_{X},\mathfrak{q})$ is a chain
map and a generic homotopy of auxiliary data $(g_{X},\mathfrak{q})$ gives rise
to a chain homotopy of $m(\mathbb{X};g_{X},\mathfrak{q})$. The argument is not
different from that of [KM07, Section 25].

###  18\. Canonical Gradings

In this section, we introduce the canonical grading of the monopole Floer
homology $\operatorname{\it HM}_{*}(\mathbb{Y},\widehat{\mathfrak{s}})$. It is
more natural to think of the grading set of $\operatorname{\it
HM}_{*}(\mathbb{Y},\widehat{\mathfrak{s}})$

$\Xi^{\pi}(\mathbb{Y},\widehat{\mathfrak{s}})$

as the space of unit-length relative spinors on $\widehat{Y}$ modulo gauge
transformations, identified also as a subset of homotopy classes of oriented
relative 2-plane fields on $Y$. In particular,

$\Xi^{\pi}(\mathbb{Y},\widehat{\mathfrak{s}}_{1})=\Xi^{\pi}(\mathbb{Y},\widehat{\mathfrak{s}}_{2})$

if $\widehat{\mathfrak{s}}_{1}$ and $\widehat{\mathfrak{s}}_{2}$ come down to
the same $spin^{c}\ $structure on $Y$.

The main result of this section is Proposition 18.6, which characterizes the
canonical grading in terms of the Index Axiom (A-I) and the Normalization
Axiom (A-II). They are inspired by the following index computation for a
closed Riemannian 4-manifold $X$:

$\dim\mathcal{M}(X,\mathfrak{s}_{X})=e(\mathfrak{s}_{X})[X]$

where $\mathcal{M}(X,\mathfrak{s}_{X})$ is the Seiberg-Witten moduli space and
$e(\mathfrak{s}_{X})$ is the Euler class of the spin bundle $S_{X}^{+}\to X$.
The canonical mod 2 grading will be discussed in Subsection 18.2.

####  18.1. Homotopy Classes of Oriented Relative 2-Plane Fields

For a closed $3$-manifold $Y$, recall that the three flavors of monopoles
Floer homology:

$\widecheck{\operatorname{\it HM}}_{\bullet}(Y),\ \widehat{\operatorname{\it
HM}}_{\bullet}(Y),\ \overline{\operatorname{\it HM}}_{\bullet}(Y)$

defined in the book [KM07] are graded by the homotopy classes of oriented
2-plane fields over $Y$. The analogous statement continues to hold in our
case, using relative oriented 2-plane fields instead, as we explain now. The
following lemma from [KM07] explains the relationship between 2-plane fields
and $spin^{c}\ $structures:

######  Lemma 18.1 ([KM07] Lemma 28.1.1).

On an oriented Riemannian 3-manifold $Y$, there is a bijection between

  1. (i)

oriented $2$-plane fields $\xi$;

  2. (ii)

1-forms $\theta$ of length $1$; and

  3. (iii)

isomorphism classes of pairs $(\mathfrak{s},\Psi)$ comprising a $spin^{c}\
$structure and a unit-length spinor $\Psi$.

Over the infinite cylinder $\mathbb{R}_{s}\times\Sigma$, we defined in (2.6) a
preferred $\mathbb{R}_{s}$-translation invariant solution

$\gamma_{*}=(B_{*},\Psi_{*})$

to the perturbed Seiberg-Witten equations $\eqref{3DSWEQ}$. The perturbation
is provided by a covariantly constant 2-form

$\omega_{*}\colonequals\mu+ds\wedge\lambda$

The correspondence in Lemma 18.1 then identifies

(18.1) $\text{the unit length 1-form }\theta_{*}\colonequals
i*_{3}\frac{\omega_{*}}{|\omega_{*}|}\leftrightarrow\text{the unit length
spinor }\frac{\Psi_{*}}{|\Psi_{*}|},$

Indeed, as $\gamma_{*}$ solves the equations (3.6),
$(\Psi_{*}\Psi_{*}^{*})_{0}=\rho_{3}(*_{3}\omega_{*})$, so

$\mathbb{C}\Psi_{*}\text{ and }\mathbb{C}(\Psi_{*})^{\perp}$

are $i$ and $-i$ eigenspaces of $\rho_{3}(\theta_{*})$ respectively. In
particular, (18.1) determines a preferred oriented 2-plane fields $\xi_{*}$ on
$\mathbb{R}_{s}\times\Sigma$ by Lemma 18.1. Now we return to a 3-manifold
$\widehat{Y}$ with cylindrical ends and state a relative version of Lemma
18.1.

######  Definition 18.2.

An oriented 2-plane field $\xi$ on $\widehat{Y}$ is called relative if $\xi$
agrees with $\xi_{*}$ over the cylindrical end $[0,\infty)_{s}\times\Sigma$.
Similarly, we define

  * •

relative 1-forms and

  * •

relative spinors

using $\theta_{*}$ and $\Psi_{*}/|\Psi_{*}|$ as the models along the end
$[0,\infty)_{s}\times\Sigma$. ∎

######  Lemma 18.3.

For any object $\mathbb{Y}\in\operatorname{Cob}_{s}$, let $\widehat{Y}$ be the
extended 3-manifold with cylindrical ends. Then there is a bijection between:

  1. (i)

oriented relative $2$-plane fields $\xi$;

  2. (ii)

1-forms relative $\theta$ of length $1$; and

  3. (iii)

isomorphism classes of pairs $(\mathfrak{s},\Psi)$ consisting of a $spin^{c}\
$structure $\mathfrak{s}$ with $c_{1}(\mathfrak{s})|_{\Sigma}=0\in
H^{2}(\Sigma,\mathbb{Z})$ and a unit-length spinor $\Psi$ that is gauge
equivalent to a relative spinor.

######  Remark 18.4.

In the last description, the identification of
$\widehat{\mathfrak{s}}|_{\Sigma}$ is not specified and a gauge transformation
does not necessarily lie in the identity component when restricted to
$\Sigma$. ∎

For each relative $spin^{c}\ $structure
$\widehat{\mathfrak{s}}\in\operatorname{Spin^{c}_{R}}(Y)$, let
$\Xi(\widehat{Y},\widehat{\mathfrak{s}})$ be the space of unit-length relative
spinors on $\widehat{Y}$. The index set for the monopole Floer homology
$\operatorname{\it HM}_{*}(Y,\widehat{\mathfrak{s}})$ will be

(18.2)
$\Xi^{\pi}(Y,\widehat{\mathfrak{s}})\colonequals\pi_{0}(\Xi(\widehat{Y},\widehat{\mathfrak{s}}))/H^{1}(Y,\partial
Y;\mathbb{Z})$

where $H^{1}(Y,\partial
Y;\mathbb{Z})=\pi_{0}(\mathcal{G}(\widehat{Y},\widehat{\mathfrak{s}}))$ acts
on $\pi_{0}(\Xi(\widehat{Y},\widehat{\mathfrak{s}}))$ by gauge
transformations. The last description in Lemma 18.3 suggests that

$\Xi^{\pi}(Y,\widehat{\mathfrak{s}}_{1})\cong\Xi^{\pi}(Y,\widehat{\mathfrak{s}}_{2})$

if $\widehat{\mathfrak{s}}_{1}$ and $\widehat{\mathfrak{s}}_{2}$ come down to
the same $spin^{c}\ $structure on $Y$. In this way,
$\Xi^{\pi}(Y,\widehat{\mathfrak{s}})$ is identified with a subset of homotopy
classes of oriented relative 2-plane fields.

Now let us introduce the axioms that characterize the canonical grading of
$\operatorname{\it HM}_{*}(\mathbb{Y},\widehat{\mathfrak{s}})$.

######  Definition 18.5.

For any configuration
$\mathfrak{a}\in\mathcal{C}_{k}(\widehat{Y},\widehat{\mathfrak{s}})$ and any
tame perturbation $\mathfrak{q}\in\mathcal{P}(Y)$, the pair
$\mathfrak{c}=(\mathfrak{a},\mathfrak{q})$ is called non-degenerate if the
extended Hessian $\widehat{\operatorname{Hess}}_{\mathfrak{a},\mathfrak{q}}$
is invertible. ∎

For any non-degenerate pair $\mathfrak{c}=(\mathfrak{a},\mathfrak{q})$, we
will assign an element

$\operatorname{\mathbf{gr}}(\mathfrak{c})\in\pi_{0}(\Xi(\widehat{Y},\widehat{\mathfrak{s}})).$

which descends to a map

(18.3)
$\operatorname{\mathbf{gr}}^{\pi}:(\mathcal{C}_{k}(\widehat{Y},\widehat{\mathfrak{s}})\times\mathcal{P})/\mathcal{G}_{k+1}(\widehat{Y})\dashrightarrow\Xi^{\pi}(\widehat{Y},\widehat{\mathfrak{s}}),[\mathfrak{c}]\mapsto[\operatorname{\mathbf{gr}}(\mathfrak{c})],$

on the “non-degenerate locus” of the quotient space. To state the axioms that
characterize the grading function $\operatorname{\mathbf{gr}}$, consider a
relative $spin^{c}\ $cobordism

$(\widehat{X},\widehat{\mathfrak{s}}_{X}):(\widehat{Y}_{1},\widehat{\mathfrak{s}}_{1})\to(\widehat{Y}_{1},\widehat{\mathfrak{s}}_{2}).$

We defined the moduli space
$\mathcal{M}_{k}(\mathfrak{a}_{1},\mathcal{X},\mathfrak{a}_{2})$ in Section
14, when $\mathfrak{a}_{i}$ is a critical point of $\hbox
to0.0pt{$-$\hss}\mathcal{L}_{\omega_{i},Y_{i}}$ for $i=1,2$. However, if we
are interested only in the linear theory, one may take $\mathfrak{a}_{1}$ and
$\mathfrak{a}_{2}$ to be any configurations. Pick a reference configuration
$\gamma$ on $\mathcal{X}$ satisfying conditions (14.1). Then the linearized
operator:

(18.4)
$\displaystyle\mathcal{Q}(\mathfrak{c}_{1},\widehat{\mathfrak{s}}_{X},\mathfrak{c}_{2})$
$\displaystyle\colonequals(\operatorname{\mathbf{d}}_{\gamma}^{*},\mathcal{D}_{\gamma}\mathfrak{F}_{\mathcal{X},\mathfrak{p}}):L^{2}_{1}(\mathcal{X},iT^{*}\mathcal{X}\oplus
S^{+})\to L^{2}(\mathcal{X},i\mathbb{R}\oplus i\Lambda^{+}\mathcal{X}\oplus
S^{-})$ $\displaystyle\text{ with }\mathfrak{p}$
$\displaystyle=(\mathfrak{q}_{1},\mathfrak{q}_{2},0,0)$

is Fredholm, by Proposition 14.1, provided that
$\mathfrak{c}_{i}=(\mathfrak{a}_{i},\mathfrak{q}_{i})$ is non-degenerate for
$i=1,2$. Any such choices of $\gamma$ will provide the same operator
$\mathcal{Q}(\mathfrak{c}_{1},\widehat{\mathfrak{s}}_{X},\mathfrak{c}_{2})$ up
to compact terms, so the underlying path $\gamma$ is omitted from our
notations.

Now we are ready to state the axioms that characterize the grading function
$\operatorname{\mathbf{gr}}$.

  1. (A-I)

(Index Axiom) The Fredholm index of
$\mathcal{Q}(\mathfrak{c}_{1},\widehat{\mathfrak{s}}_{X},\mathfrak{c}_{2})$
equals the relative Euler number:

$e(S^{+};\Psi_{1},\Psi_{*}/|\Psi_{*}|,\Psi_{2})[X,\partial X]\in\mathbb{Z}.$

where $\Psi_{i}$ is a unit-length relative spinor on $\widehat{Y}_{i}$
representing $\operatorname{\mathbf{gr}}(\mathfrak{c}_{i})$. Since
$\Psi_{1},\Psi_{*}/|\Psi_{*}|$ and $\Psi_{2}$ form a unit-length spinor of
$S^{+}$ on the boundary

$\partial X=(-Y_{1})\cup[-1,1]_{t}\times\Sigma\cup Y_{2},$

the relative Euler class $e(S^{+};\Psi_{1},\Psi_{*}/|\Psi_{*}|,\Psi_{2})\in
H^{4}(X,\partial X;\mathbb{Z})$ of this spinor is well-defined.

  1. (A-II)

(Normalization Axiom) Suppose
$\mathfrak{a}=(B,\Psi)\in\mathcal{C}_{k}(\widehat{Y},\widehat{\mathfrak{s}})$
is a configuration such that

    1. (V1)

$\Psi$ is nowhere vanishing;

    2. (V2)

$\Psi\equiv\Psi_{*}$ on $[0,+\infty)_{s}\times\Sigma$, where $\Psi_{*}$ is the
standard spinor on $\mathbb{R}_{s}\times\Sigma$;

    3. (V3)

for any $\tau\geq 1$, define the rescaled configuration
$\mathfrak{a}(\tau)\colonequals(B,\tau\Psi)$; then the extended Hessian
$\widehat{\operatorname{Hess}}_{\mathfrak{a}(\tau)}$ at $\mathfrak{a}(\tau)$
is always invertible for any $\tau\geq 1$.

We define that

$\operatorname{\mathbf{gr}}(\mathfrak{c})=[\Psi/|\Psi|]\in\pi_{0}(\Xi(\widehat{Y},\widehat{\mathfrak{s}}))\text{
if }\mathfrak{c}=(\mathfrak{a},0).$

Note that $\mathfrak{a}(\tau)$ lies in a different configuration space
obtained by rescaling the boundary date $(\lambda,\mu)$.

  1. (A-III)

(Equivariance Axiom) The grading function

$\operatorname{\mathbf{gr}}:\mathcal{C}_{k}(\widehat{Y},\widehat{\mathfrak{s}})\times\mathcal{P}\dashrightarrow\pi_{0}(\Xi(\widehat{Y},\widehat{\mathfrak{s}}))$

is equivariant under the action of $\mathcal{G}_{k+1}(\widehat{Y})$ meaning
that

$\operatorname{\mathbf{gr}}(u\cdot\mathfrak{a},\mathfrak{q})=[u]\cdot\operatorname{\mathbf{gr}}(\mathfrak{a},\mathfrak{q})$

for any non-generate pair $(\mathfrak{a},\mathfrak{q})$ and
$u\in\mathcal{G}_{k+1}(\widehat{Y})$.

The Index Axiom (A-I) can not determine the grading function
$\operatorname{\mathbf{gr}}$ completely. On the other hand, the Equivariance
Axiom (A-III) is redundant, since it follows from (A-I)(A-II). It is added to
justify the quotient map $\operatorname{\mathbf{gr}}^{\pi}$ in (18.3). Here is
the main result of this section:

######  Proposition 18.6.

There exists a unique grading function

$\operatorname{\mathbf{gr}}:\mathcal{C}_{k}(\widehat{Y},\widehat{\mathfrak{s}})\times\mathcal{P}\dashrightarrow\Xi(\widehat{Y},\widehat{\mathfrak{s}})$

satisfying axioms (A-I)(A-II)(A-III).

The proof of Proposition 18.6 will dominate the rest of this subsection. It
relies on two additional lemmas. On the one hand, we have to show the desired
configurations in the Normalization Axiom (A-II) exist at least for some
special metrics on $Y$.

######  Lemma 18.7.

For any $3$-manifold $Y$ with $\partial Y\cong\Sigma$, there exists some
cylindrical metric $g_{Y}$ and a configuration
$\mathfrak{a}\in\mathcal{C}_{k}(\widehat{Y},\widehat{\mathfrak{s}})$ that
satisfies all constraints in Axiom (A-II).

On the other hand, we have to show that Axioms (A-I) and (A-II) are
consistent.

######  Lemma 18.8.

For any relative $spin^{c}\ $cobordism
$(\widehat{X},\widehat{\mathfrak{s}}_{X}):(\widehat{Y}_{1},\widehat{\mathfrak{s}}_{1})\to(\widehat{Y}_{1},\widehat{\mathfrak{s}}_{2}),$
suppose non-generate pairs $\mathfrak{c}_{i}=(\mathfrak{a}_{i},0),i=1,2$ are
given as in (A-II), then

$\operatorname{Ind}\mathcal{Q}(\mathfrak{c}_{1},\widehat{\mathfrak{s}}_{X},\mathfrak{c}_{2})=e(S^{+};\frac{\Psi_{1}}{|\Psi_{1}|},\frac{\Psi_{*}}{|\Psi_{*}|},\frac{\Psi_{2}}{|\Psi_{2}|})[X,\partial
X],$

where $\Psi_{i}\in\Gamma(\widehat{Y}_{i},S)$ is the spinor component of
$\mathfrak{a}_{i}\in\mathcal{C}_{k}(\widehat{Y}_{i},\widehat{\mathfrak{s}}_{i})$.

###### Proof of Lemma 18.8.

This lemma is in the spirit of [KM97, Theorem 3.3] and we follow the argument
therein. When $X_{1}$ is a closed Riemannian 4-manifold, the index formula:

$\dim\mathcal{M}(X_{1},\widehat{\mathfrak{s}}_{X_{1}})=e(S^{+})[X_{1}]$

is a consequence of the Atiyah-Singer Index Theorem and [KM07, Lemma 28.2.3].
Using the excision principle, this allows us to reduce Lemma 18.8 to the
special case when

$e(S^{+};\Psi_{1},\Psi_{*},\Psi_{2})[X,\partial X]=0.$

At this point, choose a reference configuration $\gamma=(A,\Phi)$ on
$\mathcal{X}$ such that the spinor $\Phi$ is non-vanishing everywhere, and

$\gamma|_{\mathbb{H}^{2}_{+}\times\Sigma}=(A_{*},\Phi_{*})$

is the standard configuration on the planar end. By rescaling the spinor
$\Phi$, we define

$\gamma(\tau)\colonequals(A,\tau\Phi).$

which lies a different configuration space on $\mathcal{X}$. As the pair
$\mathfrak{c}_{i}(\tau)\colonequals(\mathfrak{a}_{i}(\tau),\mathfrak{q}_{i}=0),i=1,2$
are non-degenerate for any $\tau\geq 1$ by assumption (A-II)(V3), the
linearized operator at $\gamma(\tau)$ gives rise to a continuous family of
Fredholm operators:

$\mathcal{Q}(\tau)\colonequals\mathcal{Q}(\mathfrak{c}_{1}(\tau),\widehat{\mathfrak{s}}_{X},\mathfrak{c}_{2}(\tau)).$

The proof of [KM97, Lemma 3.11 & Corollary 3.12] is valid here, as
$\mathfrak{q}_{i}=0,i=1,2$. As a result, $\mathcal{Q}(\tau)$ is invertible
when $\tau\gg 1$; so

$\operatorname{Ind}\mathcal{Q}(1)=\lim_{\tau\to\infty}\operatorname{Ind}\mathcal{Q}(\tau)=0.\qed$

###### Proof of Lemma 18.7.

Following the proof of Lemma 18.8, one can easily show the extended Hessian
$\widehat{\operatorname{Hess}}_{\mathfrak{a}(\tau)}$ is invertible when
$\tau\gg 1$ for any fixed configuration $\mathfrak{a}=(B,\Psi)$ satisfying
properties (A-II)(V1) and (A-II)(V2), but we have to pick a good metric on
$\widehat{Y}$ so that this range is $[1,+\infty)$.

If $Y_{1}$ is a closed 3-manifold, one may instead rescale the metric:

$Y_{1}(\tau)=(Y_{1},\tau^{2}g_{Y_{1}}).$

and regard $\mathfrak{a}$ as a configuration on the pull-back $spin^{c}\
$structure on $Y(\tau)$. The Seiberg-Witten theory does not tell the
difference between:

$(Y(\tau),\mathfrak{a})\text{ and }(Y,\mathfrak{a}(\tau)),$

so for $\tau_{0}\gg 1$, $(Y(\tau_{0}),\mathfrak{a})$ satisfies constraints
(A-II)(V1)(A-II)(V3) in Axiom (A-II).

In our case, instead of rescaling the whole manifold

$\widehat{Y}=Y\cup[0,\infty)_{s}\times\Sigma,$

we rescale the compact region $Y$ and insert a long cylinder:

$\widehat{Y}(\tau)\colonequals
Y(\tau)\cup[0,R(\tau)]_{s}\times\Sigma\cup[0,\infty)_{s}\times\Sigma.$

The metric of $[0,R(\tau)]_{s}\times\Sigma$ interpolates the metrics
$\tau^{2}g_{\Sigma}$ and $g_{\Sigma}$ at boundary. We make this interpolation
mild enough by taking $R(\tau)\gg 1$. The extension of $\mathfrak{a}$ over the
cylinder $[0,R(\tau)]_{s}\times\Sigma$:

$(B^{\prime},\Psi^{\prime})$

must interpolate $(B_{*},\Psi_{*})$ at boundary in a mild way. One may use the
oriented relative 2-plane field $\xi_{*}$ and construct the spinor
$\Psi^{\prime}$ using Lemma 18.1. Now [KM97, Lemma 3.11] applies, and all
constraints in (A-II) are satisfied by

$(\widehat{Y}(\tau_{0}),\tilde{\mathfrak{a}})$

when $\tau_{0}\gg 1$, where $\tilde{\mathfrak{a}}$ is the extension of
$\mathfrak{a}$ on $\widehat{Y}(\tau_{0})$. ∎

###### Proof of Proposition 18.6.

The proof is modeled on that of [KM07, Subsection 28.2] which can now proceed
with no difficulties. We first deal with the existence of
$\operatorname{\mathbf{gr}}$ and divide the proof in six steps.

Step 1\. Construction. Fix a reference relative $spin^{c}\ $3-manifold
$(\widehat{Y}_{0},\widehat{\mathfrak{s}}_{0})$. Let
$\mathfrak{c}_{0}=(\mathfrak{a}_{0},0)$ be a non-generate pair constructed by
Lemma 18.7, then the value $\operatorname{\mathbf{gr}}(\mathfrak{c})$ is
determined by (A-II). Take $\Psi_{0}$ as a unit-length relative spinor on
$\widehat{Y}_{0}$ that represents $\operatorname{\mathbf{gr}}(\mathfrak{c})$.

By [KM07, Proposition 28.1.2], any two relative $spin^{c}\ $manifolds
$(\widehat{Y}_{0},\widehat{\mathfrak{s}}_{0})$ and
$(\widehat{Y}_{1},\widehat{\mathfrak{s}}_{1})$ admit a relative $spin^{c}\
$cobordism $(\widehat{X},\widehat{\mathfrak{s}}_{X})$

(18.5)
$(\widehat{X},\widehat{\mathfrak{s}}_{X}):(\widehat{Y}_{0},\widehat{\mathfrak{s}}_{0})\to(\widehat{Y}_{1},\widehat{\mathfrak{s}}_{1})$

The Index Axiom (A-I) then determines a unique homotopy class $[\Psi_{1}]$ of
unit-length relative spinors on $\widehat{Y}_{1}$ such that

$\operatorname{Ind}\mathcal{Q}(\mathfrak{c}_{0},\widehat{\mathfrak{s}}_{X},\mathfrak{c}_{1})=e(S^{+};\Psi_{0},\Psi_{*}/|\Psi_{*}|,\Psi_{1})[X,\partial
X].$

As noted in Remark 3.13, an isomorphism

$\varphi_{1}:(\widehat{X},\widehat{\mathfrak{s}}_{X})|_{\widehat{Y}_{i}}\cong(\widehat{Y}_{1},\widehat{\mathfrak{s}}_{1})$

is always encoded in a relative $spin^{c}\ $cobordism. Define
$\operatorname{\mathbf{gr}}(\mathfrak{c}_{1})\colonequals(\varphi_{1})_{*}[\Psi_{1}]\in\pi_{0}(\Xi(\widehat{Y}_{1},\widehat{\mathfrak{s}}_{1}))$.

Step 2\. $\operatorname{\mathbf{gr}}$ is well-defined. Suppose there is
another relative $spin^{c}\ $cobordism

(18.6)
$(\widehat{X}_{1},\widehat{\mathfrak{s}}_{X_{1}}):(\widehat{Y}_{0},\widehat{\mathfrak{s}}_{0})\to(\widehat{Y}_{1},\widehat{\mathfrak{s}}_{1}),$

then we reverse the orientation of
$(\widehat{X}_{1},\widehat{\mathfrak{s}}_{X_{1}})$ and form the composition:

$(\widehat{X},\widehat{\mathfrak{s}}_{X})\\#_{(\widehat{Y}_{1},\widehat{\mathfrak{s}}_{1})}((-\widehat{X}_{1}),\widehat{\mathfrak{s}}_{-X_{1}}):(\widehat{Y}_{0},\widehat{\mathfrak{s}}_{0})\to(\widehat{Y}_{0},\widehat{\mathfrak{s}}_{0}).$

By Lemma 18.8 and the additivity of Fredholm indices and relative Euler
classes, the values of $\operatorname{\mathbf{gr}}(\mathfrak{c}_{1})$ defined
using either (18.5) or (18.6) are equal.

Step 3\. Axiom (A-I) holds for $\operatorname{\mathbf{gr}}$. The proof is
similar to Step 2\. Instead of (18.6), given any $spin^{c}\ $cobordism
$(\widehat{X}_{2},\widehat{\mathfrak{s}}_{X_{2}}):(\widehat{Y}_{1},\widehat{\mathfrak{s}}_{1})\to(\widehat{Y}_{1},\widehat{\mathfrak{s}}_{2})$,
we take the pre-composition with (18.5):

$(\widehat{X},\widehat{\mathfrak{s}}_{X})\\#_{(\widehat{Y}_{1},\widehat{\mathfrak{s}}_{1})}(\widehat{X}_{2},\widehat{\mathfrak{s}}_{X_{2}}):(\widehat{Y}_{0},\widehat{\mathfrak{s}}_{0})\to(\widehat{Y}_{2},\widehat{\mathfrak{s}}_{2}).$

The rest of the argument is unchanged.

Step 4\. Axiom (A-II) holds for $\operatorname{\mathbf{gr}}$. This is by Lemma
18.8.

Step 5\. Uniqueness. This is clear from Step 1.

Step 6\. Axiom (A-III). There are two ways to proceed. In Step 1, one may
change the isomorphism $\phi_{1}$ by an automorphism of
$(\widehat{Y}_{1},\widehat{\mathfrak{s}}_{1})$, i.e a gauge transformation
$u\in\mathcal{G}_{k+1}(\widehat{Y})$. As a result, the grading function
$\operatorname{\mathbf{gr}}$ is gauge equivariant.

In the second approach, we verify the following fact: for the product manifold
$X=[-1,1]_{t}\times Y$ and $\mathcal{X}=\mathbb{R}_{t}\times\widehat{Y}$,

(18.7)
$\operatorname{Ind}\mathcal{Q}(\mathfrak{c},\widehat{\mathfrak{s}},u\cdot\mathfrak{c})=e(S^{+};\Psi,\Psi_{*},u\cdot\Psi)[X,\partial
X].$

for any non-generate pair $\mathfrak{c}$ and any gauge transformation
$u\in\mathcal{G}_{k+1}(\widehat{Y})$ such that $u\equiv 1$ on
$[0,\infty)_{s}\times\Sigma$. Here $\Psi$ is a relative spinor on
$\widehat{Y}$ representing $\operatorname{\mathbf{gr}}(\mathfrak{c})$. The
identity (18.7) now follows from Lemma 13.16. ∎

####  18.2. Canonical Mod 2 Gradings

Now we focus a single relative $spin^{c}\ $3-manifold
$(\widehat{Y},\widehat{\mathfrak{s}})$. In order to define the Euler
characteristic of the monopole Floer homology

$\chi(\operatorname{\it HM}_{*}(\widehat{Y},\widehat{\mathfrak{s}}))$

we need a mod 2 reduction of the canonical grading
$\operatorname{\mathbf{gr}}^{\pi}$. For each non-generate pair
$\mathfrak{c}=(\mathfrak{a},\mathfrak{q})$, in the sense of Definition 18.5,
we will assign a number

(18.8)
$\operatorname{\mathbf{gr^{(2)}}}(\mathfrak{c})\in\mathbb{Z}/2\mathbb{Z},$

characterized by the following axioms:

  1. (B-I)

(Reduction Axiom) Let
$(\widehat{X},\widehat{\mathfrak{s}}_{X})=[-1,1]_{t}\times(\widehat{Y},\widehat{\mathfrak{s}})$
be the product $spin^{c}\ $manifold. For any
$\mathfrak{c}_{1},\mathfrak{c}_{2}$ non-generate, we have

$\operatorname{\mathbf{gr^{(2)}}}(\mathfrak{c}_{1})-\operatorname{\mathbf{gr^{(2)}}}(\mathfrak{c}_{2})=\text{Ind}\mathcal{Q}(\mathfrak{c}_{1},\widehat{\mathfrak{s}}_{X},\mathfrak{c}_{2})\mod
2,$

  2. (B-II)

(Invariance Axiom) The mod 2 grading function

$\operatorname{\mathbf{gr^{(2)}}}:\mathcal{C}_{k}(\widehat{Y},\widehat{\mathfrak{s}})\times\mathcal{P}\dashrightarrow\mathbb{Z}/2\mathbb{Z}$

is invariant under the action of $\mathcal{G}_{k+1}(\widehat{Y})$.

Again, the Invariance Axiom (B-II) is redundant, as it follows from (B-I). One
may fix the value $\operatorname{\mathbf{gr^{(2)}}}(\mathfrak{c}_{1})$ for one
particular pair $\mathfrak{c}_{1}$ and decide the other value
$\operatorname{\mathbf{gr^{(2)}}}(\mathfrak{c}_{2})$ using the Reduction Axiom
(B-I), so such a mod 2 grading function $\operatorname{\mathbf{gr^{(2)}}}$
clearly exists. It is not unique, as the value of
$\operatorname{\mathbf{gr^{(2)}}}(\mathfrak{c}_{1})$ is arbitrary.

This ambiguity is fixed simultaneously for all relative $spin^{c}\ $structures
$\widehat{\mathfrak{s}}\in\operatorname{Spin^{c}_{R}}(Y)$, once a homological
orientation of $(Y,\partial Y)$ is chosen, as explained in [MT96], which is
also reminiscent of the case of 4-manifolds as treated in [KM07, Subsection
24.8]. Since this story has been standard nowadays, we only give a brief
sketch here.

One may alternatively think of
$\operatorname{\mathbf{gr^{(2)}}}(\mathfrak{c})$ as an orientation of the
extended Hessian

$\widehat{\operatorname{Hess}}_{\mathfrak{c}}.$

As $\mathfrak{c}$ is non-generate, an orientation of this invertible operator
$\widehat{\operatorname{Hess}}_{\mathfrak{c}}$ is equivalent to a choice of
signs in $\\{\pm 1\\}$. However, this standpoint allows us to extend the
domain of $\operatorname{\mathbf{gr^{(2)}}}$ to the whose space
$\mathcal{C}_{k}(\widehat{Y},\widehat{\mathfrak{s}})\times\mathcal{P}$.
Indeed, $\\{\widehat{\operatorname{Hess}}_{\mathfrak{c}}\\}$ forms a
continuous family of Fredholm operators, and as such gives rise to a
determinant line bundle over the base:

${\mathbb{R}\cong\det\widehat{\operatorname{Hess}}_{\mathfrak{c}}}$${L}$${\mathcal{C}_{k}(\widehat{Y},\widehat{\mathfrak{s}})\times\mathcal{P}.}$

The real line bundle $L$ is trivial as
$\mathcal{C}_{k}(\widehat{Y},\widehat{\mathfrak{s}})\times\mathcal{P}$ is
contractible. To orient $L$, it suffices to orient one particular fiber
$L_{\mathfrak{c}}$; we choose the one at $\mathfrak{c}=(\mathfrak{a},0)$ such
that $\mathfrak{a}$ agrees with the standard configuration:

$(B_{*},\Psi_{*})$

on the cylindrical end $[0,\infty)_{s}\times\Sigma$. As explained in the proof
of Proposition 12.1, the extended Hessian
$\widehat{\operatorname{Hess}}_{\mathfrak{a}}$ in this case is cast into the
form

$\sigma(\partial_{s}+\widehat{D}_{\kappa_{*}})$

on the cylindrical end $[0,\infty)_{s}\times\Sigma$, where

(18.9) $\widehat{D}_{\kappa_{*}}:L^{2}_{1}(\Sigma,i\mathbb{R}\oplus
i\mathbb{R}\oplus T^{*}\Sigma\oplus S)\to L^{2}(\Sigma,i\mathbb{R}\oplus
i\mathbb{R}\oplus T^{*}\Sigma\oplus S)$

is an invertible self-adjoint elliptic operator. For the precise expression,
see [Wan20, Subsection 7.4]. Let $H^{\pm}$ be the $(\pm)$-spectral subspaces
of $\widehat{D}_{\kappa_{*}}$. Instead of
$\widehat{\operatorname{Hess}}_{\mathfrak{a}}$, we consider the operator with
a spectral boundary projection:

(18.10) $\widehat{\operatorname{Hess}}_{\mathfrak{a}}\oplus\Pi^{-}\circ
r:L^{2}_{k}(Y,i\mathbb{R}\oplus iT^{*}Y\oplus S)\to
L^{2}_{k-1}(Y,i\mathbb{R}\oplus iT^{*}Y\oplus S)\oplus(H^{-}\cap
L^{2}_{k-1/2}).$

on the truncated 3-manifold $Y=\\{s\leq 0\\}$. At this point, we can further
deform $\mathfrak{a}$ so that $\Psi\equiv 0$, in which case

$\widehat{\operatorname{Hess}}_{\mathfrak{a}}=\begin{pmatrix}0&-d&0\\\
-d^{*}&*d&0\\\ 0&0&D_{B_{0}}\end{pmatrix}\text{ on }Y$

for a reference $spin^{c}\ $connection $B_{0}$, and

$\widehat{\operatorname{Hess}}_{\mathfrak{a}}=\sigma(\partial_{s}+\hat{D}_{0})\text{
with }D_{0}=\begin{pmatrix}D_{\operatorname{Form}}&0\\\
0&D_{\check{B}_{*}}^{\Sigma}\end{pmatrix}$

in the collar $(-1,0]_{s}\times\Sigma$. Here

$D_{\operatorname{Form}}=\begin{pmatrix}0&0&-*_{\Sigma}d_{\Sigma}\\\
0&0&-d_{\Sigma}^{*}\\\
*_{\Sigma}d_{\Sigma}&-d_{\Sigma}&0\end{pmatrix}:L^{2}_{1}(\Sigma,i\mathbb{R}\oplus
i\mathbb{R}\oplus iT^{*}\Sigma)\to L^{2}(\Sigma,i\mathbb{R}\oplus
i\mathbb{R}\oplus iT^{*}\Sigma)$

is a self-adjoint operator with kernel $H^{0}(\Sigma,i\mathbb{R})\oplus
H^{0}(\Sigma,i\mathbb{R})\oplus H^{1}(\Sigma,i\mathbb{R})$ and

$D_{\check{B}_{*}}^{\Sigma}:L^{2}_{1}(\Sigma,S)\to L^{2}(\Sigma,S)$

is the Dirac operator on the surface, which is complex linear. Consider the
projection map

$\Pi_{\operatorname{Form}}=\Pi_{1}\oplus\Pi_{\operatorname{Form}}^{-}:L^{2}(\Sigma)\to
H^{1}(\Sigma,i\mathbb{R})\oplus H^{-}_{\operatorname{Form}}.$

where $\Pi_{\operatorname{Form}}$ is the projection map onto the negative
spectral subspace of $D_{\operatorname{Form}}$ and $\Pi_{1}$ is the projection
onto $H^{1}(\Sigma,i\mathbb{R})\subset\ker D_{\operatorname{Form}}$.

######  Lemma 18.9.

The kernel and the cokernel of the operator:

(18.11) $\begin{pmatrix}0&-d\\\
-d^{*}&*d\end{pmatrix}\oplus(\Pi_{\operatorname{Form}}\circ
r):L^{2}_{1}(\mathbb{R}\oplus iT^{*}Y)\to L^{2}(\mathbb{R}\oplus
iT^{*}Y)\oplus H^{1}(\Sigma,i\mathbb{R})\oplus H^{-}_{\operatorname{Form}}.$

are isomorphic to $H^{0}(Y;i\mathbb{R})\oplus H^{1}(Y,\partial Y;\mathbb{R})$
and $H^{0}(Y,\partial Y;\mathbb{R})\oplus H^{1}(Y;i\mathbb{R})$ respectively.
In particular, an orientation of (18.11) is equivalent to a homological
orientation of $(Y,\partial Y)$.

Finally, to relate the operator (18.10) with (18.11), we have to deform the
boundary projection $\Pi^{-}$ in (18.10). Notice that the operator
$\widehat{D}_{\kappa_{*}}$ in (18.9) relies on the standard spinor $\Psi_{*}$.
The deformation is then made by taking

$\Psi_{*}\mapsto\tau\Psi_{*},\tau\to 0.$

In the limit, $\widehat{D}_{\kappa_{*}}$ will recover $\hat{D}_{0}$, which is
no longer invertible. At this point, one has to examine the deformation of
spectral projections very carefully, which is independent of relative
$spin^{c}\ $structures. In this way, an orientation of (18.11) gives rise to
an orientation of $L$.

###  19\. Floer Homology with $\mathbb{Z}$-coefficient

Let $\mathcal{R}$ be the Novikov ring of Laurent series with integral
coefficients

$\mathcal{R}=\\{\sum_{n_{i}}a_{i}q^{n_{i}}:\ a_{i}\in\mathbb{Z},\
n_{i}\in\mathbb{R},\ \lim_{i}n_{i}=-\infty\\}.$

To define the monopole Floer homology over $\mathcal{R}$, we have to orient
moduli spaces in a consistent way. Since the space
$\mathcal{C}_{k}(\widehat{Y},\widehat{\mathfrak{s}})$ does not contain any
reducible configurations, the strategy used in [KM07, Section 20] does not
work directly here. Moreover, our cobordism maps are induced from oriented
4-manifold with corners. It is not crystal clear what is meant to be a
homology orientation in this case.

We will address this problem using an analytic approach. The main result of
this section is Theorem 19.2, which leads to the replacement of homology
orientations in Definition 19.4. The proof of Theorem 19.2 relies on the
notion of relative orientations that compares the determinant line bundles of
two Fredholm operators in the excision principle. We will develop the relevant
theory in Appendix A and accomplish the proof of Theorem 19.2 in Subsection
A.10. The construction of the functor

$\operatorname{\it
HM}_{*}:\operatorname{SCob}_{s,b}\to\mathcal{R}\text{-}\mathrm{Mod}$

is explained in Subsection 19.3.

####  19.1. Determinant Line Bundles and Direct Sums

To start, let us recall the basic theory of determinant line bundles of
Fredholm operators from [KM07, Section 20.2]. Given two real Hilbert spaces
$E$ and $F$, consider a continuous family of Fredholm operators

$\mathbb{A}_{z}:E\to F,\ z\in\mathcal{Z},$

parametrized by a topological space $\mathcal{Z}$. The determinant line bundle
of this family is a real line bundle over $\mathcal{Z}$

$\det\mathbb{A}\to\mathcal{Z}$

such that the fiber $\det\mathbb{A}_{z}$ at each $z\in\mathcal{Z}$ is
identified with

$\Lambda^{\max}\ker\mathbb{A}_{z}\otimes(\Lambda^{\max}\operatorname{coker}\mathbb{A}_{z})^{*}.$

When the determinant line bundle $\det\mathbb{A}\to\mathcal{Z}$ is orientable,
denote the 2-element set of orientations by

$\Lambda(\mathbb{A})\text{ or }\Lambda(\det\mathbb{A}).$

######  Example 19.1.

Let $\mathbb{A}_{*}:E\to F$ be a reference Fredholm operator and $\mathcal{Z}$
be the space of all compact operators:

$\mathcal{Z}=\\{z:E\to F:z\text{ compact}\\}.$

Then the family $\\{\mathbb{A}_{z}=\mathbb{A}_{*}+z:z\in\mathcal{Z}\\}$ is
parametrized by a contractible space $\mathcal{Z}$. An orientation of
$\mathbb{A}_{*}$ is meant to be an orientation of this contractible family.
Denote the 2-element set of orientations by

$\Lambda(\mathbb{A}_{*})\text{ or }\Lambda(\det\mathbb{A}_{*}).\qed$

Given two families of operators $\mathbb{A}^{\prime}\to\mathcal{Z}$ and
$\mathbb{A}^{\prime\prime}\to\mathcal{Z}$ parametrized by the same space, we
form a new family by taking the point-wise direct sum of Fredholm operators

$\mathbb{A}_{z}=\mathbb{A}_{z}^{\prime}\oplus\mathbb{A}^{\prime\prime}_{z}:E^{\prime}\oplus
E^{\prime\prime}\to F^{\prime}\oplus F^{\prime\prime}.$

Then there is a natural isomorphism of real line bundles constructed in [KM07,
P.379]:

(19.1)
$q:\det\mathbb{A}^{\prime}\otimes\det\mathbb{A}^{\prime\prime}\to\det\mathbb{A}.$

Suppose $\alpha_{z}^{\prime}$ and $\alpha_{z}^{\prime\prime}$ are elements in
$\Lambda^{\max}\ker\mathbb{A}^{\prime}_{z}$ and
$\Lambda^{\max}\ker\mathbb{A}^{\prime\prime}_{z}$ respectively, while
$\beta_{z}^{\prime}$ and $\beta_{z}^{\prime\prime}$ are corresponding elements
in $\Lambda^{\max}\operatorname{coker}\mathbb{A}^{\prime}_{z}$ and
$\Lambda^{\max}\operatorname{coker}\mathbb{A}^{\prime\prime}_{z}$. Then the
bundle map $q$ is locally defined (up to a positive scalar) by the formula:

$\displaystyle\big{(}\alpha^{\prime}_{z}\otimes(\beta^{\prime}_{z})^{*}\big{)}\otimes\big{(}\alpha^{\prime\prime}_{z}\otimes(\beta^{\prime\prime}_{z})^{*}\big{)}$
$\displaystyle\mapsto(-1)^{r}(\alpha^{\prime}_{z}\wedge\alpha^{\prime\prime}_{z})\otimes(\beta^{\prime}_{z}\wedge\beta^{\prime\prime}_{z})^{*}\text{
where }$ $\displaystyle r$
$\displaystyle=\dim\operatorname{coker}\mathbb{A}^{\prime}_{z}\times\text{Ind}(\mathbb{A}^{\prime\prime}_{z}).$

The sign $(-1)^{r}$ is added here to ensure that the bundle map $q$ is
continuous as the base point $z$ varies in $\mathcal{Z}$. Moreover, the bundle
map $q$ becomes associative when we consider the direct sum of three families
of operators parametrized by the same space $\mathcal{Z}$.

For any 2-element set $\Lambda$, let $\mathbb{Z}/2\mathbb{Z}$ act on $\Lambda$
by involutions. For any $\Lambda_{1}$ and $\Lambda_{2}$ with
$\mathbb{Z}/2\mathbb{Z}$ action, we form their product set

$\Lambda_{1}\Lambda_{2}\colonequals\Lambda_{1}\times_{\mathbb{Z}/2\mathbb{Z}}\Lambda_{2}.$

As a result, by passing to the 2-element sets of orientations, the bundle map
$q$ descends to an associative multiplication, denoted also by $q$:

$q:\Lambda(\mathbb{A}^{\prime})\times\Lambda(\mathbb{A}^{\prime\prime})\to\Lambda(\mathbb{A}^{\prime}\oplus\mathbb{A}^{\prime\prime}),$

or an isomorphism preserving the $\mathbb{Z}/2\mathbb{Z}$-action:

$q:\Lambda(\mathbb{A}^{\prime})\Lambda(\mathbb{A}^{\prime\prime})\xrightarrow{\cong}\Lambda(\mathbb{A}^{\prime}\oplus\mathbb{A}^{\prime\prime}).$

####  19.2. Homology Orientations

Having discussed the abstract properties of determinant line bundles, let us
explain now the primary application in gauge theory. Given a morphism
$\mathbb{X}:(\mathbb{Y}_{1},\widehat{\mathfrak{s}}_{1})\to(\mathbb{Y}_{2},\widehat{\mathfrak{s}}_{2})$
in the strict cobordism category $\operatorname{SCob}_{s}$, consider non-
degenerate pairs (in the sense of Definition 18.5)

$\mathfrak{c}_{i}=(\mathfrak{a}_{i},\mathfrak{q}_{i})\in\mathcal{C}_{k}(\widehat{Y}_{i},\widehat{\mathfrak{s}}_{i})\times\mathcal{P}(Y_{i}),i=1,2.$

By looking at the linearized Seiberg-Witten map and the linearized gauge
fixing equation on the complete Riemannian 4-manifold $\mathcal{X}$, we
obtained in $\eqref{E23.8}$ a Fredholm operator
$\mathcal{Q}(\mathfrak{c}_{1},\widehat{\mathfrak{s}}_{X},\mathfrak{c}_{2})$
for any relative $spin^{c}\ $cobordism
$(\widehat{X},\widehat{\mathfrak{s}}_{X}):(\widehat{Y}_{1},\widehat{\mathfrak{s}}_{1})\to(\widehat{Y}_{2},\widehat{\mathfrak{s}}_{2})$.
Define

$\Lambda(\mathfrak{c}_{1},\widehat{\mathfrak{s}}_{X},\mathfrak{c}_{2})\colonequals\Lambda(\mathcal{Q}(\mathfrak{c}_{1},\widehat{\mathfrak{s}}_{X},\mathfrak{c}_{2}))$

for any non-degenerate pairs $\mathfrak{c}_{1},\mathfrak{c}_{2}$ and any
$\widehat{\mathfrak{s}}_{X}\in\operatorname{Spin^{c}_{R}}(X;\widehat{\mathfrak{s}}_{1},\widehat{\mathfrak{s}}_{2})$.
The 2-element set
$\Lambda(\mathfrak{c}_{1},\widehat{\mathfrak{s}}_{X},\mathfrak{c}_{2})$ is
understood in the sense of Example 19.1. Since the different choices of the
reference configuration $\gamma$ will give rise to the same operator
$\mathcal{Q}(\mathfrak{c}_{1},\widehat{\mathfrak{s}}_{X},\mathfrak{c}_{2})$ up
to compact terms,
$\Lambda(\mathfrak{c}_{1},\widehat{\mathfrak{s}}_{X},\mathfrak{c}_{2})$ is
independent of the choice of $\gamma$.

Our goal is to identify these 2-element sets
$\Lambda(\mathfrak{c}_{1},\widehat{\mathfrak{s}}_{X},\mathfrak{c}_{2})$ in a
canonical way for all relative $spin^{c}\ $cobordisms
$\widehat{\mathfrak{s}}_{X}\in\operatorname{Spin^{c}_{R}}(X;\widehat{\mathfrak{s}}_{1},\widehat{\mathfrak{s}}_{2})$.
As a result, if the orientation is fixed for one particular
$\widehat{\mathfrak{s}}_{X}$, then it automatically fixes the choice for any
other relative $spin^{c}\ $cobordisms.

Recall that
$\operatorname{Spin^{c}_{R}}(X;\widehat{\mathfrak{s}}_{1},\widehat{\mathfrak{s}}_{2})$
is a torsor over $H^{2}(X,\partial X;\mathbb{Z})$.

######  Theorem 19.2.

For any isomorphism classe of relative line bundles $[L]\in H^{2}(X,\partial
X;\mathbb{Z})$, there exists a natural bijection

$e_{L}:\Lambda(\mathfrak{c}_{1},\widehat{\mathfrak{s}}_{X},\mathfrak{c}_{2})\to\Lambda(\mathfrak{c}_{1},\widehat{\mathfrak{s}}_{X}\otimes
L,\mathfrak{c}_{2}),$

for any
$\widehat{\mathfrak{s}}_{X}\in\operatorname{Spin^{c}_{R}}(X;\widehat{\mathfrak{s}}_{1},\widehat{\mathfrak{s}}_{2})$
satisfying the following two properties:

  1. (U1)

$e_{L_{1}}\circ e_{L_{2}}=e_{L_{1}\otimes L_{2}}$;

  2. (U2)

the collection $\\{e_{L}\\}$ is compatible with the concatenation map $q$
meaning that the diagram

(19.2)
${\Lambda(\mathfrak{c}_{1},\widehat{\mathfrak{s}}_{12},\mathfrak{c}_{2})\Lambda(\mathfrak{c}_{2},\widehat{\mathfrak{s}}_{23},\mathfrak{c}_{3})}$${\Lambda(\mathfrak{c}_{1},\widehat{\mathfrak{s}}_{13},\mathfrak{c}_{3})}$${\Lambda(\mathfrak{c}_{1},\widehat{\mathfrak{s}}_{12}\otimes
L_{12},\mathfrak{c}_{2})\Lambda(\mathfrak{c}_{2},\widehat{\mathfrak{s}}_{23}\otimes
L_{23},\mathfrak{c}_{3})}$${\Lambda(\mathfrak{c}_{1},\widehat{\mathfrak{s}}_{13}\otimes
L_{13},\mathfrak{c}_{3})}$$\scriptstyle{q}$$\scriptstyle{e_{L_{12}}\otimes
e_{L_{23}}}$$\scriptstyle{e_{L_{13}}}$$\scriptstyle{q}$

is commutative for any relative $spin^{c}\ $cobordisms:

$\displaystyle(\widehat{X}_{12},\widehat{\mathfrak{s}}_{12}):(Y_{1},\widehat{\mathfrak{s}}_{1})$
$\displaystyle\to(Y_{2},\widehat{\mathfrak{s}}_{2}),$
$\displaystyle(\widehat{X}_{23},\widehat{\mathfrak{s}}_{23}):(Y_{2},\widehat{\mathfrak{s}}_{3})$
$\displaystyle\to(Y_{3},\widehat{\mathfrak{s}}_{3}).$

Here
$(\widehat{X}_{13},\widehat{\mathfrak{s}}_{13})=(\widehat{X}_{12}\\#\widehat{X}_{23},\widehat{\mathfrak{s}}_{12}\\#\widehat{\mathfrak{s}}_{23})$
is the concatenation of relative cobordisms and $L_{13}=L_{12}\\#L_{23}$ is
the concatenation of relative line bundles.

######  Remark 19.3.

The proof of Theorem 19.2 is constructive: we will construct each $e_{L}$
explicitly and verify properties (U1)(U2) by hands. The key ingredient is the
notion of relative orientations, which allows us to reduce the problem from a
non-compact manifold $\mathcal{X}$ to a closed 4-manifold. In the latter case,
we know how to construct $e_{L}$, since the Dirac operator and the self-dual
operator are now decoupled. The relevant theory is developed in Appendix A.
The proof of Theorem 19.2 will be accomplished in Subsection A.10. ∎

The horizontal maps $q$ in the diagram (19.2) require some further
explanations. Take non-degenerate pairs $\mathfrak{c}_{i}$ on
$\widehat{Y}_{i}$ for $1\leq i\leq 3$. Instead of $\mathcal{Q}$, we look at
operators on $\widehat{X}_{ij}$ with spectral projections:

(19.3)
$\mathcal{Q}^{\prime}(\mathfrak{c}_{i},\widehat{\mathfrak{s}}_{ij},\mathfrak{c}_{j})\colonequals
D_{ij}\oplus(\Pi^{+}_{\mathbb{A}_{i}},\Pi^{-}_{\mathbb{A}_{j}})\circ(r_{i},r_{j}),1\leq
i<j\leq 3,$

understood in the sense of Proposition 13.9 and Subsection 13.4 adapted to the
case of general cobordisms. In particular, $\Pi^{\pm}_{\mathbb{A}_{i}}$ are
spectral projections of the extended Hessians at $\mathfrak{c}_{i}$:

$\widehat{\operatorname{Hess}}_{\mathfrak{c}_{i}}:L^{2}_{k}(\widehat{Y}_{i},i\mathbb{R}\oplus
iT^{*}Y_{i}\oplus S)\to L^{2}_{k}(\widehat{Y}_{i},i\mathbb{R}\oplus
iT^{*}Y_{i}\oplus S),\ 1\leq i\leq 3.$

The 2-element set
$\Lambda(\mathfrak{c}_{1},\widehat{\mathfrak{s}}_{12},\mathfrak{c}_{2})$ can
be defined using
$\mathcal{Q}^{\prime}(\mathfrak{c}_{i},\widehat{\mathfrak{s}}_{ij},\mathfrak{c}_{j})$
instead. As explained in [KM07, P. 384], there is a canonical bundle
isomorphism defined using the map $\eqref{E24.3}$,

(19.4)
$q:\det\mathcal{Q}^{\prime}(\mathfrak{c}_{1},\widehat{\mathfrak{s}}_{12},\mathfrak{c}_{2})\otimes\det\mathcal{Q}^{\prime}(\mathfrak{c}_{2},\widehat{\mathfrak{s}}_{23},\mathfrak{c}_{3})\to\det\mathcal{Q}^{\prime}(\mathfrak{c}_{1},\widehat{\mathfrak{s}}_{13},\mathfrak{c}_{3}).$

which descends to an associative multiplication:

$q:\Lambda(\mathfrak{c}_{1},\widehat{\mathfrak{s}}_{12},\mathfrak{c}_{2})\Lambda(\mathfrak{c}_{2},\widehat{\mathfrak{s}}_{23},\mathfrak{c}_{3})\to\Lambda(\mathfrak{c}_{1},\widehat{\mathfrak{s}}_{13},\mathfrak{c}_{3}).$

Our construction of homology orientations is based upon Theorem 19.2.

######  Definition 19.4.

Following the notations in Theorem 19.2, for any triple
$(\mathfrak{c}_{1},\mathbb{X},\mathfrak{c}_{2})$, define the 2-element set of
homology orientations as the quotient space