for some $t\in\mathbb{R}$, where $c\colonequals-2\pi i[\omega]$ is the period
class of this perturbation. Furthermore, it is called balanced, positively or
negatively monotone if $t=0$, $t>0$ or $<0$ respectively. We write

$\operatorname{\it
HM}_{*}(Y,\mathfrak{s},c;\Gamma)\colonequals\operatorname{Im}j_{*}\subset\widehat{\operatorname{\it
HM}}_{*}(Y,\mathfrak{s},c;\Gamma)$

for the reduced monopole Floer homology. The third group
$\overline{\operatorname{\it HM}}_{*}(Y,\mathfrak{s},c;\Gamma)$ in (6.2) is
always trivial unless the period class
$c=c_{b}\colonequals-2\pi^{2}c_{1}(\mathfrak{s})$ is balanced. Thus
$\operatorname{\it HM}_{*}(Y,\mathfrak{s},c;\Gamma)=\widehat{\operatorname{\it
HM}}_{*}(Y,\mathfrak{s},c;\Gamma)$ if $c\neq c_{b}$.

We focus on the local coefficient systems $\mathcal{R}_{\omega}$ on
$\mathcal{B}^{\sigma}(Y,\mathfrak{s})$, whose fiber at every point is always
the mod 2 Novikov ring $\mathcal{R}$. The fundamental group of
$\mathcal{B}^{\sigma}(Y,\mathfrak{s})$ is
$H^{1}(Y,\mathbb{Z})=\pi_{0}(\mathcal{G}(Y))$. The monodromy of
$\mathcal{R}_{\omega}$ is then defined by sending $z\in H^{1}(Y,\mathbb{Z})$
to $q^{f(z)}$ with

(6.3) $\displaystyle
f(z)\colonequals\langle(2\pi^{2}c_{1}(\mathfrak{s})+c)\cup
z,[Y]\rangle\in\mathbb{R}.$

For the adjunction inequality, it suffices to consider $spin^{c}\ $structures
with non-torsion $c_{1}(\mathfrak{s})$. The problem already occurs for the
3-torus $\mathbb{T}^{3}=\mathbb{T}^{2}\times S^{1}$ in Lemma 3.8. Let $d\in
H^{2}(\mathbb{T}^{3},\mathbb{Z})$ be the Poincaré dual of $\\{pt\\}\times
S^{1}$. Suppose $\omega=i\delta\cdot d$ and consider the $spin^{c}\ $structure
$\mathfrak{s}$ on $\mathbb{T}^{3}$ with $c_{1}(\mathfrak{s})=2\cdot d$. If
$\delta<-2\pi$, then this perturbation is negatively monotone. In this case,
the moduli space of the Seiberg-Witten equations (3.2) is non-empty, and is
diffeomorphic to $\mathbb{T}^{2}\times\\{pt\\}$. As one may verify, for either
$\Gamma=\mathbb{Z}$ or $\mathcal{R}_{\omega}$, the monopole Floer homology
group $\operatorname{\it HM}_{*}(\mathbb{T}^{3},\mathfrak{s},c;\Gamma)$ is
non-trivial, and so the adjunction inequality in Theorem 6.1 is violated in
this case.

However, by Proposition 6.2, negatively monotone perturbations are the only
exceptions.

######  Proposition 6.2.

Let $Y$ be any closed oriented 3-manifold with $b_{1}(Y)\geq 2$ and
$\mathfrak{s}$ be any non-torsion $spin^{c}\ $structure. Suppose that
$[\omega]\in H^{2}(Y,i\mathbb{R})$ is neither negatively monotone nor balanced
with respect to $\mathfrak{s}$, then there is an isomorphism

$\operatorname{\it
HM}_{*}(Y,\mathfrak{s},c;\mathcal{R}_{\omega})\cong\operatorname{\it
HM}_{*}(Y,\mathfrak{s};\mathcal{R}_{\omega})$

where the second group is defined using an exact perturbation with local
system $\mathcal{R}_{\omega}$. In particular, the adjunction inequality from
Theorem 6.1 holds also for the group $\operatorname{\it
HM}_{*}(Y,\mathfrak{s},c;\mathcal{R}_{\omega})$: it is trivial, whenever
$\langle c_{1}(\mathfrak{s}),\kappa\rangle>\|\kappa\|_{Th}$ for some integral
class $\kappa\in H_{2}(Y;\mathbb{Z})$.

The non-vanishing result is more robust: it suffices to rule out exact
perturbations:

######  Proposition 6.3.

Let $Y$ be any irreducible closed oriented 3-manifold with $b_{1}(Y)\geq 1$.
If the period class $c=-2\pi i[\omega]\in H^{2}(Y;\mathbb{R})$ is non-exact,
then for any integral class $\kappa\in H^{2}(Y;\mathbb{Z})$, there is a
$spin^{c}\ $structure $\mathfrak{s}$ such that

$\langle c_{1}(\mathfrak{s}),\kappa\rangle=\|\kappa\|_{Th}\text{ and
}\operatorname{\it HM}_{*}(Y,\mathfrak{s},c;\mathcal{R}_{\omega})\neq\\{0\\}.$

To ease our notation, we introduce the following shorthand:

######  Definition 6.4.

For any closed 2-form $\omega\in\Omega^{2}(Y;i\mathbb{R})$ and any integral
class $\kappa\in H_{2}(Y;\mathbb{Z})$, we write $\operatorname{\it
HM}(Y,[\omega])$ for the direct sum

$\bigoplus_{\mathfrak{s}}\operatorname{\it
HM}_{*}(Y,\mathfrak{s},c;\mathcal{R}_{\omega})$

and $\operatorname{\it HM}(Y,[\omega]|\kappa)$ for the subgroup

$\bigoplus_{\langle
c_{1}(\mathfrak{s}),\kappa\rangle=x(\kappa)}\operatorname{\it
HM}_{*}(Y,\mathfrak{s},c;\mathcal{R}_{\omega}).$

For any connected oriented subsurface $F\subset Y$, define similarly

$\operatorname{\it HM}(Y,[\omega]|F)=\bigoplus_{\langle
c_{1}(\mathfrak{s}),\kappa\rangle=x(F)}\operatorname{\it
HM}_{*}(Y,\mathfrak{s},c;\mathcal{R}_{\omega}).\qed$

######  Remark 6.5.

When $[\omega]=0\in H^{2}(Y;i\mathbb{R})$ and $c_{1}(\mathfrak{s})\neq 0$, the
local system $\mathcal{R}_{\omega}$ is not trivial. Nevertheless, by [KM07,
P.288 (16.5)], we still have an isomorphism:

$\operatorname{\it
HM}_{*}(Y,\mathfrak{s},c;\mathcal{R}_{\omega})\cong\operatorname{\it
HM}_{*}(Y,\mathfrak{s};\mathcal{R}).$

The latter group is defined using the trivial system with coefficients
$\mathcal{R}$. Set

$\operatorname{\it HM}_{*}(Y|\kappa)\colonequals\bigoplus_{\langle
c_{1}(\mathfrak{s}),\kappa\rangle=x(\kappa)}\operatorname{\it
HM}_{*}(Y,\mathfrak{s};\mathcal{R}).$

Then $\operatorname{\it HM}_{*}(Y|\kappa)\cong\operatorname{\it
HM}_{*}(Y,[0]|\kappa)$. We define similarly the group $\operatorname{\it
HM}_{*}(Y|F)$. ∎

######  Corollary 6.6.

If $Y$ is irreducible with $b_{1}(Y)>0$ and $[\omega]\neq 0\in
H^{2}(Y;i\mathbb{R})$, then $\operatorname{\it
HM}(Y,[\omega]|\kappa)\neq\\{0\\}$ for any integral class $\kappa\in
H^{2}(Y;\mathbb{Z})$.

The proof of Proposition 6.2 and 6.3 will dominate the rest of this section.

####  6.2. Completions of Chain Complexes

The proof of Proposition 6.2 is achieved in two steps: the first step is to
relate the group $\operatorname{\it
HM}_{*}(Y,\mathfrak{s},c;\mathcal{R}_{\omega})$ with the Floer homology
associated to a balanced class $c_{b}=-2\pi^{2}c_{1}(s)$. For monotone
perturbations, this is already done in [KM07, Theorem 31.1.1 & 31.5.1]. We
shall describe a slightly different setup that allows generalization for any
non-balanced perturbations.

Consider the monopole Floer chain complexes of $(Y,\mathfrak{s},c_{b})$
associated to the local system $\mathcal{R}_{\omega}$:

$\widecheck{C}_{*}(Y,\mathfrak{s},c_{b};\mathcal{R}_{\omega}),\
\widehat{C}_{*}(Y,\mathfrak{s},c_{b};\mathcal{R}_{\omega}),\
\overline{C}_{*}(Y,\mathfrak{s},c_{b};\mathcal{R}_{\omega}).$

As modules, they are free over the Novikov ring $\mathcal{R}$, but none of
them is finitely generated. Since the period class $c_{b}$ is balanced, we
have to blow up the configuration space in order to define the Floer homology.
Suppose an admissible perturbation of the Chern-Simons-Dirac functional
$\mathcal{L}_{\omega}$ is chosen, then each reducible solution $[\alpha]$ of
the perturbed 3-dimensional Seiberg-Witten equations will contribute
infinitely many generators to each of
$\widecheck{C}_{*},\widehat{C}_{*},\overline{C}_{*}$, corresponding to the
eigenvectors of the Dirac operator at $[\alpha]$.

As noted in [KM07, Section 30.1], we can form a chain level completion using
the filtration of eigenvalues. Label the reducible solutions in the quotient
configuration space $\mathcal{B}(Y,\mathfrak{s})$ as
$[\alpha^{1}],\cdots,[\alpha^{p}]$ and label the corresponding critical points
in the blown-up space as $[\alpha^{r}_{i}]$ with $i\in\mathbb{Z}$ and $1\leq
r\leq p$, so that $[\alpha^{r}_{i}]$ corresponds to the eigenvalue
$\lambda^{r}_{i}$ of the perturbed Dirac operator at $[\alpha^{r}]$ and

$\cdots<\lambda^{r}_{i}<\lambda^{r}_{i+1}<\cdots,$

with $\lambda^{r}_{0}$ being the first positive one. For any $m\geq 1$, let

(6.4)
$\widehat{C}_{*}(Y,\mathfrak{s},c_{b};\mathcal{R}_{\omega})_{m}\subset\widehat{C}_{*}(Y,\mathfrak{s},c_{b};\mathcal{R}_{\omega}),$

be the subgroup generated by $[\alpha^{r}_{i}]$ with $i\leq-m$. This defines a
filtration on $\widehat{C}_{*}$, called the $\lambda$-filtration. We form the
completion

$\widehat{C}_{\bullet}(Y,\mathfrak{s},c_{b};\mathcal{R}_{\omega})\supset\widehat{C}_{*}(Y,\mathfrak{s},c_{b};\mathcal{R}_{\omega}).$

The same construction applies also to the bar-version; so we obtain

$\overline{C}_{\bullet}(Y,\mathfrak{s},c_{b};\mathcal{R}_{\omega})\supset\overline{C}_{*}(Y,\mathfrak{s},c_{b};\mathcal{R}_{\omega}).$

On the other hand, $\widehat{C}_{*}$ carries an additional filtration arising
from the base ring $\mathcal{R}$, called the $\mathcal{R}$-filtration. One may
formally write

$\widehat{C}_{*}(Y,\mathfrak{s},c_{b};\mathcal{R}_{\omega})=\bigoplus_{n\geq
0}\mathcal{R}e_{n},$

by identifying the fibers of $\mathcal{R}_{\omega}$ at different critical
points using a collection of paths in the blown-down space
$\mathcal{B}(Y,\mathfrak{s})$. We may start with the group ring
$\mathbb{F}_{2}[\mathbb{R}]$ and obtain $\mathcal{R}$ by taking the completion
in the negative direction. This filtration on
$\mathcal{R}=\mathbb{F}_{2}[\mathbb{R}]^{-}$ then induces the
$\mathcal{R}$-filtration on the free module
$\widehat{C}_{*}(Y,\mathfrak{s},c_{b};\mathcal{R}_{\omega})$; let

$\widehat{C}_{\diamond}(Y,\mathfrak{s},c_{b};\mathcal{R}_{\omega})\subset\widehat{C}_{\bullet}(Y,\mathfrak{s},c_{b};\mathcal{R}_{\omega}),$

be the resulting completion. Any element $\sum_{n\geq 0}a_{n}\cdot
e_{n}\in\widehat{C}_{\diamond}(Y,\mathfrak{s},c_{b};\mathcal{R}_{\omega})$ may
have infinitely many non-zero coefficients $a_{n}\in\mathcal{R}$, but under
the topology of $\mathcal{R}$, we must have

(6.5) $\lim_{n\to\infty}a_{n}=0.$

The bar-version analogue is constructed in a slightly different way. First,
take the $\mathcal{R}$-completion of its $m$-th filtered subgroup
$\overline{C}_{*}(Y,\mathfrak{s},c_{b};\mathcal{R}_{\omega})_{m}\subset\overline{C}_{*}(Y,\mathfrak{s},c_{b};\mathcal{R}_{\omega})$
for each $m\in\mathbb{Z}$, denoted by

$\overline{C}_{\diamond}(Y,\mathfrak{s},c_{b};\mathcal{R}_{\omega})_{m}.$

Next, we form the union:

$\overline{C}_{\diamond}(Y,\mathfrak{s},c_{b};\mathcal{R}_{\omega})\colonequals\bigcup_{m\in\mathbb{Z}}\overline{C}_{\diamond}(Y,\mathfrak{s},c_{b};\mathcal{R}_{\omega})_{m}\subset\overline{C}_{\bullet}(Y,\mathfrak{s},c_{b};\mathcal{R}_{\omega}).$

An element in
$\overline{C}_{\diamond}(Y,\mathfrak{s},c_{b};\mathcal{R}_{\omega})$ has only
finitely many non-zero coefficients for critical points with
$\lambda^{r}_{i}>0$, while a infinite sum may occur in the negative direction
satisfying the convergence condition (6.5). The upshot is that the homology
groups of

$\widehat{C}_{*}(Y,\mathfrak{s},c_{b};\mathcal{R}_{\omega}),\widehat{C}_{\diamond}(Y,\mathfrak{s},c_{b};\mathcal{R}_{\omega}),\overline{C}_{\diamond}(Y,\mathfrak{s},c_{b};\mathcal{R}_{\omega})$

form a long exact sequence (by the proof of [KM07, Proposition 22.2.1]), which
fits into a diagram below:

(6.6) ${\cdots}$${\widecheck{\operatorname{\it
HM}}_{*}(Y,\mathfrak{s},c_{b};\mathcal{R}_{\omega})}$${\widehat{\operatorname{\it
HM}}_{*}(Y,\mathfrak{s},c_{b};\mathcal{R}_{\omega})}$${\overline{\operatorname{\it
HM}}_{*}(Y,\mathfrak{s},c_{b};\mathcal{R}_{\omega})}$${\cdots}$${\cdots}$${\widecheck{\operatorname{\it
HM}}_{*}(Y,\mathfrak{s},c_{b};\mathcal{R}_{\omega})}$${\widehat{\operatorname{\it
HM}}_{\diamond}(Y,\mathfrak{s},c_{b};\mathcal{R}_{\omega})}$${\overline{\operatorname{\it
HM}}_{\diamond}(Y,\mathfrak{s},c_{b};\mathcal{R}_{\omega})}$${\cdots}$${\cdots}$${\widecheck{\operatorname{\it
HM}}_{*}(Y,\mathfrak{s},c_{b};\mathcal{R}_{\omega})}$${\widehat{\operatorname{\it
HM}}_{\bullet}(Y,\mathfrak{s},c_{b};\mathcal{R}_{\omega})}$${\overline{\operatorname{\it
HM}}_{\bullet}(Y,\mathfrak{s},c_{b};\mathcal{R}_{\omega})}$${\cdots}$$\scriptstyle{i_{*}}$$\scriptstyle{j_{*}}$$\scriptstyle{p_{*}}$$\scriptstyle{x_{*}}$$\scriptstyle{i_{*}}$$\scriptstyle{i_{\diamond}}$$\scriptstyle{j_{\diamond}}$$\scriptstyle{p_{\diamond}}$$\scriptstyle{y_{\diamond}}$$\scriptstyle{i_{\diamond}}$$\scriptstyle{i_{\bullet}}$$\scriptstyle{j_{\bullet}}$$\scriptstyle{p_{\bullet}}$$\scriptstyle{i_{\bullet}}$

The next proposition says that $\operatorname{\it
HM}_{*}(Y,\mathfrak{s},c;\mathcal{R}_{\omega})$ can be computed in terms of
the group $\widehat{\operatorname{\it HM}}_{\diamond}$ associated the balanced
period class $c_{b}\colonequals-2\pi^{2}c_{1}(\mathfrak{s})$.

######  Proposition 6.7.

For any $(Y,\mathfrak{s})$ and any non-balanced perturbation with period class
$c=-2\pi i[\omega]$, we have an isomorphism

$\displaystyle\widehat{\operatorname{\it
HM}}_{*}(Y,\mathfrak{s},c;\mathcal{R}_{\omega})$
$\displaystyle\cong\widehat{\operatorname{\it
HM}}_{\diamond}(Y,\mathfrak{s},c_{b};\mathcal{R}_{\omega}),$

###### Proof.

This result follows from the proof of [KM07, Theorem 31.1.1]. See [KM07,
Section 31.2]. Since we have used a $c$-complete local coefficient system
$\mathcal{R}_{\omega}$ here, there is no need to estimate the topological
energy $\mathcal{E}_{1}^{top}$ in [KM07, (31.5) P.611]. The argument is even
simpler. ∎

The second step in the proof of Proposition 6.2 is to relate
$\widehat{\operatorname{\it
HM}}_{\diamond}(Y,\mathfrak{s},c_{b};\mathcal{R}_{\omega})$ with the bullet-
version $\widehat{\operatorname{\it
HM}}_{\bullet}(Y,\mathfrak{s},c_{b};\mathcal{R}_{\omega})$ using the vertical
map $y_{\diamond}$ in (6.6). In fact, more is true when the class $[\omega]$
is not monotone with respect to $c_{1}(\mathfrak{s})$:

######  Proposition 6.8.

Let $Y$ be any closed oriented 3-manifold with $b_{1}(Y)\geq 2$ and
$\mathfrak{s}$ be any non-torsion $spin^{c}\ $structure. Suppose that the
period class $c=-2\pi i[\omega]$ is not monotone, then the vertical maps
$x_{*}$ and $y_{\diamond}$ in digram (6.6) are isomorphisms

$\widehat{\operatorname{\it
HM}}_{*}(Y,\mathfrak{s},c_{b};\mathcal{R}_{\omega})\xrightarrow[\cong]{x_{*}}\widehat{\operatorname{\it
HM}}_{\diamond}(Y,\mathfrak{s},c_{b};\mathcal{R}_{\omega})\xrightarrow[\cong]{y_{\diamond}}\widehat{\operatorname{\it
HM}}_{\bullet}(Y,\mathfrak{s},c_{b};\mathcal{R}_{\omega}).$

Now we ready to prove Proposition 6.2.

###### Proof of Proposition 6.2.

When $c$ is not monotone, we can combine Proposition 6.7 & 6.8 with the
following isomorphism

(6.7) $\widehat{\operatorname{\it
HM}}_{\bullet}(Y,\mathfrak{s},c_{b};\mathcal{R}_{\omega})\cong\operatorname{\it
HM}_{*}(Y,\mathfrak{s};\mathcal{R}_{\omega})$

from [KM07, Theorem 31.1.1] to conclude. The case when $c$ is positively
monotone is already addressed in [KM07, Theorem 31.1.2]. ∎

###### Proof of Proposition 6.8.

Since the left vertical maps in digram (6.6) are identity maps, the statement
then follows from the fact that for any $\circ\in\\{*,\diamond,\bullet\\}$,

$\overline{\operatorname{\it
HM}}_{\circ}(Y,\mathfrak{s},c_{b};\mathcal{R}_{\omega})=\\{0\\}.$

The case when $\circ=\bullet$ is addressed already in [KM07, Theorem 31.1.1];
in fact, the group $\overline{\operatorname{\it
HM}}_{\bullet}(Y,\mathfrak{s},c_{b};\Gamma)$ always vanishes, no matter which
local system $\Gamma$ we use.

In general, for any $\circ\in\\{*,\diamond,\bullet\\}$, the group
$\overline{\operatorname{\it
HM}}_{\circ}(Y,\mathfrak{s},c_{b};\mathcal{R}_{\omega})$ is an instance of
coupled Morse homology (see [KM07, Section 34]) associated to the Picard torus
of $Y$

$\mathbb{T}^{b}\colonequals H^{1}(Y;\mathbb{R})/H^{1}(Y;\mathbb{Z}),$

with $b\colonequals b_{1}(Y)\geq 2$. In particular, we may use [KM07, Theorem
35.1.6] to compute $\overline{\operatorname{\it
HM}}_{\circ}(Y,\mathfrak{s},c_{b};\mathcal{R}_{\omega})$ using a standard
chain complex, which we recall below. Consider the local coefficient system
$\Gamma_{\omega}$ on $\mathbb{T}^{b}$ with fiber $\mathcal{R}[T,T^{-1}]$ and
with monodromy given by

(6.8) $\displaystyle\pi_{1}(\mathbb{T}^{b})\cong H^{1}(Y,\mathbb{Z})$
$\displaystyle\to\mathcal{R}[T,T^{-1}]^{\times}$ $\displaystyle z$
$\displaystyle\mapsto q^{f(z)}T^{g(z)}$

where $f(z)$ is as in (6.3) and
$g(z)=\langle\frac{1}{2}c_{1}(\mathfrak{s})\cup z,[Y]\rangle$. Then the group
$\overline{\operatorname{\it
HM}}_{*}(Y,\mathfrak{s},c_{b};\mathcal{R}_{\omega})$ is isomorphic to the
homology of $(C,\partial=\partial_{1}+\partial_{3})$, where

$C=C(\mathbb{T}^{b},h;\Gamma_{\omega})\colonequals\bigoplus_{n\geq
0}^{N}\mathcal{R}[T,T^{-1}]x_{n}$

is the Morse complex of $\mathbb{T}^{b}$ with local coefficients
$\Gamma_{\omega}$, defined using a suitable Morse function
$h:\mathbb{T}^{b}\to\mathbb{R}$ and $\partial_{1}$ is the Morse differential.
As a result, $C$ is a finite-rank free module over the ring
$\mathcal{R}[T,T^{-1}]$, generated by critical points $\\{x_{n}\\}$ of $h$.
The Morse index then induces an additional grading on $C$ such that
$\deg\partial_{i}=-i$ for $i=1,3$. The $m$-th $\lambda$-filtered subgroup of
$C$ is given by

$C_{m}\colonequals T^{-m}\bigoplus_{n\geq
0}^{N}\mathcal{R}[T^{-1}]x_{n},m\in\mathbb{Z}.$

The upshot is that te differential $\partial$ is
$\mathcal{R}[T,T^{-1}]$-linear. The bar-version chain complex

$(\overline{C}_{*}(Y,\mathfrak{s},c_{b};\mathcal{R}_{\omega}),\bar{\partial})$

which defines $\overline{\operatorname{\it
HM}}_{*}(Y,\widehat{\mathfrak{s}},c;\mathcal{R}_{\omega})$ admits a very
similar structure to $(C,\partial)$, but $\bar{\partial}$ is only linear in
$\mathcal{R}$ and it may have higher components with respect to the Morse
grading:
$\bar{\partial}=\bar{\partial}_{1}+\bar{\partial}_{3}+\bar{\partial}_{5}+\cdots$.
By [KM07, Proposition 34.4.1 & 33.3.8, Theorem 35.1.6],
$(\overline{C}_{*}(Y,\mathfrak{s},c_{b};\mathcal{R}_{\omega}),\bar{\partial})$
is homotopic to such a standard complex $(C,\partial)$, preserving the
topology induced by $\\{(\overline{C}_{*})_{m}\\}$ and $\\{C_{m}\\}$
respectively.

To compute the homology of $(C,\partial)$, we exploit the spectral sequence
induced from the Morse grading, which abuts to $H(C,\partial)$ and whose
$E_{1}$-page is $H(C,\partial_{1})$. Since the period class $c=-2\pi
i[\omega]$ is not monotone, one verifies immediately that
$H(C,\partial_{1})=\\{0\\}$. Indeed, we may write
$\mathbb{T}^{b}=S^{1}\times\mathbb{T}^{b-1}$ such that a integral multiple of
$\\{pt\\}\times\mathbb{T}^{b-1}$ is dual to the map $g\in
H^{1}(\mathbb{T}^{b},\mathbb{Z})=\operatorname{Hom}(H_{1}(\mathbb{T}^{b},\mathbb{Z}),\mathbb{Z})$.
One may further decompose $\mathbb{T}^{b-1}$ into a product of circles and let
$h$ be the sum of Morse functions on circles. It follows that the homology of
$\mathbb{T}^{b-1}$ with local coefficients $\Gamma_{\omega}$ is trivial, since
the holonomy of $\Gamma_{\omega}$ along some $S^{1}$-factor is non-trivial and
lies in $\mathcal{R}$. Finally, we deduce that $H(C,\partial_{1})=\\{0\\}$
using the Künneth formula.

This finishes the proof when $\circ=*$. When $\circ=\diamond$ or $\bullet$, it
suffices to replace the base ring $\mathcal{R}[T,T^{-1}]$ in (6.8) by

$\mathcal{R}[T,T^{-1}]_{\diamond}\text{ or }\mathcal{R}[T,T^{-1}]]$

respectively. The ring $\mathcal{R}[T,T^{-1}]_{\diamond}$ is obtained by
taking the completion of $\mathcal{R}[T^{-1}]$ with respect to the topology of
$\mathcal{R}$ and then inverting $T^{-1}$. Any element in
$\mathcal{R}[T,T^{-1}]_{\diamond}$ is then of the form $\sum_{n\geq
m}^{\infty}a_{n}T^{-n}$ for some $m\in\mathbb{Z}$ and $a_{n}\in\mathcal{R}$
such that $\lim_{n\to\infty}a_{n}=0$ in $\mathcal{R}$; so

$\mathcal{R}[T,T^{-1}]\subset\mathcal{R}[T,T^{-1}]_{\diamond}\subset\mathcal{R}[T,T^{-1}]].$

One may now apply the universal coefficient theorem to conclude. ∎

####  6.3. Proof of Proposition 6.3

To deal with the non-vanishing result, we first look at a non-torsion
$spin^{c}\ $structure $\mathfrak{s}$ on $Y$.

If the period class $c$ is non-balanced with respect to $\mathfrak{s}$, then
the map $j_{\bullet}=y_{\diamond}\circ j_{\diamond}$ in diagram (6.6) is an
isomorphism, since the group $\overline{\operatorname{\it
HM}}_{\bullet}(Y,\mathfrak{s},c_{b};\mathcal{R}_{\omega})$ vanishes by [KM07,
Theorem 31.1.1]. This shows that the vertical map

$y_{\diamond}:\widehat{\operatorname{\it
HM}}_{\diamond}(Y,\mathfrak{s},c_{b};\mathcal{R}_{\omega})\to\widehat{\operatorname{\it
HM}}_{\bullet}(Y,\mathfrak{s},c_{b};\mathcal{R}_{\omega}).$

is always surjective. By (6.7) and Proposition 6.7, we conclude that

(6.9) $\operatorname{rank}_{\mathcal{R}}\operatorname{\it
HM}_{*}(Y,\mathfrak{s},c;\mathcal{R}_{\omega})\geq\operatorname{rank}_{\mathcal{R}}\operatorname{\it
HM}_{*}(Y,\mathfrak{s};\mathcal{R}_{\omega}).$

If the period class $c$ is balanced with respect to $\mathfrak{s}$, then
$\operatorname{\it
HM}_{*}(Y,\mathfrak{s},c;\mathcal{R}_{\omega})\colonequals\operatorname{Im}j^{*}$
in the digram (6.6). We apply the same argument to $y_{\diamond}\circ x_{*}$;
so the inequality (6.9) still holds.

When $c_{1}(\mathfrak{s})$ is torsion and $c$ is non-exact, the inequality
(6.9) follows from [KM07, Theorem 31.1.3].

It remains to verify that for the special $spin^{c}\ $structure $\mathfrak{s}$
given by Theorem 6.1 $(2)$, we have $\operatorname{\it
HM}_{*}(Y,\mathfrak{s};\mathcal{R}_{\omega})\neq\\{0\\}$. Combined with (6.9),
this will complete the proof of Proposition 6.3.

The rest of the argument is completely formal and is inspired by [KM07,
Section 32.3]. The group $\operatorname{\it
HM}_{*}(Y,\mathfrak{s};\mathcal{R}_{\omega})$ can be understood using a
smaller local coefficient system $\Pi_{\omega}$, whose fiber at any point in
$\mathcal{B}(Y,\mathfrak{s})$ is the group ring
$R\colonequals\mathbb{Z}[\mathbb{Z}]$ and whose monodromy is again (6.3) (note
that the image of $f$ lies in a $\mathbb{Z}$-subgroup in $\mathbb{R}$). Let
$(\widehat{C}_{*},\hat{\partial})$ be the resulting chain complex defined over
$R$. There are a few other base rings that we will look at:

${R=\mathbb{Z}[\mathbb{Z}]}$${\mathbb{R}[\mathbb{Z}]}$${\mathbb{R}}$${\mathbb{F}_{2}[\mathbb{Z}]}$${\mathcal{K}}$${\mathcal{R}.}$$\scriptstyle{g_{\alpha}}$

where $\mathcal{K}$ is the rational field of $\mathbb{F}_{2}[\mathbb{Z}]$ and
for any $\alpha\in\mathbb{R}$, $g_{\alpha}$ is the ring homomorphism sending
the generator $t\in\mathbb{Z}$ to $e^{\alpha}\in\mathbb{R}$. Note that both
$\mathbb{F}_{2}[\mathbb{Z}]$ and $\mathbb{R}[\mathbb{Z}]$ are principal ideal
domains. Moreover, $\mathcal{K}\to\mathcal{R}$ is a field extension. The goal
is to show that
$H(\widehat{C}_{*}\otimes_{R}\mathbb{F}_{2}[\mathbb{Z}],\hat{\partial})$
contains at least one free summand, so that $\operatorname{\it
HM}_{*}(Y,\mathfrak{s};\mathcal{R}_{\omega})=H(\widehat{C}_{*}\otimes_{R}\mathcal{R},\hat{\partial})\neq\\{0\\}$.

Consider a finite presentation of $H(\widehat{C}_{*})$ over the Noetherian
ring $R$:

(6.10) $R^{n}\xrightarrow{A}R^{k}\to H(\widehat{C}_{*})\to 0,$

and let $I\subset R$ be the ideal generated by all $k\times k$ minors of $A$,
called the Fitting ideal of $H(\widehat{C}_{*})$. By [Eis95, Corollary 20.4],
$I$ is independent of the resolution that we choose.

Suppose on the contrary that the finitely generated
$\mathbb{F}_{2}[\mathbb{Z}]$-module
$H(\widehat{C}_{*}\otimes_{R}\mathbb{F}_{2}[\mathbb{Z}],\hat{\partial})$ is
torsion. Then we claim that $I\neq\\{0\\}$; otherwise, the extension
$I_{\mathbb{F}_{2}}$ of $I$ in $\mathbb{F}_{2}[\mathbb{Z}]$ is trivial.
$I_{\mathbb{F}_{2}}$ is also the Fitting ideal of
$H(\widehat{C}_{*})\otimes_{\mathbb{Z}}\mathbb{F}_{2}$, which can be seen by
taking the tensor product of (6.10) with $\mathbb{F}_{2}$ (over $\mathbb{Z}$).
As a result, $H(\widehat{C}_{*})\otimes_{\mathbb{Z}}\mathbb{F}_{2}$ contains a
free $\mathbb{F}_{2}[\mathbb{Z}]$-summand. The universal coefficient theorem
[Hat02, Theorem 3A.3] then provides an injection:

$0\to H(\widehat{C}_{*})\otimes_{\mathbb{Z}}\mathbb{F}_{2}\to
H(\widehat{C}_{*}\otimes_{\mathbb{Z}}\mathbb{F}_{2},\hat{\partial}),$

which contradicts the assumption that we start with.

Now consider the tensor product of $\eqref{E4.10}$ with $\mathbb{R}$. The
Fitting ideal $I_{\mathbb{R}}$ of
$H(\hat{C}_{*})\otimes_{\mathbb{Z}}\mathbb{R}$ is then the extension of $I$ in
$\mathbb{R}[\mathbb{Z}]$. Since $I\neq\\{0\\}$, $I_{\mathbb{R}}\neq\\{0\\}$.
Using the universal coefficient theorem, we conclude that

$H(\hat{C}_{*})\otimes_{\mathbb{Z}}\mathbb{R}\cong
H(\hat{C}_{*}\otimes_{\mathbb{Z}}\mathbb{R})$

is a torsion $\mathbb{R}[\mathbb{Z}]$-module and is therefore annihilated by
some polynomial $u(t)\in\mathbb{R}[\mathbb{Z}]$.

On the one hand, one may pick $\alpha\in\mathbb{R}$ such that
$g_{\alpha}(u(t))=u(e^{\alpha})\neq 0\in\mathbb{R}$; as a result, the group
$H(\widehat{C}_{*}\otimes_{g_{\alpha}}\mathbb{R},\hat{\partial})$ is trivial
for this particular ring homomorphism
$g_{\alpha}:\mathbb{R}[\mathbb{Z}]\to\mathbb{R}$.

On the other hand, the local coefficient system
$\Pi_{\omega}\otimes_{g_{\alpha}}\mathbb{R}$ agrees with $\Gamma_{\xi}$ for
some 1-cycle $\xi$ with $[\xi]\neq 0\in H_{1}(Y;\mathbb{Z})$, since $[\omega]$
is non-balanced. Theorem 6.1 (2) then asserts that $\operatorname{\it
HM}_{*}(Y,\mathfrak{s};\Gamma_{\xi})$ is non-trivial. A contradiction.∎

####  6.4. Computation for Product 3-manifolds

For future reference, we recall a classical result for the 3-manifold
$\Sigma_{g}\times S^{1}$, where $\Sigma_{g}$ is a closed surface of genus
$g\geq 2$.

######  Lemma 6.9.

Following the shorthands from Definition 6.4, for any closed 2-form
$\omega\in\Omega^{2}(\Sigma_{g}\times S^{1},i\mathbb{R})$ with

(6.11) $i\langle[\omega],[\Sigma_{g}]\rangle<2\pi(g-1),$

we have $\operatorname{\it HM}(\Sigma_{g}\times
S^{1},[\omega]|\Sigma_{g})\cong\mathcal{R}.$

###### Proof of Lemma 6.9.

Let $\mathfrak{s}$ be any $spin^{c}\ $structures contributing to the group
$\operatorname{\it HM}(\Sigma_{g}\times S^{1},[\omega]|\Sigma_{g})$. Under the
assumption (6.11), the period class $c=-2\pi i[\omega]$ is neither negatively
monotone nor balanced with respect to $\mathfrak{s}$; indeed,

$\langle 2\pi^{2}c_{1}(\mathfrak{s})+c,[\Sigma_{g}]\rangle=4\pi^{2}(g-1)-2\pi
i\langle[\omega],[\Sigma_{g}]\rangle>0.$

Using Proposition 6.2, we reduce the computation to the case when the
perturbation is exact, but the coefficient system is still
$\mathcal{R}_{\omega}$. Now we use [KM10, Lemma 2.2] to conclude the proof. ∎

###  7\. Thurston Norm Detection

From now on, we will always take $Y$ to be an oriented 3-manifold with
toroidal boundary.

######  Definition 7.1.

For any $T$-surface $\mathbf{\Sigma}=(\Sigma,g_{\Sigma},\lambda,\mu)$ and any
1-cell $\mathbb{Y}\in\mathbf{T}(\emptyset,\mathbf{\Sigma})$, consider the set
of monopole classes:

$\mathfrak{M}(\mathbb{Y})\colonequals\\{c_{1}(\widehat{\mathfrak{s}}):\operatorname{\it
HM}_{*}(\mathbb{Y},\widehat{\mathfrak{s}})\neq\\{0\\}\\}\subset
H^{2}(Y,\partial Y;\mathbb{Z}),$

and for any $\kappa\in H_{2}(Y,\partial Y;\mathbb{Z})$, define

$\varphi_{\mathbb{Y}}(\kappa)\colonequals\max_{c_{1}(\widehat{\mathfrak{s}})\in\mathfrak{M}(\mathbb{Y})}\langle
c_{1}(\widehat{\mathfrak{s}}),\kappa\rangle,\ \kappa\in H_{2}(Y,\partial
Y;\mathbb{Z}).$

Our convention here is that $\varphi_{\mathbb{Y}}\equiv-\infty$, if
$\mathfrak{M}(\widehat{Y})=\emptyset$. ∎

When $Y$ is connected and irreducible, it is tempting to generalize Theorem
6.1 and relate $\mathfrak{M}(\mathbb{Y})$ with the Thurston norm on
$H_{2}(Y,\partial Y;\mathbb{R})$. However, the author was not able to show
that $\mathfrak{M}(\mathbb{Y})$ is symmetric about the origin; so only a
weaker statement is obtained in this paper.

######  Theorem 7.2.

For any $T$-surface $\mathbf{\Sigma}=(\Sigma,g_{\Sigma},\lambda,\mu)$, let
$\mathbb{Y}\in\mathbf{T}(\emptyset,\mathbf{\Sigma})$ be any 1-cell with $Y$
connected and irreducible. Then the set of monopole classes
$\mathfrak{M}(\mathbb{Y})$ is non-empty and determines the Thurston norm on
$H_{2}(Y,\partial Y;\mathbb{R})$ in the following sense:

$x(\kappa)=\frac{1}{2}(\varphi_{\mathbb{Y}}(\kappa)+\varphi_{\mathbb{Y}}(-\kappa)),\
\forall\kappa\in H_{2}(Y,\partial Y;\mathbb{Z}).$

In general, we only have an inequality
$\varphi_{\mathbb{Y}}(\kappa)+\varphi_{\mathbb{Y}}(-\kappa)\leq 2x(\kappa)$.

One may approach the more desirable statement

$x(\kappa)=\varphi_{\mathbb{Y}}(\kappa)=\varphi_{\mathbb{Y}}(-\kappa).$

by either proving the symmetry of $\mathfrak{M}(\mathbb{Y})$ or the adjunction
inequality:

$\varphi_{\mathbb{Y}}(\kappa)\leq x(\kappa).$

But the author was unable to verify either of them directly. This Thurston
norm detection result is accompanied with a fiberness detection result:

######  Theorem 7.3.

For any $T$-surface $\mathbf{\Sigma}=(\Sigma,g_{\Sigma},\lambda,\mu)$, let
$\mathbb{Y}\in\mathbf{T}(\emptyset,\mathbf{\Sigma})$ be any 1-cell with $Y$
connected and irreducible, and $\kappa\in H_{2}(Y,\partial Y;\mathbb{Z})$ be
any integral class. Consider the subgroup

$\operatorname{\it HM}_{*}(\mathbb{Y}|\kappa)\colonequals\bigoplus_{\langle
c_{1}(\widehat{\mathfrak{s}}),\kappa\rangle=\varphi_{\mathbb{Y}}(\kappa)}\operatorname{\it
HM}_{*}(\mathbb{Y},\widehat{\mathfrak{s}}).$

If $\operatorname{rank}_{\mathcal{R}}\operatorname{\it
HM}_{*}(\mathbb{Y}|\kappa)=\operatorname{rank}_{\mathcal{R}}\operatorname{\it
HM}_{*}(\mathbb{Y}|-\kappa)=1$, then $\kappa$ can be represented by a Thurston
norm minimizing surface $F$ and $Y$ fibers over $S^{1}$ with $F$ as fiber.

The rest of this section is devoted to the proof of Theorem 7.2, while the
proof of Theorem 7.3 is deferred to Section 9, after a digression into sutured
monopole Floer homology in Section 8.

###### Proof of Theorem 7.2.

We focus on the case when $Y$ is irreducible. The strategy is to apply the
Gluing Theorem 2.7 and reduce the problem to the double of $Y$:

$\tilde{Y}\colonequals Y\ \bigcup_{\Sigma}\ (-Y).$

Since $Y$ is irreducible, so is $\tilde{Y}$. Then we can conclude using
Proposition 6.2 and 6.3.

For any 1-cell $\mathbb{Y}=(Y,g_{Y},\omega,\cdots)$, consider its orientation
reversal:

$(-\mathbb{Y})=(-Y,g_{Y},\omega,\cdots)\in\mathbf{T}(\mathbf{\Sigma}^{\prime},\emptyset).$

The problem here is that the $T$-surface
$\mathbf{\Sigma}^{\prime}=(\Sigma,g_{\Sigma},-\lambda,\mu)$ is different from
$\mathbf{\Sigma}$ in that the sign of $\lambda$ is reversed. We can not form
the horizontal composition $\mathbb{Y}\circ_{h}(-\mathbb{Y})$.

This problem is circumvented by changing the 2-form $\omega$ on $(-Y)$. Since
$\operatorname{\it HM}_{*}(\mathbb{Y},\widehat{\mathfrak{s}})$ is independent
of the metric $g_{Y}$, we assume that it is cylindrical on
$(-2,1]_{s}\times\Sigma\subset Y$. Pick a cut-off function
$\chi_{1}:(-2,1]_{s}\to\mathbb{R}$ such that $\chi_{1}\equiv 0$ if $s\leq-3/2$
and $\equiv 1$ if $s\geq-1$. Write
$\omega=\overline{\omega}+\chi_{1}(s)ds\wedge\lambda$ and set

$\mathbb{Y}^{\prime}\colonequals(Y,g_{Y},\omega^{\prime},\cdots)\in\mathbf{T}(\emptyset,\mathbf{\Sigma}^{\prime})\text{
with
}\omega^{\prime}\colonequals\overline{\omega}-\chi_{1}(s)ds\wedge\lambda.$

We shall work instead with the orientation reversal
$(-\mathbb{Y}^{\prime})\in\mathbf{T}(\mathbf{\Sigma},\emptyset)$. At this
point, we need a lemma relating the Floer homology of $\mathbb{Y}^{\prime}$
with that of $\mathbb{Y}$:

######  Lemma 7.4.

For any $T$-surface $\mathbf{\Sigma}$, let
$\mathbb{Y}\in\mathbf{T}(\emptyset,\mathbf{\Sigma})$ be any 1-cell and
$\widehat{\mathfrak{s}}\in\operatorname{Spin^{c}_{R}}(Y)$ be any relative
$spin^{c}\ $structure. Then we have the following isomorphisms:

  1. (1)

(Poincaré Duality) $\operatorname{\it
HM}_{*}((-\mathbb{Y}),\widehat{\mathfrak{s}})\cong\operatorname{\it
HM}^{*}(\mathbb{Y},\widehat{\mathfrak{s}})$.

  2. (2)

(Reversing the sign of $\lambda$) $\operatorname{\it
HM}_{*}(\mathbb{Y},\widehat{\mathfrak{s}})\cong\operatorname{\it
HM}_{*}(\mathbb{Y}^{\prime},\widehat{\mathfrak{s}})$.

Let us finish the proof of Theorem 7.2 assuming Lemma 7.4. Consider the
horizontal composition of $\mathbb{Y}$ and $-\mathbb{Y}^{\prime}$:

$(\tilde{Y},\tilde{\omega},\cdots)\colonequals\mathbb{Y}\circ_{h}(-\mathbb{Y}^{\prime}).$

We verify that the closed 2-form $\tilde{\omega}$ is neither negatively
monotone nor balanced with respect to any $spin^{c}\ $structures on the double
$\tilde{Y}$. Indeed, by (T3) and (T4),
$|\langle\tilde{\omega},[\Sigma^{(i)}]\rangle|<2\pi$ and $\neq 0$ for any
component $\Sigma^{(i)}\subset\Sigma$.

Any integral class $\kappa\in H_{2}(Y,\partial Y;\mathbb{Z})$ can be
represented by a properly embedded oriented surface $F\subset Y$ minimizing
the Thurston norm. Combined with its orientation reversal $(-F)\subset(-Y)$,
they form a closed surface

$\tilde{F}\colonequals F\cup(-F)\subset\tilde{Y}.$

Since $Y$ is irreducible and $Y\neq S^{1}\times D^{2}$, the surface $F$ has no
sphere or disk components. By [Gab83, Lemma 6.15], the double
$\tilde{F}\subset\tilde{Y}$ is also norm-minimizing. By Theorem 3.7, we have

(7.1) $\operatorname{\it
HM}_{*}(\tilde{Y},\mathfrak{s},\tilde{c};\mathcal{R}_{\tilde{\omega}})\cong\bigoplus_{\widehat{\mathfrak{s}}_{1}\circ_{h}\widehat{\mathfrak{s}}_{2}=\mathfrak{s}}\operatorname{\it
HM}_{*}(\mathbb{Y},\widehat{\mathfrak{s}}_{1})\otimes_{\mathcal{R}}\operatorname{\it
HM}_{*}(-\mathbb{Y}^{\prime},\widehat{\mathfrak{s}}_{2}).$

where the sum is over all pairs
$(\widehat{\mathfrak{s}}_{1},\widehat{\mathfrak{s}}_{2})\in\operatorname{Spin^{c}_{R}}(Y)\times\operatorname{Spin^{c}_{R}}(-Y)$
with
$\widehat{\mathfrak{s}}_{1}\circ_{h}\widehat{\mathfrak{s}}_{2}=\mathfrak{s}$.
Note that

(7.2) $\langle
c_{1}(\widehat{\mathfrak{s}}_{1}\circ_{h}\widehat{\mathfrak{s}}_{2}),[\tilde{F}]\rangle=\langle
c_{1}(\widehat{\mathfrak{s}}_{1}),[F]\rangle+\langle
c_{1}(\widehat{\mathfrak{s}}_{2}),[-F]\rangle,$

By the adjunction inequality from Proposition 6.2, the left hand side of (7.1)
vanishes whenever

$\langle c_{1}(\mathfrak{s}),[\tilde{F}]\rangle>x(\tilde{F})=2x(F).$

Combined with Corollary 6.6 and (7.1)(7.2), this implies that the group

(7.3) $\operatorname{\it
HM}(\tilde{Y},[\tilde{\omega}]|[\tilde{F}])\cong\operatorname{\it
HM}_{*}(\mathbb{Y}|\kappa)\otimes_{\mathcal{R}}\operatorname{\it
HM}_{*}(-\mathbb{Y}^{\prime}|-\kappa),$

is non-vanishing. By Lemma 7.4, we have

(7.4) $\operatorname{rank}_{\mathcal{R}}\operatorname{\it
HM}_{*}(-\mathbb{Y}^{\prime}|-\kappa)=\operatorname{rank}_{\mathcal{R}}\operatorname{\it
HM}_{*}(\mathbb{Y}|-\kappa),$

so

$2x(\kappa)=x(\tilde{F})=\varphi_{\mathbb{Y}}(\kappa)+\varphi_{(-\mathbb{Y}^{\prime})}(-\kappa)=\varphi_{\mathbb{Y}}(\kappa)+\varphi_{\mathbb{Y}}(-\kappa).$

This completes the proof of Theorem 7.2 when $Y$ is irreducible. In the
general case, the inequality

$\varphi_{\mathbb{Y}}(\kappa)+\varphi_{\mathbb{Y}}(-\kappa)\leq 2x(\kappa),$

follows from the vanishing result that we used earlier. ∎

###### Proof of Lemma 7.4.

The first isomorphism (1) is due to Poincaré duality. Changing the orientation
of $\widehat{Y}$ has the same effect as changing the sign of the perturbed
Chern-Simons-Dirac functional $\mathcal{L}_{\omega}$ in (3.3), while keeping
the orientation fixed; see [KM07, Section 22.5] for more details. Since we
have worked with a ring $\mathcal{R}$ over $\mathbb{F}_{2}$, there is no need
to deal with orientations. Our Floer homology $\operatorname{\it
HM}_{*}(\mathbb{Y})$ is defined using a local coefficient system with fibers
$\mathcal{R}$. Thus the most relevant analogue for closed 3-manifolds is
[KM07, P.624 (32.2)].

The second isomorphism (2) is induced from a concrete strict cobordism

$\mathbb{X}:\mathbb{Y}\coprod e_{\mathbf{\Sigma}}\to\mathbb{Y}^{\prime},$

as we describe now. Similar to the construction in Section 3, we start with a
hexagon $\Omega_{1}$ with boundary consisting of geodesic segments of length
$2$ and whose internal angles are always $\pi/2$. Also, the metric of
$\Omega_{1}$ is flat near the boundary.

\begin{overpic}[scale={.09}]{Pic12.png} \put(37.0,41.0){\small$\gamma_{12}$}
\put(56.0,41.0){\small$\gamma_{23}$} \put(53.0,28.0){\small$t_{13}$}
\put(27.0,21.0){\small$\widehat{Y}$} \put(68.0,21.0){\small$\widehat{Y}$}
\put(41.0,58.0){\small$\mathbb{R}_{s_{2}}\times\Sigma$}
\put(5.0,46.0){\small$s_{1}$} \put(94.0,47.0){\small$s_{3}$}
\put(15.0,69.0){\small$s_{2}$} \put(80.0,69.0){\small$s_{2}$}
\put(46.0,18.0){\small$\gamma_{13}$} \put(27.0,46.0){\small$t_{12}$}
\put(67.0,46.0){\small$t_{32}$} \put(48.0,46.0){$\Omega_{1}$} \end{overpic}
Figure 12.

The desired cobordism $X$ is then obtained from $\Omega_{1}\times\Sigma$ by
attaching $[-1,1]_{t}\times Y_{-}$ to $\gamma_{13}\times\Sigma$ in Figure 12,
where $Y_{-}\colonequals\\{s\leq-1\\}\subset\widehat{Y}$. In this case, we use
$s_{1},s_{2},s_{3}$ for spatial coordinates and $t_{12},t_{13},t_{32}$ for
time coordinates; their relations are indicated as in Figure 12.

We have to specify the closed 2-form $\omega_{X}$ on $X$ in order to construct
the cobordism map. Recall that the 2-form $\omega$ on $Y$ is decomposed as

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

First, pull back $\overline{\omega}$ to $[-1,1]_{t}\times Y_{-}$ and extend it
on $\Omega_{1}\times\Sigma$ constantly as the harmonic 2-form
$\mu\in\Omega^{1}_{h}(\Sigma,i\mathbb{R})$. Second, consider a function
$f:(-2,\infty)_{s}\to\mathbb{R}$ such that $f(s)\equiv s$ when $s\geq-1$ and
$f^{\prime}(s)=\chi_{1}(s)$ when $s\in(-2,1]$. We shall regard $f$ as a
function on $\widehat{Y}$ by extending $f$ constantly when $s\leq-3/2$. Now
consider a function $h:\widehat{X}\to\mathbb{R}$ with the following
properties:

  * •

$h\equiv f$ on a neighborhood $[-1,-1+\epsilon)_{t_{13}}\times\widehat{Y}$ and
$\equiv-f$ on $(1-\epsilon,1]_{t_{13}}\times\widehat{Y}$;

  * •

$h\equiv s_{1}\equiv s_{2}$ on
$\gamma_{12}\times[1,\infty)_{s_{1}}\times\Sigma$ and $\equiv-s_{3}\equiv
s_{2}$ on $\gamma_{23}\times[1,\infty)_{s_{3}}\times\Sigma$;

  * •

$h\equiv s_{2}$ on a neighborhood
$(1-\epsilon,1]_{t_{12}}\times\mathbb{R}_{s_{2}}\times\Sigma$.

The extension of $h$ over the interior of $X$ can be arbitrary. Finally, set

$\omega_{X}=\overline{\omega}+dh\wedge\lambda\text{ on }\widehat{X}.$

The strict cobordism $\mathbb{X}=(X,\omega_{X})$ then induces a map:

$\alpha_{1}:\operatorname{\it
HM}_{*}(\mathbb{Y},\widehat{\mathfrak{s}})\xrightarrow{\operatorname{Id}\otimes
1}\operatorname{\it
HM}_{*}(\mathbb{Y},\widehat{\mathfrak{s}})\otimes\operatorname{\it
HM}_{*}(e_{\mathbf{\Sigma}},\widehat{\mathfrak{s}}_{std})\xrightarrow{\operatorname{\it
HM}_{*}(\mathbb{X})}\operatorname{\it
HM}_{*}(\mathbb{Y}^{\prime},\widehat{\mathfrak{s}}).$

Switching the roles of $\mathbb{Y}$ and $\mathbb{Y}^{\prime}$ produces a
strict cobordism $\mathbb{X}^{\prime}:\mathbb{Y}^{\prime}\coprod
e_{\mathbf{\Sigma}}\to\mathbb{Y}$ and a map in the opposite direction:

$\alpha_{1}^{\prime}:\operatorname{\it
HM}_{*}(\mathbb{Y}^{\prime},\widehat{\mathfrak{s}})\xrightarrow{\operatorname{Id}\otimes
1}\operatorname{\it
HM}_{*}(\mathbb{Y}^{\prime},\widehat{\mathfrak{s}})\otimes\operatorname{\it
HM}_{*}(e_{\mathbf{\Sigma}^{\prime}},\widehat{\mathfrak{s}}_{std})\xrightarrow{\operatorname{\it
HM}_{*}(\mathbb{X}^{\prime})}\operatorname{\it
HM}_{*}(\mathbb{Y},\widehat{\mathfrak{s}}).$

It remains to verify that $\alpha_{1}$ and $\alpha_{1}^{\prime}$ are mutual
inverses to each other (up to a non-zero scalar). To see this, compose
$\mathbb{X}$ with $\mathbb{X}^{\prime}$ along the common boundary
$\mathbb{Y}^{\prime}$. The resulting cobordism

$\mathbb{Y}\ \coprod\ e_{\mathbf{\Sigma}}\ \coprod\
e_{\mathbf{\Sigma}^{\prime}}\to\mathbb{Y}$

induces the gluing map $\alpha$ in Subsection 3.6, which is an isomorphism by
Theorem 3.6. Indeed, one can change the 2-form
$\omega_{X}^{\prime}\circ\omega_{X}$ into the standard one in (3.4) by adding
an exact 2-form. Thus the map
$\alpha_{1}^{\prime}\circ\alpha_{1}:\operatorname{\it
HM}_{*}(\mathbb{Y},\widehat{\mathfrak{s}})\to\operatorname{\it
HM}_{*}(\mathbb{Y},\widehat{\mathfrak{s}})$ is invertible. The other
composition $\alpha_{1}\circ\alpha_{1}^{\prime}$ is dealt with in a similar
manner. This completes the proof of Lemma 7.4. ∎

###  8\. Relations with Sutured Floer Homology

Before we prove the fiberness detection result, Theorem 7.3, we add a
digression to explain the relations of $\operatorname{\it HM}_{*}(\mathbb{Y})$
with the sutured Floer homology $\operatorname{\it SHM}$ [KM10] and
$\operatorname{SFH}$ [Juh06, Juh08]. In the original definition of Gabai
[Gab83, Definition 2.6], any 3-manifolds with toroidal boundary are sutured
manifolds with only toral components. However, they are not examples of
balanced sutured manifolds in the sense of Juhász [Juh06, Definition 2.2];
thus the sutured Floer homology, either $\operatorname{\it SHM}$ or
$\operatorname{SFH}$, is never defined for this class of sutured manifolds.

One may regard our construction as a natural extension of the existing sutured
Floer theory, and ask if the suture manifold decomposition theorem, e.g.
[Juh08, Theorem 1.3] and [KM10, Proposition 6.9], continue to hold in our
case. In this section, we will only prove a preliminary result towards this
direction. Since our Floer homology $\operatorname{\it HM}_{*}(\mathbb{Y})$
also relies on the closed 2-form $\omega$, it is not clear to the author
whether a general result is available.

It is worth mentioning that the sutured monopole Floer homology
$\operatorname{\it SHM}$ introduced by Kronheimer-Mrowka [KM10] is defined
over $\mathbb{Z}$, but the construction extends naturally to the mod 2 Novikov
ring $\mathcal{R}$, as explained in [Siv12, Section 2.2]. We shall work with
the latter case.

####  8.1. Cutting along a surface

For any $T$-surface $\mathbf{\Sigma}=(\Sigma,g_{\Sigma},\lambda,\mu)$ and any
1-cell $\mathbb{Y}\in\mathbf{T}(\emptyset,\mathbf{\Sigma})$, we listed a few
cohomological conditions on $\omega$ in (P4) (P5). In this section, we shall
think of them geometrically and work with a more restrictive setup:

  1. (P5)

there exists a properly embedded oriented surface $F\subset Y$ such that
$\partial F$ intersects each component of $\Sigma$ in parallel circles, and
$i[*_{2}\lambda]$ is dual to $z[\partial F]\in H_{1}(\Sigma,\mathbb{Z})$ for
some $z\in\mathbb{R}$; moreover, $F$ has no closed component or disk
components. In particular, $\chi(F)\leq 0$.

  2. (P6)

the Poincaré dual of $-2\pi i[\omega]\in H^{2}(Y;\mathbb{R})$ can be
represented by a real 1-cycle $\eta$ that lies on the surface $F$.

If $\mathbb{Y}$ satisfies the additional properties (P5)(P6), then we can cut
$Y$ along the surface $F$ to obtain a balanced sutured manifold, denoted by
$M(Y,F)$. Let

$\operatorname{\it SHM}(M(Y,F))$

be the sutured monopole Floer homology of $M(Y,F)$ defined over $\mathcal{R}$
with trivial coefficient system; cf. [Siv12, Section 2.2].

######  Theorem 8.1.

For any $T$-surface $\mathbf{\Sigma}$ and any 1-cell
$\mathbb{Y}\in\mathbf{T}(\emptyset,\mathbf{\Sigma})$ satisfying (P5)(P6), we
have

$\operatorname{\it SHM}(M(Y,F))\cong\bigoplus_{\langle
c_{1}(\widehat{\mathfrak{s}}),[F]\rangle=x(F)}\operatorname{\it
HM}_{*}(\mathbb{Y},\widehat{\mathfrak{s}}).$

Moreover, $\varphi_{\mathbb{Y}}([F])\leq x(F)$ with $\varphi_{\mathbb{Y}}$
defined as in Definition 7.1.

For the proof of Theorem 8.1, we have to first understand the special case
when $F$ is connected and $Y$ fibers over $S^{1}$ with $F$ as a fiber. Thus
$F$ is a genus $g$ surface with $n$ boundary circles, and $Y$ is the mapping
torus of a self-diffeomorphism $\phi:F\to F$ such that
$\phi|_{\pi_{0}(\partial F)}$ has at least two orbits.

######  Lemma 8.2.

For any $T$-surface $\mathbf{\Sigma}$ and any 1-cell
$\mathbb{Y}\in\mathbf{T}(\emptyset,\mathbf{\Sigma})$, if $F$ is connected and
$Y$ is a mapping torus over $F$, then for $\kappa=\pm[F]$,
$\varphi_{\mathbb{Y}}(\kappa)=-\chi(F)$ and

$\operatorname{\it HM}(\mathbb{Y}|\kappa)\cong\mathcal{R}.$

###### Proof of Lemma 8.2.

If $\chi(F)=0$, then $F$ is an annulus and $Y=[-3,3]_{s}\times\mathbb{T}^{2}$
is a product. This case is addressed already in Lemma 5.3. We focus on the
case when $\chi(F)<0$.

With loss of generality, let $\kappa=[F]$. Note that $Y$ is irreducible and
$F$ minimizes the Thurston norm. We follow the notations and arguments in the
proof of Theorem 7.2. Let $(\tilde{Y},\tilde{F})$ denote the double of
$(Y,F)$, then $\tilde{Y}$ is a mapping torus over $\tilde{F}$ with
$g(\tilde{F})\geq 2$. By (7.3), we have

(8.1) $\operatorname{\it
HM}(\tilde{Y},[\tilde{\omega}]|\tilde{F})=\operatorname{\it
HM}(\mathbb{Y}|\kappa)\otimes_{\mathcal{R}}\operatorname{\it
HM}(-\mathbb{Y}^{\prime}|-\kappa).$

The rest of the proof is divided into four steps.

Step 1\. We can arrange the 2-form $\tilde{\omega}$ so that
$\langle[\tilde{\omega}],[\tilde{F}]\rangle=0$.

Instead of $\mathbb{Y}\circ_{h}(-\mathbb{Y}^{\prime})$, we consider the
horizontal composition

$(\tilde{Y},\tilde{\omega},\cdots)=\mathbb{Y}\circ_{h}\mathbb{Y}_{1}\circ_{h}(-\mathbb{Y}^{\prime}),$

where
$\mathbb{Y}_{1}=([-3,3]_{s}\times\Sigma,\omega_{1},\cdots)\in\mathbf{T}(\mathbf{\Sigma},\mathbf{\Sigma})$.
Here $\omega_{1}=\mu+ds\wedge\lambda+\omega_{2}$ and
$\omega_{2}\in\Omega^{2}_{c}(Y_{1},i\mathbb{R})$ is compactly supported. By
Lemma 5.3, inserting this extra 1-cell $\mathbb{Y}_{1}$ does not affect the
identity (8.1), but changing the class $[\omega_{2}]\in H^{2}(Y_{1},\partial
Y_{1};i\mathbb{R})$ can effectively alter the pairing
$\langle[\tilde{\omega}],[\tilde{F}]\rangle$. From now on, we shall always
assume that $\langle[\tilde{\omega}],[\tilde{F}]\rangle=0$.

Step 2\. The group $\operatorname{\it
HM}(\tilde{Y},[\tilde{\omega}]|\tilde{F})$ in (8.1) has rank $1$.

Since $g(\tilde{F})\geq 2$, this computation for mapping tori is [KM10, Lemma
4.7], if $\tilde{\omega}=0$. The general case is not really different. Since
$\langle[\tilde{\omega}],[\tilde{F}]\rangle=0$, we can still apply Floer’s
excision theorem [KM10, Theorem 3.1] to reduce the problem to the case when
$\tilde{Y}=\tilde{F}\times S^{1}$ is a product. The same trick is used also in
the proof of Theorem 8.1 below. Now the statement follows from Lemma 6.9.

Step 3\. We conclude from (8.1), Lemma 7.4 and Step 2 that

$\operatorname{rank}_{\mathcal{R}}\operatorname{\it
HM}(\mathbb{Y}|\kappa)=\operatorname{rank}_{\mathcal{R}}\operatorname{\it
HM}(\mathbb{Y}|-\kappa)=1.$

Step 4\. By Theorem 7.2,
$\varphi_{\mathbb{Y}}(\kappa)+\varphi_{\mathbb{Y}}(-\kappa)=2x(\kappa)$. We
have to verify that
$\varphi_{\mathbb{Y}}(\kappa)=\varphi_{\mathbb{Y}}(-\kappa)$. This equality
now follows from the symmetry of the graded Euler characteristics:

$\operatorname{\underline{SW}}(Y):\widehat{\mathfrak{s}}\mapsto\chi(\operatorname{\it
HM}_{*}(\mathbb{Y},\widehat{\mathfrak{s}}));$

see Theorem 3.4. This function is invariant under the conjugacy of relative
$spin^{c}\ $structures:
$\widehat{\mathfrak{s}}\leftrightarrow\widehat{\mathfrak{s}}^{*}$ by [MT96,
Tau01]. The computation of $\operatorname{\it HM}(\mathbb{Y}|\kappa)$ shows
that $\chi(\operatorname{\it HM}_{*}(\mathbb{Y},\widehat{\mathfrak{s}}))\neq
0$ for exactly one $\widehat{\mathfrak{s}}$ with $\langle
c_{1}(\widehat{\mathfrak{s}}),\kappa\rangle\geq\varphi_{\mathbb{Y}}(\kappa)$.
This proves the equality
$\varphi_{\mathbb{Y}}(\kappa)=\varphi_{\mathbb{Y}}(-\kappa)$ and completes the
proof of Lemma 8.2. ∎

###### Proof of Theorem 8.1.

In [KM10], the group $\operatorname{\it SHM}(M(Y,F))$ is defined as the
monopole Floer homology of a suitable closure of $M(Y,F)$, which can be
described as follows.

Consider an 1-cell $\mathbb{Y}_{1}\in\mathbf{T}(\mathbf{\Sigma},\emptyset)$
such that $Y_{1}$ is a mapping torus over a connected surface $F_{1}$. We
require that

  * •

$g(F_{1})\geq 2$;

  * •

$[\partial F_{1}]=-[\partial F]\in H_{1}(\Sigma,\mathbb{Z})$, and

  * •

the 1-cell $\mathbb{Y}_{1}$ satisfies property (P5)(P6) for the surface
$F_{1}$.

Consider the closed 3-manifold obtained by gluing $\mathbb{Y}$ and
$\mathbb{Y}_{1}$:

$(Y_{2},\omega_{2},\cdots)\colonequals\mathbb{Y}\circ_{h}\mathbb{Y}_{1},$

We can arrange so that $\partial F$ is identical to $\partial F_{1}$ on
$\Sigma$; so $Y_{2}$ contains a closed oriented surface $F_{2}\colonequals
F\cup F_{1}$ with $g(F_{2})\geq 2$. Moreover, $F_{2}$ is connected. Following
the shorthands from Definition 6.4, the sutured monopole Floer homology of
$M(Y,K)$ is then defined as

$\operatorname{\it SHM}(M(Y,K))\colonequals\operatorname{\it
HM}_{*}(Y_{2}|F_{2});$

see [KM10, Definition 4.3] and [Siv12, Section 2.2]. For the latter group, we
have

$\operatorname{\it HM}_{*}(Y_{2}|F_{2})\cong\operatorname{\it
HM}_{*}(Y_{2},[0]|F_{2}),$

by Remark 6.5. Let $c_{2}=-2\pi i[\omega_{2}]$ be the period class of
$\omega_{2}\in\Omega^{2}(Y_{2},i\mathbb{R})$. Using Property (P6), we can
arrange so that the Poincaré dual of $c_{2}$ is represented by a real 1-cycle
$\eta_{2}$ lying over $F_{2}\subset Y_{2}$. Now consider the subgroup

$G_{y}\colonequals\bigoplus_{\langle
c_{1}(\widehat{\mathfrak{s}}),[F]\rangle=y}\operatorname{\it
HM}_{*}(\mathbb{Y},\widehat{\mathfrak{s}})\subset\operatorname{\it
HM}_{*}(\mathbb{Y}).$

Theorem 3.7 and Lemma 8.2 then imply that

(8.2) $\bigoplus_{\langle
c_{1}(\mathfrak{s}),[F_{2}]\rangle=y+x(F_{1})}\operatorname{\it
HM}_{*}(Y_{2},\mathfrak{s},c_{2};\mathcal{R}_{\omega_{2}})=G_{b}\otimes_{\mathcal{R}}\operatorname{\it
HM}(\mathbb{Y}_{1}|[F_{1}])\cong G_{b},$

for any $y\geq\varphi_{\mathbb{Y}}([F])$. By the adjunction inequality from
Proposition 6.2, the left hand side of (8.2) vanishes whenever
$b+x(F_{1})>x(F_{2})$. As a result,

$\varphi_{\mathbb{Y}}([F])\leq x(F)=-\chi(F).$

Let $y=x(F)$ in (8.2), then

$\operatorname{\it HM}(Y_{2},[\omega_{2}]|F_{2})\cong G_{x(F)}.$

To complete the proof of Theorem 8.1, it remains to verify that

(8.3) $\operatorname{\it HM}(Y_{2},[0]|F_{2})\cong\operatorname{\it
HM}(Y_{2},[\omega_{2}]|F_{2}).$

This isomorphism, which involves only the closed 3-manifold $Y_{2}$, is
similar to the one in [KM10, Corollary 3.4], except that $\omega_{2}$ is used
for non-exact perturbations here. The proof of (8.3) relies on the property
that the real 1-cycle $\eta_{2}$ that represents $c_{2}$ lies on the surface
$F_{2}$, so we can pick $\omega_{2}^{\prime}\in[\omega_{2}]$ such that
$\omega_{2}^{\prime}$ is supported on a tubular neighborhood of $F_{2}$:

$[-1,1]\times F_{2}\subset Y_{2}.$

By identifying $\\{\pm 1\\}\times F_{2}$, $\omega_{2}^{\prime}$ becomes a
closed 2-form on $F_{2}\times S^{1}$. The same process applied to
$Y_{2}\setminus[-1,1]\times F_{2}$ yields another copy of $Y_{2}$, but the
2-form is zero now.

As in the proof of [KM10, Corollary 3.4], we apply Floer’s excision theorem
[KM10, Theorem 3.1] to obtain that

$\operatorname{\it HM}(Y_{2},[\omega_{2}]|F_{2})=\operatorname{\it
HM}(Y_{2},[0]|F_{2})\otimes_{\mathcal{R}}\operatorname{\it HM}(F_{2}\times
S^{1},[\omega_{2}^{\prime}]|F_{2}).$

By Lemma 6.9, $\operatorname{\it HM}(F_{2}\times
S^{1},[\omega_{2}^{\prime}]|F_{2})\cong\mathcal{R}$. This completes the proof
of Theorem 8.1. ∎

The proof of Theorem 8.1 has an immediate corollary:

######  Corollary 8.3.

For any $T$-surface $\mathbf{\Sigma}$ and any 1-cell
$\mathbb{Y}\in\mathbf{T}(\emptyset,\mathbf{\Sigma})$ satisfying (P5),
$\varphi_{\mathbb{Y}}([F])\leq x(F)$.

######  Remark 8.4.

The property (P5) is crucial to the proof of Theorem 8.1 for the following
reason: for the gluing argument to work, we have to make sure that

  1. (i)

$[\partial F]=-[\partial F_{1}]\in H_{1}(\Sigma;\mathbb{Z})$;

  2. (ii)

there exists classes $a\in H_{2}(Y,\partial Y;\mathbb{R})$ and $a_{1}\in
H_{2}(Y_{1},\partial Y_{1};\mathbb{R})$ such that

$[\partial a]=-[\partial a_{1}]=\text{the Poincar\'{e} dual of
}i[*_{\Sigma}\lambda]\in H_{1}(\Sigma;\mathbb{R});$

In general, it is not clear to the author whether $Y$ and $Y_{1}$ can be
always glued. However, Property (P5) reduces (i)(ii) into a single condition,
which is easier to verify in practice. ∎

####  8.2. Relations with Link Floer Homology

As a special case of Theorem 8.1, our construction recovers the monopole knot
Floer homology $\operatorname{\it KHM}$ introduced by Kronheimer and Mrowka
[KM10]. This statement is also true for the link Floer homology; let us expand
on its meaning.

For any link $L=\\{L_{i}\\}_{i=1}^{n}$ inside a closed oriented 3-manifold
$Z_{0}$, consider the balanced sutured manifold

$Z_{0}(L)\colonequals(Z_{0}\setminus N(L),\gamma)$

where $s(\gamma)\cap N(L_{i})$ are two meridional sutures on $\partial
N(L_{i})$ oriented in opposite ways. The link Floer homology of $(Z_{0},L)$ is
defined as the sutured monopole Floer homology of $Z_{0}(L)$:

$\operatorname{\it LHM}(Z_{0},L)\colonequals\operatorname{\it SHM}(Z_{0}(L)),$

and we shall work with the mod 2 Novikov ring $\mathcal{R}$.

On the other hand, pick a meridian $m_{i}$ for each link component $L_{i}$ and
consider the link complement

$Y(Z_{0},L)\colonequals Z_{0}\setminus N(L\cup m_{1}\cup m_{2}\cup\cdots\cup
m_{n}).$

Each $m_{i}$ bounds a disk in $Z_{0}$, which becomes an annulus $A_{i}$ in
$Y(Z_{0},L)$. Let $F$ be the union $\bigcup_{i=1}^{n}A_{i}$ (with any fixed
orientation). The balanced sutured manifold $Z_{0}(L)$ is then obtained from
$Y(Z_{0},L)$ by cutting along $F$.

To apply Theorem 8.1, we have to specify the choice of $\omega$:

  * •

let $\Sigma=\partial Y(Z_{0},L)$. Pick a flat metric $g_{\Sigma}$ and
$\lambda\in\Omega^{1}_{h}(\Sigma,i\mathbb{R})$ such that
$i[*_{\Sigma}\lambda]$ is dual to $z[\partial F]\in H_{1}(\Sigma,\mathbb{R})$
for some $z\neq 0\in\mathbb{R}$;

  * •

$\eta$ is a real 1-cycle on $F$ such that $\eta\cap A_{i}$ is a segment
joining two components of $\partial A_{i}$. Let $-2\pi i[\omega]\in
H^{2}(Y(Z_{0},L),\mathbb{R})$ be the dual of $\eta$.

Finally, let
$\mathbb{Y}(Z_{0},L)=(Y(Z_{0},L),\omega,\cdots)\in\mathbf{T}(\emptyset,\mathbf{\Sigma})$
be any 1-cell satisfying these properties. As a corollary of Theorem 8.1, we
have

######  Corollary 8.5.

For any 1-cell $\mathbb{Y}(Z_{0},L)$ constructed above, we have an isomorphism

$\operatorname{\it HM}_{*}(\mathbb{Y}(Z_{0},L))\cong\operatorname{\it
SHM}(Z_{0}(L))=\operatorname{\it LHM}(Z_{0},L).$

Moreover, if $\operatorname{\it
HM}_{*}(\mathbb{Y}(Z_{0},L),\widehat{\mathfrak{s}})\neq\\{0\\}$, then $\langle
c_{1}(\widehat{\mathfrak{s}}),[A_{i}]\rangle=0$ for any $1\leq i\leq n$.

###### Proof of Corollary 8.5.

If $\operatorname{\it
HM}_{*}(\mathbb{Y}(Z_{0},L),\widehat{\mathfrak{s}})\neq\\{0\\}$, then Theorem
8.1 implies that

$\langle
c_{1}(\widehat{\mathfrak{s}}),[F]\rangle\leq\varphi_{\mathbb{Y}}([F])\leq
x(F)=0.$

Applying the same argument for $[-F]$, we conclude that $\langle
c_{1}(\widehat{\mathfrak{s}}),[F]\rangle=0$. The desired isomorphism then
follows from the first part of Theorem 8.1.

It remains to verify the stronger statement: $\langle
c_{1}(\widehat{\mathfrak{s}}),[A_{i}]\rangle=0$. In the proof of Theorem 8.1,
we composed $\mathbb{Y}$ with a mapping torus over a connected surface
$F_{1}$. We now make $F_{1}$ disconnected: take $F_{1}=(-F)$ and
$Y_{1}=(-F)\times S^{1}$. When $Y(Z_{0},L)$ are glued with $Y_{1}$, we require
that

  * •

$\partial N(L_{i}\cup m_{i})\subset\partial Y(Z_{0},L)$ is identified with
$\partial(-A_{i})\times S^{1}\subset\partial Y_{1}$;

  * •

$\partial A_{i}\subset\partial Y(Z_{0},L)$ is identified with
$\partial(-A_{i})\times\\{pt\\}\subset\partial Y_{1}$.

The closed 3-manifold $\mathbb{Y}\circ_{h}\mathbb{Y}_{1}$ contains a
collection of 2-tori $\mathbb{T}^{2}_{i}$, which are doubles of $A_{i}$’s. Now
we use the adjunction inequality in Proposition 6.2 and Lemma 5.3 to conclude.
∎

###  9\. Fiberness Detection

####  9.1. Some Preparations

In this section, we complete the proof of Theorem 7.3. We start with a
preliminary result:

######  Lemma 9.1.

Under the assumption of Theorem 7.3, if $\kappa\in H_{2}(Y,\partial
Y;\mathbb{Z})$ is primitive, then $\kappa$ can be represented by a connected
Thurston norm minimizing surface $F$ that intersects each component of
$\Sigma$ non-trivially. Moreover,
$\varphi_{\mathbb{Y}}(\kappa)=\varphi_{\mathbb{Y}}(-\kappa)$.

###### Proof.

This lemma follows from [McM02, Theorem 4.1 & Proposition 6.1]. Let

$\phi\in H^{1}(Y;\mathbb{Z})=\operatorname{Hom}(\pi_{1}(Y),\mathbb{Z})$

be the Poincaré dual of $\kappa$ and $b_{1}(\ker\phi)$ be the first Betti
number of the subgroup $\ker\phi\subset\pi_{1}(Y)$. Then [McM02, Proposition
6.1] states that $\kappa$ can be represented by such a norm-minimizing surface
$F$ if

(9.1) $b_{1}(\ker\phi)<\infty.$

To verify that $F$ intersects each component of $\Sigma$ non-trivially, one
has to go through the proof of [McM02, Proposition 6.1]. The condition (9.1)
will follow from [McM02, Theorem 4.1], if we can verify its assumptions.
Consider the set of Alexander classes

$\Delta(Y)\colonequals\\{c_{1}(\widehat{\mathfrak{s}}):\chi(\operatorname{\it
HM}_{*}(\mathbb{Y},\widehat{\mathfrak{s}}))\neq 0\\}\subset H^{2}(Y,\partial
Y;\mathbb{Z}).$

By Theorem 3.4, $\Delta(Y)$ is precisely the support of the Alexander
polynomial of $Y$ and is symmetric about the origin. Since

$\operatorname{rank}_{\mathcal{R}}\operatorname{\it
HM}_{*}(\mathbb{Y}|\kappa)=\operatorname{rank}_{\mathcal{R}}\operatorname{\it
HM}_{*}(\mathbb{Y}|-\kappa)=1,$

we conclude that $\Delta(Y)\neq\emptyset$ and
$\varphi_{\mathbb{Y}}(\kappa)=\varphi_{\mathbb{Y}}(-\kappa)$. Moreover, the
maximum

$\max_{a,b\in\Delta(Y)}\phi(a-b)$

is achieved for exactly one pair of elements $(a,b)$. The other assumption of
[McM02, Theorem 4.1] is certified by [McM02, Theorem 5.1]. This completes the
proof of Lemma 9.1. ∎

####  9.2. Proof of Theorem 7.3

The proof of Theorem 7.3 is based on the fiberness detection result for
balanced sutured manifolds, adapted to the case of mod 2 Novikov ring
$\mathcal{R}$:

######  Theorem 9.2 ([KM10, Theorem 6.1]).

Suppose that a balanced sutured manifold $(M,\gamma)$ is taut and a homology
product. Then $(M,\gamma)$ is a product sutured manifold if and only if
$\operatorname{\it SHM}(M,\gamma)\cong\mathcal{R}$.

For the proof of Theorem 7.3, it suffices to deal with the case when $\kappa$
is primitive. Let $F$ be the surface given by Lemma 9.1. We will address the
cases when $\chi(F)<0$ and when $\chi(F)=0$ separately.

###### Proof of Theorem 7.3 when $\chi(F)<0$.

Since $Y$ is irreducible, by Theorem 7.3 and Lemma 9.1, we have

$\varphi_{\mathbb{Y}}(\kappa)=\varphi_{\mathbb{Y}}(-\kappa)=x(F).$

Let $(\tilde{Y},\tilde{F})$ be the double of $(Y,F)$, then $g(\tilde{F})\geq
2$. Following the notations in the proof of Theorem 7.3, we have

(9.2) $\operatorname{\it
HM}(\tilde{Y},[\tilde{\omega}]|\tilde{F})=\operatorname{\it
HM}_{*}(\mathbb{Y}|\kappa)\otimes_{\mathcal{R}}\operatorname{\it
HM}_{*}(-\mathbb{Y}^{\prime}|-\kappa)\cong\mathcal{R}.$

The rest of the proof is divided into six steps.

Step 1\. We can effectively change the 2-form $\tilde{\omega}$ so that
$\langle[\tilde{\omega}],[\tilde{F}]\rangle=0$. This is Step 1 in the proof of
Lemma 8.2.

Step 2\. Let $\tilde{M}$ be the 3-manifold with boundary obtained by cutting
$\tilde{Y}$ along $\tilde{F}$. Write
$\partial\tilde{M}=\tilde{F}_{+}\cup\tilde{F}_{-}$. Then
$(\tilde{M},\tilde{F}_{+})$ is a homology product, i.e.
$H_{*}(\tilde{F}_{+};\mathbb{Z})\to H_{*}(\tilde{M};\mathbb{Z})$ is an
isomorphism. This follows from the fact [Tur98, Theorem 1] that for the graded
Euler characteristic of $\operatorname{\it
HM}_{*}(\tilde{Y},[\tilde{\omega}])$ recovers the Minor-Turaev torsion
invariant $T(\tilde{Y})$. Then we apply [Ni09a, Proposition 3.1] and (9.2).

Step 3\. $\operatorname{\it HM}(\tilde{Y}|\tilde{F})\cong\mathcal{R}$.

Let $N\colonequals[-1,1]\times\tilde{F}\subset\tilde{Y}$ be a tubular
neighborhood of $\tilde{F}$. Then $\tilde{Y}\setminus N\cong\tilde{M}$. We
claim that $[\tilde{\omega}]$ is represented by a 2-form $\tilde{\omega}_{1}$
supported in $[-1,1]\times\tilde{F}$. This follows from Step 1, Step 2 and a
diagram-chasing:

${H^{2}(\tilde{Y},\\{1\\}\times\tilde{F})}$${H^{2}(\tilde{Y})}$${H^{2}(\tilde{F})}$${H^{2}(\tilde{Y},\tilde{M})}$${H^{2}(N,\partial
N).}$$\scriptstyle{[\tilde{\omega}]\mapsto
0}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$

Given such a 2-form $\tilde{\omega}_{1}$, we use Floer’s excision theorem as
in the proof of Theorem 8.1 to deduce that $\operatorname{\it
HM}(\tilde{Y}|\tilde{F})\cong\operatorname{\it
HM}(\tilde{Y},[\tilde{\omega}]|\tilde{F})\cong\mathcal{R}$.

Step 4\. Let $(M(Y,F),\gamma)$ be the balanced sutured manifold obtained by
cutting $Y$ along $F$. Then $M(Y,F)$ is a homology product.

Note that $\tilde{M}$ is the double of $M(Y,F)$ along the annuli $A(\gamma)$.
Write $\tilde{M}=M_{1}\cup M_{2}$ and $F_{i}\colonequals
M_{i}\cap\tilde{F}_{+}$ for $i=1,2$. Then the statement follows by examining
the long exact sequences:

${\cdots}$${H_{*}(F_{1}\cap F_{2})}$${H_{*}(F_{1})\oplus
H_{*}(F_{2})}$${H_{*}(\tilde{F}_{+})}$${\cdots}$${\cdots}$${H_{*}(M_{1}\cap
M_{2})}$${H_{*}(M_{1})\oplus
H_{*}(M_{2})}$${H_{*}(\tilde{M})}$${\cdots}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$

By Step 3 and the five lemma, the middle vertical map is also an isomorphism.

Step 5\. If properties (P5)(P6) hold for $(\mathbf{\Sigma},\mathbb{Y},F)$,
then Theorem 7.3 holds.

In this special case, we can use Theorem 8.1 and Lemma 9.1 to obtain that

$\operatorname{\it SHM}(M(Y,F),\gamma)\cong\operatorname{\it
HM}(\mathbb{Y}|[F])\cong\mathcal{R}.$

Since $Y$ is irreducible and $F$ is norm-minimizing, $(M(Y,F),\gamma)$ is
taut. By Step 4 and Theorem 9.2, $M(Y,F)$ is a product sutured manifold.

Step 6\. Reduce the general case to Step 5.

Let $\mathbf{\Sigma}_{2}$ be another $T$-surface with the same underlying
oriented surface as $\mathbf{\Sigma}$ and
$\mathbb{Y}_{2}\in\mathbf{T}(\emptyset,\mathbf{\Sigma}_{2})$ has the same
underlying 3-manifold as $\mathbb{Y}$. We require that properties (P5)(P6)
hold for $(\mathbf{\Sigma}_{2},\mathbb{Y}_{2},F)$. The goal is to show that
$\operatorname{\it HM}(\mathbb{Y}_{2}|[F])\cong\mathcal{R}$. Let

$(\tilde{Y},\tilde{\omega}_{2},\cdots)=\mathbb{Y}_{2}\circ_{h}(-\mathbb{Y}_{2}^{\prime}).$

By Step 1, we may assume that $\langle[\tilde{\omega}_{2}],[F]\rangle=0$. By
Step 3, we have

$\mathcal{R}\cong\operatorname{\it
HM}(\tilde{Y},[\tilde{\omega}]|\tilde{F})\cong\operatorname{\it
HM}(\tilde{Y}|\tilde{F})\cong\operatorname{\it
HM}(\tilde{Y},[\tilde{\omega}_{2}]|\tilde{F}).$

By (9.2), $\operatorname{\it HM}(\mathbb{Y}_{2}|[F])\cong\mathcal{R}$. Now we
use Step 5 to complete the proof of Theorem 7.3 when $\chi(F)<0$. ∎

###### Proof of Theorem 7.3 when $\chi(F)=0$.

In this case, $F$ is an annulus, $\Sigma$ has 2-components and the double
$\tilde{F}$ is a 2-torus. By Lemma 9.1,
$\varphi_{\mathbb{Y}}(\kappa)=\varphi_{\mathbb{Y}}(-\kappa)=0$. Our
assumptions then imply that $\operatorname{\it
HM}_{*}(\mathbb{Y})\cong\mathcal{R}$.

The proof when $\chi(F)<0$ carries over with no essential changes. Let us
explain where the differences arise:

  * •

In Step 1, we require instead that $\langle i[\tilde{\omega}],[F]\rangle=a$
for some fixed $a\in\mathbb{R}$ with $a\neq 0$ and $|a|<2\pi$;

  * •

In Step 2, Ni’s result [Ni09a, Proposition 3.1b] is stated for a closed
surface with genus $\geq 2$; but its proof in [Ni09a, Section 3.3] relies only
on the property of the Milnor-Turaev torsion invariant $T(\tilde{Y})$ (note
also that $b_{1}(\tilde{Y})\geq 3$). Thus we can still conclude from (9.2)
that $(\tilde{M},\tilde{F}_{+})$ is a homology product;

  * •

Step 3 is replaced by the isomorphism

$\operatorname{\it
HM}(\tilde{Y},[\tilde{\omega}]|\tilde{F})\cong\operatorname{\it
HM}(\tilde{Y},[\tilde{\omega}_{2}]|\tilde{F})$

for any classes $[\tilde{\omega}]=[\tilde{\omega}_{2}]\in
H^{2}(\tilde{Y},i\mathbb{R})$ with $\langle
i[\tilde{\omega}],[F]\rangle=\langle i[\tilde{\omega}_{2}],[F]\rangle=a$. Then
the difference $[\tilde{\omega}]-[\tilde{\omega}_{2}]$ is represented by a
2-form $\tilde{\omega}_{1}$ supported in the neighborhood
$N=[-1,1]\times\tilde{F}$. As in the proof of Gluing Theorem 3.6, one may
adapt Floer’s excision theorem [KM10, Theorem 3.1] to the case of a 2-torus
using non-exact perturbations,

The rest of proof can now proceed with no difficulty. The conclusion says that
$Y$ is a mapping torus over an annulus; so $Y=[-3,3]_{s}\times\mathbb{T}^{2}$
is in fact a cylinder. ∎

As an immediate corollary of Theorem 7.3, we have

######  Corollary 9.3.

For any $T$-surface $\mathbf{\Sigma}=(\Sigma,g_{\Sigma},\lambda,\mu)$ and any
1-cell $\mathbb{Y}\in\mathbf{T}(\emptyset,\mathbf{\Sigma})$, if $Y$ is
connected, irreducible and $\operatorname{\it
HM}_{*}(\mathbb{Y},\widehat{\mathfrak{s}})\cong\mathcal{R}$, then
$Y=[-1,1]_{s}\times\mathbb{T}^{2}$.

###### Proof of Corollary 9.3.

Let $\kappa\in H_{2}(Y,\partial Y;\mathbb{Z})$ be any primitive class, i.e.,
$\kappa$ is not divisible by any other integral classes non-trivially. By the
symmetry of the graded Euler characteristic $\operatorname{\underline{SW}}(Y)$
in Theorem 3.4, the conditions of Theorem 7.3 are verified with
$\varphi_{\mathbb{Y}}(\kappa)=\varphi_{\mathbb{Y}}(-\kappa)=0$. By Theorem
7.3, $Y$ is a mapping torus over an annulus; so
$Y=[-1,1]_{s}\times\mathbb{T}^{2}$ is a product. ∎

###  10\. Connected Sum Formulae

Having discussed irreducible 3-manifolds with toroidal boundary, we focus on
reducible 3-manifolds in this section and derive the connected sum formulae.

####  10.1. Connected Sums with 3-Manifolds with Toroidal Boundary

For $i=1,2$, let $\mathbf{\Sigma}_{i}$ be any $T$-surface and
$\mathbb{Y}_{i}\in\mathbf{T}(\emptyset,\mathbf{\Sigma}_{i})$ be any 1-cell.
Then we can form an 1-cell

$\mathbb{Y}_{3}=(Y_{3},\omega_{3},\cdots)\in\mathbf{T}(\emptyset,\mathbf{\Sigma}_{1}\cup\mathbf{\Sigma}_{2})$

with the following properties:

  1. (C1)

the underlying 3-manifold $Y_{3}$ is $Y_{1}\\#Y_{2}$; let $S^{2}\subset Y_{3}$
be the 2-sphere separating $Y_{1}$ and $Y_{2}$;

  2. (C2)

$[\omega_{3}]\in H^{2}(Y_{3};i\mathbb{R})$ is determined by the relations:
$k_{i}([\omega_{3}])=l_{i}([\omega_{i}])$ for $i=1,2$ in the digram below. As
a result, $\langle[\omega_{3}],[S^{2}]\rangle=0$.

${0}$${H^{2}(Y_{1}\\#Y_{2};i\mathbb{R})}$${H^{2}(Y_{1}\setminus
B^{3}_{1};i\mathbb{R})\oplus H^{2}(Y_{2}\setminus
B^{3}_{2};i\mathbb{R})}$${H^{2}(S^{2};i\mathbb{R})}$${([\omega_{1}],[\omega_{2}])\in
H^{2}(Y_{1};i\mathbb{R})\oplus
H^{2}(Y_{2};i\mathbb{R}).}$$\scriptstyle{(k_{1},k_{2})}$$\scriptstyle{l_{1}\oplus
l_{2}}$

Although $\mathbb{Y}_{3}$ is not uniquely determined by these properties, we
say that $\mathbb{Y}_{3}$ is a connected sum of $\mathbb{Y}_{1}$ and
$\mathbb{Y}_{2}$. By Theorem 5.2, the isomorphism type of $\operatorname{\it
HM}_{*}(\mathbb{Y}_{3})$ is determined by (C1)(C2).

######  Proposition 10.1.

The monopole Floer homology of $\mathbb{Y}_{3}$ can be computed as follows:

$\operatorname{\it HM}_{*}(\mathbb{Y}_{3})\cong\operatorname{\it
HM}_{*}(\mathbb{Y}_{1})\otimes_{\mathcal{R}}\operatorname{\it
HM}_{*}(\mathbb{Y}_{2})\otimes_{\mathcal{R}}V,$

where $V$ is a 2-dimensional vector space over $\mathcal{R}$.

###### Proof.

By Theorem 5.2, we can compute the group $\operatorname{\it
HM}_{*}(\mathbb{Y}_{3})$ using a convenient metric on $Y_{1}\\#Y_{2}$ and a
2-form $\omega_{3}$. Take a component $\Sigma_{i}^{\prime}\subset\Sigma_{i}$
for each $i=1,2$. Consider the 3-manifold

$([-3,3]_{s}\times\Sigma_{i}^{\prime},\
\mu_{i}|_{\Sigma_{i}^{\prime}}+ds\wedge\lambda_{i}|_{\Sigma_{i}^{\prime}}),i=1,2.$

Let
$\mathbb{Y}_{4}=(Y_{4},\omega_{4},\cdots)\in\mathbf{T}(\mathbf{\Sigma}_{1}^{\prime}\cup\mathbf{\Sigma}_{2}^{\prime},\mathbf{\Sigma}_{1}^{\prime}\cup\mathbf{\Sigma}_{2}^{\prime})$
be their connected sum. The 1-cell $\mathbb{Y}_{3}$ can be then obtained as
the horizontal composition

$(\mathbb{Y}_{1}\coprod\mathbb{Y}_{2})\circ_{h}\mathbb{Y}_{4}.$

Using the Gluing Theorem 3.6, we obtain that

$\operatorname{\it HM}_{*}(\mathbb{Y}_{3})\cong\operatorname{\it
HM}_{*}(\mathbb{Y}_{1})\otimes_{\mathcal{R}}\operatorname{\it
HM}_{*}(\mathbb{Y}_{2})\otimes_{\mathcal{R}}\operatorname{\it
HM}_{*}(\mathbb{Y}_{4}).$

It remains to verify that $\operatorname{rank}_{\mathcal{R}}\operatorname{\it
HM}_{*}(\mathbb{Y}_{4})=2$. Regard $\mathbb{Y}_{4}$ as a 1-cell in
$\mathbf{T}(\emptyset,(-\mathbf{\Sigma}_{1}^{\prime})\cup(-\mathbf{\Sigma}_{2}^{\prime})\cup\mathbf{\Sigma}_{1}^{\prime}\cup\mathbf{\Sigma}_{2}^{\prime})$
and consider the horizontal composition with

$e_{\mathbf{\Sigma}_{1}^{\prime}}\in\mathbf{T}((-\mathbf{\Sigma}_{1}^{\prime})\cup\mathbf{\Sigma}_{1}^{\prime},\emptyset)\text{
and
}e_{\mathbf{\Sigma}_{2}^{\prime}}\in\mathbf{T}((-\mathbf{\Sigma}_{2}^{\prime})\cup\mathbf{\Sigma}_{2}^{\prime},\emptyset).$

Another application of Theorem 3.6 implies that

$\operatorname{\it HM}_{*}(\mathbb{Y}_{4})\cong\operatorname{\it
HM}_{*}((\Sigma^{\prime}_{1}\times S^{1})\\#(\Sigma^{\prime}_{2}\times
S^{1}),[\omega_{1}^{\prime}]\\#[\omega_{2}^{\prime}]),$

where
$\omega_{i}^{\prime}=\mu_{i}|_{\Sigma_{i}^{\prime}}+d\theta\wedge\lambda_{i}|_{\Sigma_{i}^{\prime}}$
and $\theta$ denotes the coordinate of $S^{1}$.
$[\omega_{1}^{\prime}]\\#[\omega_{2}^{\prime}]$ is obtained from
$[\omega_{1}^{\prime}]$ and $[\omega_{2}^{\prime}]$ using (C2). Here we have
used the shorthands from Definition 6.4. By Lemma 5.4, we have already known
that

$\operatorname{\it HM}_{*}(\Sigma_{i}^{\prime}\times
S^{1},[\omega_{i}^{\prime}])\cong\mathcal{R}\text{ for }i=1,2.$

To conclude, we apply the connected sum formula [Lin17, Theorem 5] for closed
3-manifolds. As a result, the group $\operatorname{\it
HM}_{*}(\mathbb{Y}_{4})$ is concentrated in a single $spin^{c}\ $grading and
has rank $2$. ∎

####  10.2. Connected Sums with Closed 3-Manifolds

Let us first review the definition of $\widetilde{\operatorname{\it
HM}}_{*}(M)$ for any closed 3-manifold $M$.

######  Definition 10.2.

For any closed 3-manifold $Z$, we obtain a balanced sutured manifold
$(Z(1),\delta)$ by taking $Z(1)=Z\setminus B^{3}$ and the suture
$s(\delta)\subset\partial Z(1)$ to be the equator. Define

$\widetilde{\operatorname{\it HM}}(Z)\colonequals\operatorname{\it
SHM}(Z(1),\delta).\qed$

Let $\mathbf{\Sigma}$ be any $T$-surface,
$\mathbb{Y}\in\mathbf{T}(\emptyset,\mathbf{\Sigma})$ be any 1-cell and $Z$ be
any closed 3-manifold. Consider an 1-cell
$\mathbb{Y}_{5}=(Y_{5},\omega_{5},\cdots)\in\mathbf{T}(\emptyset,\mathbf{\Sigma})$
that satisfies the following properties:

  * •

the underlying 3-manifold $Y_{5}$ is $Y\\#Z$;

  * •

$[\omega_{5}]\in H^{2}(Y_{5};i\mathbb{R})$ is obtained from $[\omega]$ using
(C2) and the zero form on $Z$.

######  Proposition 10.3.

The monopole Floer homology of $\mathbb{Y}_{5}$ can be computed as follows:

$\operatorname{\it HM}_{*}(\mathbb{Y}_{5})\cong\operatorname{\it
HM}_{*}(\mathbb{Y})\otimes_{\mathcal{R}}\widetilde{\operatorname{\it HM}}(Z).$

###### Proof.

Following the proof of Proposition 10.1, it suffices to verify that

(10.1) $\widetilde{\operatorname{\it HM}}(M)\cong\operatorname{\it
HM}_{*}(M\\#(\Sigma^{\prime}\times S^{1}),[0]\\#[\omega^{\prime}])$

where $\Sigma^{\prime}$ is a connected component of $\Sigma$ and
$\omega^{\prime}\colonequals\mu|_{\Sigma^{\prime}}+d\theta\wedge\lambda|_{\Sigma^{\prime}}$.
One can argue as in Step 1 of the proof of Lemma 8.2 and work instead with the
2-form

$\omega^{\prime\prime}\equiv\mu|_{\Sigma^{\prime}}\text{ on
}S^{1}\times\Sigma^{\prime}.$

Then the Poincaré dual of $[\omega^{\prime\prime}]$ is proportional to the
1-cycle $\\{pt\\}\times S^{1}$. Since $[0]\\#[\omega^{\prime\prime}]$ is
neither balanced nor negatively monotone for any $spin^{c}\ $structure on
$M\\#(S^{1}\times\Sigma^{\prime})$, one may apply Proposition 6.2 to compute
the right hand side of (10.1) using exact perturbations. Now the isomorphism
(10.1) follows from [KM10, Proposition 4.6]. ∎

### References

  * [BD95] P. J. Braam and S. K. Donaldson.  Floer’s work on instanton homology, knots and surgery.  In The Floer memorial volume, volume 133 of Progr. Math., pages 195–256. Birkhäuser, Basel, 1995.
  * [Ben67] Jean Benabou.  Introduction to bicategories.  In Reports of the Midwest Category Seminar, pages 1–77. Springer, Berlin, 1967.
  * [EE67] C. J. Earle and J. Eells.  The diffeomorphism group of a compact Riemann surface.  Bull. Amer. Math. Soc., 73:557–559, 1967.
  * [Eis95] David Eisenbud.  Commutative algebra, volume 150 of Graduate Texts in Mathematics.  Springer-Verlag, New York, 1995.  With a view toward algebraic geometry.
  * [Gab83] David Gabai.  Foliations and the topology of $3$-manifolds.  J. Differential Geom., 18(3):445–503, 1983.
  * [Ghi08] Paolo Ghiggini.  Knot Floer homology detects genus-one fibred knots.  Amer. J. Math., 130(5):1151–1169, 2008.
  * [GL19] Sudipta Ghosh and Zhenkun Li.  Decomposing sutured monopole and instanton floer homologies, 2019.
  * [Hat02] Allen Hatcher.  Algebraic topology.  Cambridge University Press, Cambridge, 2002.
  * [Juh06] András Juhász.  Holomorphic discs and sutured manifolds.  Algebr. Geom. Topol., 6:1429–1457, 2006.
  * [Juh08] András Juhász.  Floer homology and surface decompositions.  Geom. Topol., 12(1):299–350, 2008.
  * [KM97] P. B. Kronheimer and T. S. Mrowka.  Scalar curvature and the Thurston norm.  Math. Res. Lett., 4(6):931–937, 1997.
  * [KM07] Peter Kronheimer and Tomasz Mrowka.  Monopoles and three-manifolds, volume 10 of New Mathematical Monographs.  Cambridge University Press, Cambridge, 2007.
  * [KM10] Peter Kronheimer and Tomasz Mrowka.  Knots, sutures, and excision.  J. Differential Geom., 84(2):301–364, 2010.
  * [Li18] Zhenkun Li.  Contact structures, excisions, and sutured monopole floer homology, 2018\.
  * [Lin17] Francesco Lin.  ${\rm Pin}(2)$-monopole Floer homology, higher compositions and connected sums.  J. Topol., 10(4):921–969, 2017.
  * [Man16] Ciprian Manolescu.  An introduction to knot Floer homology.  In Physics and mathematics of link homology, volume 680 of Contemp. Math., pages 99–135. Amer. Math. Soc., Providence, RI, 2016.
  * [McM02] Curtis T. McMullen.  The Alexander polynomial of a 3-manifold and the Thurston norm on cohomology.  Ann. Sci. École Norm. Sup. (4), 35(2):153–171, 2002.
  * [MT96] Guowu Meng and Clifford Henry Taubes.  $\underline{\rm SW}=$ Milnor torsion.  Math. Res. Lett., 3(5):661–674, 1996.
  * [Ni07] Yi Ni.  Knot Floer homology detects fibred knots.  Invent. Math., 170(3):577–608, 2007.
  * [Ni08] Yi Ni.  Addendum to: ”knots, sutures and excision”, 2008.
  * [Ni09a] Yi Ni.  Heegaard Floer homology and fibred 3-manifolds.  Amer. J. Math., 131(4):1047–1063, 2009.
  * [Ni09b] Yi Ni.  Link Floer homology detects the Thurston norm.  Geom. Topol., 13(5):2991–3019, 2009.
  * [OS04] Peter Ozsváth and Zoltán Szabó.  Holomorphic disks and knot invariants.  Adv. Math., 186(1):58–116, 2004.
  * [OS08] Peter Ozsváth and Zoltán Szabó.  Link Floer homology and the Thurston norm.  J. Amer. Math. Soc., 21(3):671–709, 2008.
  * [Ras03] Jacob Rasmussen.  Floer homology and knot complements.  arXiv:math/0306378, 2003.
  * [Sei08] Paul Seidel.  Fukaya categories and Picard-Lefschetz theory.  Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich, 2008.
  * [Siv12] Steven Sivek.  Monopole Floer homology and Legendrian knots.  Geom. Topol., 16(2):751–779, 2012.
  * [Tau01] Clifford Henry Taubes.  The Seiberg-Witten invariants and 4-manifolds with essential tori.  Geom. Topol., 5:441–519, 2001.
  * [Thu86] William P. Thurston.  A norm for the homology of $3$-manifolds.  Mem. Amer. Math. Soc., 59(339):i–vi and 99–130, 1986.
  * [Tur98] Vladimir Turaev.  A combinatorial formulation for the Seiberg-Witten invariants of $3$-manifolds.  Math. Res. Lett., 5(5):583–598, 1998.
  * [Wan20a] Donghao Wang.  Monopoles and landau-ginzburg models i.  arXiv:2004.06227, 2020.
  * [Wan20b] Donghao Wang.  Monopoles and landau-ginzburg models ii: Floer homology.  arXiv:2005.04333, 2020.