Link Floer homology also detects split links

Joshua Wang

Abstract

Inspired by work of Lipshitz-Sarkar, we show that the module structure on link Floer homology detects split links. Using results of Ni, Alishahi-Lipshitz, and Lipshitz-Sarkar, we establish an analogous detection result for sutured Floer homology. ^†^†2010 Mathematics Subject Classification 57M27

1 Introduction

A remarkable feature of modern homology theories in low-dimensional topology is their ability to detect many topological properties of interest. We refer to the introduction of [LS19] for a list of such detection results in Khovanov homology and Heegaard Floer homology. The main theorem of [LS19] is an additional detection result for Khovanov homology: that the module structure on Khovanov homology detects split links. In this short paper, we add one more item to the list: that the analogous module structure on link Floer homology also detects split links.

If $L$ is a two-component link in $S^{3}$ , then its link Floer homology $\smash{\widehat{\mathrm{HFL}}}(L)$ [OS08a], which takes the form of a finite-dimensional vector space over $\mathbf{F}_{2}=\mathbf{Z}/2$ , is naturally equipped with an endomorphism $X$ satisfying $X\circ X=0$ . Such an endomorphism gives $\smash{\widehat{\mathrm{HFL}}}(L)$ the structure of a module over $\mathbf{F}_{2}[X]/X^{2}$ . The map $X$ is defined to be the homological action [Ni14] of a generator of the first relative homology group of the exterior of $L$ . We review the definition of this action in Section 2.

Theorem 1.1.

Let $L$ be a two-component link in $S^{3}$ . Then $L$ is split if and only if $\smash{\widehat{\mathrm{HFL}}}(L)$ is a free $\mathbf{F}_{2}[X]/X^{2}$ -module.

Remark 1.2.

Before the work of this paper began, Tye Lidman observed and informed the author that link Floer homology should detect split links in this way using results of [Ni13] and arguments similar to those in [LS19]. The proof of Theorem 1.1 appearing in this short paper is due to the author and is independent of the results and arguments in [Ni13] and [LS19]. In particular, it uses sutured manifold hierarchies and does not require citing any deep results in symplectic geometry. However, the proof of Theorem 1.4 appearing here does use such a citation.

Remark 1.3.

Link Floer homology detects the Thurston norm of its exterior [OS08b, Theorem 1.1], [Ni09b, Theorem 1.1]. This by itself does not imply that link Floer homology detects split links. For example, the exterior of the Whitehead link has the same Thurston norm as the exterior of a split union of two genus $1$ knots.

More generally, there is a homological action $X_{\zeta}$ [Ni14] on the sutured Floer homology $\operatorname{SFH}(M,\gamma)$ [Juh06] of a balanced sutured manifold $(M,\gamma)$ satisfying $X_{\zeta}\circ X_{\zeta}=0$ for each $\zeta\in H_{1}(M,\partial M)$ . We prove the following generalization of Theorem 1.1 using a result of Lipshitz-Sarkar [LS19], which builds on results of Alishahi-Lipshitz [AL19] and Ni [Ni13].

Theorem 1.4.

Let $(M,\gamma)$ be a balanced sutured manifold, let $\zeta\in H_{1}(M,\partial M)$ , and assume that $\operatorname{SFH}(M,\gamma)\neq 0$ . Then $\operatorname{SFH}(M,\gamma)$ is a free $\mathbf{F}_{2}[X]/X^{2}$ -module with respect to the homological action of $\zeta$ if and only if there is an embedded $2$ -sphere $S$ in $M$ for which the algebraic intersection number $S\cdot\zeta$ is odd.

Remark 1.5.

Let $L$ be a link in $S^{3}$ with at least two components, and let $C_{0},C_{1}$ be two distinct components of $L$ . Let $S^{3}(L)$ denote the sutured exterior of $L$ , and let $\zeta\in H_{1}(S^{3}(L),\partial S^{3}(L))$ be the relative homology class of a path from $C_{0}$ and $C_{1}$ in $S^{3}(L)$ . The sutured Floer homology of $S^{3}(L)$ can be identified with $\smash{\widehat{\mathrm{HFL}}}(L)$ by [Juh06, Proposition 9.2]. By Theorem 1.4, $\smash{\widehat{\mathrm{HFL}}}(L)$ is a free $\mathbf{F}_{2}[X]/X^{2}$ -module with respect to the homological action of $\zeta$ if and only if there exists an embedded $2$ -sphere in the complement of $L$ which separates $C_{0}$ from $C_{1}$ .

Homological actions on Heegaard Floer homology for closed oriented $3$ -manifolds were originally defined in [OS04, Section 4.2.5]. These homological actions on Heegaard Floer homology and the closely-related construction of twisted Heegaard Floer homology are well-studied. See [Ni09a, HN10, Ni13, HN13, AL19, LS19, HL20] for further connections to non-separating spheres and to the module structure on Khovanov homology.

Acknowledgments.

I thank Tye Lidman, Robert Lipshitz, and Maggie Miller for helpful discussions. I also thank my advisor Peter Kronheimer for his continued guidance, support, and encouragement. This material is based upon work supported by the NSF GRFP through grant DGE-1745303.

2 Preliminaries

See [Gab83, Gab87, Juh06, Juh08, Juh10] for the definitions of balanced sutured manifolds, nice surface decompositions, and sutured Floer homology. In this paper, we use sutured Floer homology with coefficients in the field $\mathbf{F}_{2}=\mathbf{Z}/2$ . We first provide some examples of sutured manifolds which also serve to set notation.

Examples.

A product sutured manifold is a sutured manifold of the form $([-1,1]\times\Sigma,[-1,1]\times\partial\Sigma)$ where $\Sigma$ is a compact oriented surface. It is balanced if $\Sigma$ has no closed components.

Let $L$ be a link in a closed oriented connected $3$ -manifold $Y$ . The sutured exterior $Y(L)$ of the link is the balanced sutured manifold obtained from $Y$ by deleting a regular neighborhood of $L$ and adding two oppositely oriented meridional sutures on each boundary component.

If $Y$ is a closed oriented connected $3$ -manifold, then let $Y(n)$ be the balanced sutured manifold obtained by deleting $n$ disjoint open balls from $Y$ and adding a suture to each boundary component. We similarly define $(M,\gamma)(n)$ when $(M,\gamma)$ is a connected balanced sutured manifold.

Definition ([Gab83, Definition 2.10]).

A sutured manifold $(M,\gamma)$ is taut if $M$ is irreducible and $R(\gamma)$ is norm-minimizing in $H_{2}(M,\gamma)$ .

Remark 2.1.

Balanced product sutured manifolds are taut. If $L$ is a two-component link in $S^{3}$ , then $S^{3}(L)$ is taut if and only if $L$ is not split. Except for $S^{3}(1)$ , any balanced sutured manifold of the form $Y(n)$ or $(M,\gamma)(n)$ for $n\geq 1$ is not irreducible and therefore is not taut.

Theorem 2.2 ([Gab83, Theorem 4.2], [Juh08, Theorem 8.2]).

If $(M,\gamma)$ is a taut balanced sutured manifold, then there is a sequence of nice surface decompositions

(M,\gamma)\overset{S_{1}}{\rightsquigarrow}(M_{1},\gamma_{1})\overset{S_{2}}{\rightsquigarrow}\cdots\overset{S_{n}}{\rightsquigarrow}(M_{n},\gamma_{n})

where $(M_{n},\gamma_{n})$ is a balanced product sutured manifold.

Let $(M,\gamma)$ be a balanced sutured manifold. We review Ni’s definition of the homological action of a relative homology class $\zeta\in H_{1}(M,\partial M)$ on $\operatorname{SFH}(M,\gamma)$ [Ni14, Section 2.1]. These actions are an extension of the homological actions on Heegaard Floer homology defined in [OS04, Section 4.2.5]. We then recall Ni’s result that these homological actions are compatible with nice surface decompositions [Ni14, Theorem 1.1].

Definition ((Homological actions on sutured Floer homology)).

Let $(\Sigma,\boldsymbol{\alpha},\boldsymbol{\beta})$ be an admissible balanced diagram for $(M,\gamma)$ , and let $\omega=\sum k_{i}c_{i}$ be a formal finite sum of properly embedded oriented curves $c_{i}$ on $\Sigma$ with integer coefficients $k_{i}$ . Each $c_{i}$ is required to intersect the $\alpha$ - and $\beta$ -curves transversely and to be disjoint from every intersection point of the $\alpha$ - and $\beta$ -curves. Let $\zeta\in H_{1}(M,\partial M)$ denote the relative homology class that $\omega$ represents. Any relative first homology class of $M$ is represented by such a relative $1$ -cycle on $\Sigma$ .

Let $\operatorname{SFC}(\Sigma,\boldsymbol{\alpha},\boldsymbol{\beta})$ be the sutured Floer chain complex whose differential $\partial$ is defined with respect to a suitable family of almost complex structures. In particular

\partial\mathbf{x}=\sum_{\mathbf{y}\in\mathbf{T}_{\alpha}\cap\mathbf{T}_{\beta}}\sum_{\begin{subarray}{c}\phi\in\pi_{2}(\mathbf{x},\mathbf{y})\\ \mu(\phi)=1\end{subarray}}\#\widehat{\mathscr{M}}(\phi)\cdot\mathbf{y}

for each $\mathbf{x}\in\mathbf{T}_{\alpha}\cap\mathbf{T}_{\beta}$ . Any Whitney disc $\phi\in\pi_{2}(\mathbf{x},\mathbf{y})$ has an associated $2$ -chain on $\Sigma$ called its domain $D(\phi)$ . Let $\partial_{\alpha}D(\phi)$ be the part of $\partial D(\phi)$ lying in the $\alpha$ -circles, thought of as an oriented multi-arc from $\mathbf{x}$ to $\mathbf{y}$ . Define $X_{\zeta}\colon\operatorname{SFC}(\Sigma,\boldsymbol{\alpha},\boldsymbol{\beta})\to\operatorname{SFC}(\Sigma,\boldsymbol{\alpha},\boldsymbol{\beta})$ by

X_{\zeta}\cdot\mathbf{x}=\sum_{\mathbf{y}\in\mathbf{T}_{\alpha}\cap\mathbf{T}_{\beta}}\sum_{\begin{subarray}{c}\phi\in\pi_{2}(\mathbf{x},\mathbf{y})\\ \mu(\phi)=1\end{subarray}}(\omega\cdot\partial_{\alpha}D(\phi))\>\#\widehat{\mathscr{M}}(\phi)\cdot\mathbf{y}

where $\omega\cdot\partial_{\alpha}D(\phi)$ is the algebraic intersection number mod $2$ .

It is shown in [Ni14, OS04] that $X_{\zeta}$ is a chain map, and its induced map on homology, also denoted $X_{\zeta}$ , squares to zero. Furthermore, the map on homology is independent of the choice of $\omega$ representing $\zeta\in H_{1}(M,\partial M)$ and the choice of admissible Heegaard diagram. This induced map on $\operatorname{SFH}(M,\gamma)$ is called the homological action of $\zeta$ . Unless otherwise stated, $X_{\zeta}$ refers to the map on homology. Since $X_{\zeta}\circ X_{\zeta}=0$ , we may view $\operatorname{SFH}(M,\gamma)$ as a module over $\mathbf{F}_{2}[X]/X^{2}$ where the action of $X$ is $X_{\zeta}$ . Note that if $\zeta=2\zeta^{\prime}$ is an even homology class, then $X_{\zeta}=0$ . More generally, $X_{\zeta}$ is additive in $\zeta$ , which is to say that $X_{\zeta+\zeta^{\prime}}=X_{\zeta}+X_{\zeta^{\prime}}$ .

Example.

We will make use of the following direct computation. Let $\Sigma$ denote an annulus, and let $\boldsymbol{\alpha},\boldsymbol{\beta}$ be embedded essential curves which intersect transversely in two points. Then $(\Sigma,\boldsymbol{\alpha},\boldsymbol{\beta})$ is an admissible diagram for the balanced sutured manifold $S^{3}(2)$ . Then the two points in the intersection $\mathbf{T}_{\alpha}\cap\mathbf{T}_{\beta}$ can be labeled $\mathbf{x},\mathbf{y}$ so that there are two Whitney discs from $\mathbf{x}$ to $\mathbf{y}$ . Each has a unique holomorphic representative, and they cancel in the differential of $\operatorname{SFC}(\Sigma,\boldsymbol{\alpha},\boldsymbol{\beta})$ so $\dim_{\mathbf{F}_{2}}\operatorname{SFH}(S^{3}(2))=2$ . Let $\omega$ be an embedded oriented arc in $\Sigma$ whose endpoints lie on different boundary components and which intersects $\boldsymbol{\alpha}\cup\boldsymbol{\beta}$ transversely in exactly two points. Note that $\omega$ represents a generator $\zeta$ of $H_{1}(S^{3}(2),\partial S^{3}(2))$ . Then $X_{\zeta}\cdot\mathbf{x}=\mathbf{y}$ so $\operatorname{SFH}(S^{3}(2))=\mathbf{F}_{2}[X]/X^{2}$ as a module with respect to the action of $\zeta$ .

Essentially the same computation shows that $\smash{\widehat{\mathrm{HF}}}(S^{1}\times S^{2})=\mathbf{F}_{2}[X]/X^{2}$ as a module with respect to the action of a generator of $H_{1}(S^{1}\times S^{2})$ .

Remark 2.3.

Let $Y$ be a closed oriented connected $3$ -manifold. Then there is an identification $\smash{\widehat{\mathrm{HF}}}(Y)=\operatorname{SFH}(Y(1))$ [Juh06, Proposition 9.1]. The homological actions on $\smash{\widehat{\mathrm{HF}}}(Y)$ defined in [OS04] correspond to the homological actions on $\operatorname{SFH}(Y(1))$ using the natural identification $H_{1}(Y)=H_{1}(Y(1),\partial Y(1))$ .

Remark 2.4.

Let $(M,\gamma)$ and $(N,\beta)$ be balanced sutured manifolds, and let $\zeta\in H_{1}(M,\partial M)$ and $\xi\in H_{1}(N,\partial N)$ . Consider the disjoint union $(M\amalg N,\gamma\cup\beta)$ and the relative homology class

\zeta\oplus\xi\in H_{1}(M,\partial M)\oplus H_{1}(N,\partial N)=H_{1}(M\amalg N,\partial(M\amalg N)).

The homological action $X_{\zeta\oplus\xi}$ on $\operatorname{SFH}(M\amalg N,\gamma\cup\beta)=\operatorname{SFH}(M,\gamma)\otimes_{\mathbf{F}_{2}}\operatorname{SFH}(N,\beta)$ is given by

X_{\zeta\oplus\xi}(a\otimes b)=(X_{\zeta}a)\otimes b+a\otimes(X_{\xi}b).

Theorem 2.5 ([Ni14, Theorem 1.1]).

Let $(M,\gamma)\overset{S}{\rightsquigarrow}(M^{\prime},\gamma^{\prime})$ be a nice surface decomposition of balanced sutured manifolds. Let $i_{*}\colon H_{1}(M,\partial M)\to H_{1}(M,(\partial M)\cup S)\cong H_{1}(M^{\prime},\partial M^{\prime})$ be the map induced by the inclusion map $i\colon(M,\partial M)\to(M,(\partial M)\cup S)$ .

View $\operatorname{SFH}(M,\gamma)$ as an $\mathbf{F}_{2}[X]/X^{2}$ -module with respect to the homological action of some $\zeta\in H_{1}(M,\partial M)$ . View $\operatorname{SFH}(M^{\prime},\gamma^{\prime})$ as an $\mathbf{F}_{2}[X]/X^{2}$ -module with respect to the homological action of $i_{*}(\zeta)\in H_{1}(M^{\prime},\partial M^{\prime})$ . Then $\operatorname{SFH}(M^{\prime},\gamma^{\prime})$ is isomorphic as an $\mathbf{F}_{2}[X]/X^{2}$ -module to a direct summand of $\operatorname{SFH}(M,\gamma)$ as an $\mathbf{F}_{2}[X]/X^{2}$ -module.

The following lemma addresses the special case where the decomposing surface $S$ is a product disc. Recall that a decomposing surface $D$ in a sutured manifold $(M,\gamma)$ is called a product disc [Gab87, Definitions 0.1] if $D$ is a disc and $|D\cap s(\gamma)|=2$ .

Lemma 2.6 ([Juh06, Lemma 9.13]).

Let $(M,\gamma)$ be a balanced sutured manifold, let $(M,\gamma)\overset{D}{\rightsquigarrow}(M^{\prime},\gamma^{\prime})$ be a product disc decomposition, and let $\zeta\in H_{1}(M,\partial M)$ . Then there is an isomorphism $\operatorname{SFH}(M,\gamma)\cong\operatorname{SFH}(M^{\prime},\gamma^{\prime})$ of $\mathbf{F}_{2}[X]/X^{2}$ -modules where $X$ acts on $\operatorname{SFH}(M,\gamma)$ and $\operatorname{SFH}(M^{\prime},\gamma^{\prime})$ by the homological actions of $\zeta$ and $i_{*}(\zeta)$ , respectively.

Proof.

The lemma follows directly from the definition of the homological action and the proof of [Juh06, Lemma 9.13]. ∎

We will use the following two connected-sum formulas for the homological action on sutured Floer homology.

Lemma 2.7.

Let $(M,\gamma)$ be a balanced sutured manifold, and let $Y$ be a closed oriented $3$ -manifold. Fix $\zeta\in H_{1}(M\#Y,\partial(M\#Y))$ and write $\zeta=\zeta^{\prime}+\zeta^{\prime\prime}$ according to the decomposition $H_{1}(M\#Y,\partial(M\#Y))=H_{1}(M,\partial M)\oplus H_{1}(Y)$ . Then there is an isomorphism of $\mathbf{F}_{2}[X]/X^{2}$ -modules

\operatorname{SFH}(M\#Y,\gamma)\cong\operatorname{SFH}(M,\gamma)\otimes_{\mathbf{F}_{2}}\smash{\widehat{\mathrm{HF}}}(Y)

where $X$ acts on $\operatorname{SFH}(M\#Y,\gamma)$ by $X_{\zeta}$ while $X$ acts on the right-hand side by

X(a\otimes b)=(X_{\zeta^{\prime}}a)\otimes b+a\otimes(X_{\zeta^{\prime\prime}}b).

Proof.

As observed in [Juh06, Proposition 9.15], there is a product disc decomposition

(M\#Y,\gamma)\overset{D}{\rightsquigarrow}(M,\gamma)\amalg Y(1).

The result now follows from Remark 2.4 and Lemma 2.6. ∎

Lemma 2.8.

Let $(M,\gamma)$ be a connected sum of balanced sutured manifolds $(N_{1},\beta_{1})$ and $(N_{2},\beta_{2})$ , and let $\zeta\in H_{1}(M,\partial M)$ . Then there is an isomorphism of $\mathbf{F}_{2}[X]/X^{2}$ -modules

\operatorname{SFH}(M,\gamma)\cong\operatorname{SFH}(N_{1},\beta_{1})\otimes_{\mathbf{F}_{2}}\operatorname{SFH}(N_{2},\beta_{2})\otimes_{\mathbf{F}_{2}}\operatorname{SFH}(S^{3}(2))

where $X$ acts on $\operatorname{SFH}(M,\gamma)$ by the homological action of $\zeta$ , and $X$ acts on the right-hand side by

X(a\otimes b\otimes c)=(X_{\zeta_{1}}a)\otimes b\otimes c+a\otimes(X_{\zeta_{2}}b)\otimes c+a\otimes b\otimes(X_{\xi}c)

for certain classes $\zeta_{1}\in H_{1}(N_{1},\partial N_{1}),\zeta_{2}\in H_{1}(N_{2},\partial N_{2})$ , and $\xi\in H_{1}(S^{3}(2),\partial S^{3}(2))$ .

Let $S$ be the $2$ -sphere in $M$ along which the connected sum is formed. If $S\cdot\zeta$ is odd, then $\operatorname{SFH}(S^{3}(2))\cong\mathbf{F}_{2}[X]/X^{2}$ with respect to the homological action of $\xi$ . If $S\cdot\zeta$ is even, then $X_{\xi}=0$ .

Proof.

Again as observed in [Juh06, Proposition 9.15], there are product disc decompositions

(M,\gamma)\overset{D}{\rightsquigarrow}(N_{1},\beta_{1})\amalg(N_{2},\beta_{2})(1)\overset{D^{\prime}}{\rightsquigarrow}(N_{1},\beta_{1})\amalg(N_{2},\beta_{2})\amalg S^{3}(2)

Let $\zeta_{1}\oplus\zeta_{2}\oplus\xi$ be the image of $\zeta$ in

H_{1}(M_{1},\partial M_{1})\oplus H_{1}(M_{2},\partial M_{2})\oplus H_{1}(S^{3}(2),\partial(S^{3}(2))).

Note that $\xi$ is an even class if and only if $S\cdot\zeta$ is even. Furthermore, a direct computation gives an isomorphism $\operatorname{SFH}(S^{3}(2))\cong\mathbf{F}_{2}[X]/X^{2}$ as modules when $X$ is the homological action of a generator. The result now follows from Remark 2.4 and Lemma 2.6. ∎

Before turning to the main results, we record here an algebraic lemma.

Lemma 2.9.

Let $M_{1},\ldots,M_{k}$ be a collection of finitely-generated $\mathbf{F}_{2}[X]/X^{2}$ -modules, and view $M=M_{1}\otimes_{\mathbf{F}_{2}}\cdots\otimes_{\mathbf{F}_{2}}M_{k}$ as an $\mathbf{F}_{2}[X]/X^{2}$ -module where the action of $X$ is defined by

X(m_{1}\otimes\cdots\otimes m_{k})=\sum_{i=1}^{k}m_{1}\otimes\cdots\otimes m_{i-1}\otimes(Xm_{i})\otimes m_{i+1}\otimes\cdots\otimes m_{k}.

Then $M$ is free if and only if at least one of the $M_{i}$ is free.

Proof.

View $\mathbf{F}_{2}$ as an $\mathbf{F}_{2}[X]/X^{2}$ -module where $X$ acts by zero. Then any finitely-generated $\mathbf{F}_{2}[X]/X^{2}$ -module is isomorphic to $(\mathbf{F}_{2}[X]/X^{2})^{\oplus n}\oplus(\mathbf{F}_{2})^{\oplus m}$ for some nonnegative integers $n,m$ . The result follows from the computations that

\frac{\mathbf{F}_{2}[X]}{X^{2}}\otimes_{\mathbf{F}_{2}}\frac{\mathbf{F}_{2}[X]}{X^{2}}\qquad\text{ and }\qquad\frac{\mathbf{F}_{2}[X]}{X^{2}}\otimes_{\mathbf{F}_{2}}\mathbf{F}_{2}

are free while $\mathbf{F}_{2}\otimes_{\mathbf{F}_{2}}\mathbf{F}_{2}$ is not. ∎

3 Main results

The following lemma contains the main argument of this short paper. Using this lemma, we provide a quick proof of Theorem 1.1 before turning to its generalization.

Lemma 3.1.

Let $(M,\gamma)$ be a taut balanced sutured manifold with $\zeta\in H_{1}(M,\partial M)$ . Then $\operatorname{SFH}(M,\gamma)$ is not a free $\mathbf{F}_{2}[X]/X^{2}$ -module with respect to the homological action of $\zeta$ .

Proof.

By Theorem 2.2, we may find a sequence of nice surface decompositions from $(M,\gamma)$ to a product sutured manifold $(N,\beta)$ . Then $\operatorname{SFH}(N,\beta)$ is isomorphic to a direct summand of $\operatorname{SFH}(M,\gamma)$ as an $\mathbf{F}_{2}[X]/X^{2}$ -module by Theorem 2.5 where $X$ acts on $\operatorname{SFH}(N,\beta)$ by some homological action. Since $\dim_{\mathbf{F}_{2}}\operatorname{SFH}(N,\beta)=1$ by [Juh06, Proposition 9.4], it cannot be a free $\mathbf{F}_{2}[X]/X^{2}$ -module. Thus $\operatorname{SFH}(M,\gamma)$ is not a free $\mathbf{F}_{2}[X]/X^{2}$ -module. ∎

Proof of Theorem 1.1.

Recall that $\smash{\widehat{\mathrm{HFL}}}(L)=\operatorname{SFH}(S^{3}(L))$ [Juh06, Proposition 9.2] where $S^{3}(L)$ is the sutured exterior of $L$ . If $L$ is split, then $\operatorname{SFH}(S^{3}(L))$ is a free $\mathbf{F}_{2}[X]/X^{2}$ -module by a computation in a carefully chosen Heegaard diagram. This computation can be formalized in the following way. If $L$ is the split union of the knots $K$ and $J$ , then $S^{3}(L)=S^{3}(K)\>\#\>S^{3}(J)$ . By Lemma 2.8, there is an isomorphism of $\mathbf{F}_{2}[X]/X^{2}$ -modules

\operatorname{SFH}(S^{3}(L))\cong\operatorname{SFH}(S^{3}(K))\otimes_{\mathbf{F}_{2}}\operatorname{SFH}(S^{3}(J))\otimes_{\mathbf{F}_{2}}\operatorname{SFH}(S^{3}(2))

where the action of $X$ on the right-hand side is given by $\operatorname{Id}\otimes\operatorname{Id}\otimes\,X_{\xi}$ where $\xi$ is a generator of $H_{1}(S^{3}(2),\partial S^{3}(2))$ . Since $\operatorname{SFH}(S^{3}(2))\cong\mathbf{F}_{2}[X]/X^{2}$ as modules with respect to the action of $\xi$ , it follows from Lemma 2.9 that $\operatorname{SFH}(S^{3}(L))$ is a free $\mathbf{F}_{2}[X]/X^{2}$ -module.

If $L$ is not split, then $S^{3}(L)$ is taut, so $\operatorname{SFH}(S^{3}(L))$ is not free by Lemma 3.1. ∎

The next lemma is a direct consequence of [LS19, Lemma 5.6], which Lipshitz-Sarkar prove using [AL19, Theorem 1.1]. This result of Alishahi-Lipshitz builds on work of Ni [Ni13]. See also [HN13, Theorem 4 and Corollary 5.2].

Lemma 3.2.

Let $Y$ be an irreducible closed oriented $3$ -manifold with $\zeta\in H_{1}(Y)$ . Then $\smash{\widehat{\mathrm{HF}}}(Y)$ is not a free $\mathbf{F}_{2}[X]/X^{2}$ -module with respect to the homological action of $\zeta$ .

Proof.

We use the notation in [LS19] without reintroducing it. Since $S\cdot\zeta=0$ for all embedded $2$ -spheres $S$ in $Y$ , the unrolled homology of $\smash{\widehat{\mathrm{CF}}}(Y)$ with respect to $\zeta$ is nontrivial by [LS19, Lemma 5.6]. The $E^{1}$ -page of the spectral sequence associated to the horizontal filtration on the unrolled complex of $\smash{\widehat{\mathrm{CF}}}(Y)$ is the unrolled complex of $\smash{\widehat{\mathrm{HF}}}(Y)$ viewed as an $\mathbf{F}_{2}[X]/X^{2}$ -module with respect to $\zeta$ . Since the $E^{\infty}$ -page is nonzero, the $E^{2}$ -page is also nonzero so $\smash{\widehat{\mathrm{HF}}}(Y)$ is not a free $\mathbf{F}_{2}[X]/X^{2}$ -module. ∎

Proof of Theorem 1.4.

We first prove the result under the assumption that $(M,\gamma)$ is strongly-balanced. We then prove the general statement by reducing to this case. A balanced sutured manifold $(M,\gamma)$ is strongly-balanced [Juh08, Definition 3.5] if for each component $F$ of $\partial M$ , we have the equality $\chi(F\cap R_{+}(\gamma))=\chi(F\cap R_{-}(\gamma))$ .

Under the assumption that $(M,\gamma)$ is strongly-balanced, suppose there exists a $2$ -sphere $S$ in $M$ for which $S\cdot\zeta$ is odd. There are two cases.

(a)

The sphere $S$ is non-separating.

Then $(M,\gamma)$ is the connected sum of a strongly-balanced sutured manifold $(N,\beta)$ with $S^{1}\times S^{2}$ , where $S$ is a copy of $\mathrm{pt}\times S^{2}$ in the $S^{1}\times S^{2}$ summand. If $\zeta=\zeta^{\prime}\oplus\zeta^{\prime\prime}$ under the natural identification $H_{1}(M,\partial M)=H_{1}(N,\partial N)\oplus H_{1}(S^{1}\times S^{2})$ , then $\zeta^{\prime\prime}$ is an odd multiple of a generator $\xi$ of $H_{1}(S^{1}\times S^{2})$ by the assumption that $S\cdot\zeta$ is odd. By Lemma 2.7, there is an isomorphism of $\mathbf{F}_{2}[X]/X^{2}$ -modules

$\operatorname{SFH}(M,\gamma)\cong\operatorname{SFH}(N,\beta)\otimes_{\mathbf{F}_{2}}\operatorname{SFH}(S^{1}\times S^{2}),$

where the actions of $X$ on the right-hand side is $X_{\zeta^{\prime}}\otimes\operatorname{Id}+\operatorname{Id}\otimes\,X_{\zeta^{\prime\prime}}$ . Since $\zeta^{\prime\prime}-\xi$ is even, we know that $X_{\zeta^{\prime\prime}}=X_{\xi}$ . By a direct computation, $\operatorname{SFH}(S^{1}\times S^{2})\cong\mathbf{F}_{2}[X]/X^{2}$ as an $\mathbf{F}_{2}[X]/X^{2}$ -module with respect to the action of $\xi$ . It follows that $\operatorname{SFH}(M,\gamma)$ is free by Lemma 2.9.

(b)

The sphere $S$ is separating.

Then $(M,\gamma)$ is the connected sum of sutured manifolds $(N_{1},\beta_{1})$ and $(N_{2},\beta_{2})$ along the sphere $S$ . Since $(M,\gamma)$ is strongly-balanced, both $(N_{1},\beta_{1}),(N_{2},\beta_{2})$ are as well. By Lemma 2.8, there is an isomorphism of $\mathbf{F}_{2}[X]/X^{2}$ -modules

\operatorname{SFH}(M,\gamma)\cong\operatorname{SFH}(N_{1},\beta_{1})\otimes_{\mathbf{F}_{2}}\operatorname{SFH}(N_{2},\beta_{2})\otimes_{\mathbf{F}_{2}}\operatorname{SFH}(S^{3}(2))

where the action of $X$ on the right-hand side is given by

X_{\zeta_{1}}\otimes\operatorname{Id}\otimes\operatorname{Id}+\operatorname{Id}\otimes\,X_{\zeta_{2}}\otimes\operatorname{Id}+\operatorname{Id}\otimes\operatorname{Id}\otimes\,X_{\xi}

for classes $\zeta_{i}\in H_{1}(N_{i},\partial N_{i})$ and $\xi\in H_{1}(S^{3}(2),\partial S^{3}(2))$ . Furthermore, by the assumption that $S\cdot\zeta$ is odd, Lemma 2.8 implies that $\operatorname{SFH}(S^{3}(2))\cong\mathbf{F}_{2}[X]/X^{2}$ as modules with respect to the action of $\xi$ . Thus $\operatorname{SFH}(M,\gamma)$ is a free $\mathbf{F}_{2}[X]/X^{2}$ -module by Lemma 2.9.

Now assume that $S\cdot\zeta$ is even for every embedded $2$ -sphere $S$ in $M$ , where $(M,\gamma)$ is strongly-balanced. To show that $\operatorname{SFH}(M,\gamma)$ is not a free $\mathbf{F}_{2}[X]/X^{2}$ -module with respect to the action of $\zeta$ , we use Lemmas 2.7, 2.8, and 2.9 to reduce to the irreducible case, which is then handled by Lemmas 3.1 and 3.2. Write $M$ as a connected sum

M=N_{1}\>\#\>\cdots\>\#\>N_{k}\>\#\>Y_{1}\>\#\>\cdots\>\#\>Y_{\ell}\>\#\>(S^{1}\times S^{2})^{\#m}

where the $N_{i}$ are irreducible compact $3$ -manifolds with nonempty boundary and the $Y_{i}$ are irreducible closed $3$ -manifolds. A quick way to see that such a decomposition exists uses the Grushko-Neumann theorem on the ranks of free products of finitely-generated groups and the Poincaré conjecture (for example, see [Mil62]). The assumption that $(M,\gamma)$ is strongly-balanced implies $(N_{i},\beta_{i})$ is also strongly-balanced where $\beta_{i}$ are the sutures inherited from $\gamma$ . Because $\operatorname{SFH}(M,\gamma)\neq 0$ , it follows that $\operatorname{SFH}(N_{i},\beta_{i})\neq 0$ so $(N_{i},\beta_{i})$ is taut by [Juh06, Proposition 9.18]. Note that $S\cdot\zeta$ is even for each sphere along which the connected sums are formed and for each sphere of the form $\mathrm{pt}\times S^{2}$ in each $S^{1}\times S^{2}$ summand. The result now follows from Lemmas 2.7, 2.8, 2.9, 3.1, and 3.2.

We now reduce to the case that $(M,\gamma)$ is strongly-balanced. As explained in [Juh08, Remark 3.6], we may construct a strongly-balanced sutured manifold $(M^{\prime},\gamma^{\prime})$ from a given balanced sutured manifold $(M,\gamma)$ so that there is a sequence of product disc decompositions from $(M^{\prime},\gamma^{\prime})$ to $(M,\gamma)$ . To construct $(M^{\prime},\gamma^{\prime})$ , we repeat the following procedure, which is sometimes referred to as a contact $1$ -handle attachment: fix two components $F_{1},F_{2}$ of $\partial M$ for which $\chi(F_{i}\cap R_{+}(\gamma))\neq\chi(F_{i}\cap R_{-}(\gamma))$ , choose discs $D_{i}$ centered at points on $s(\gamma)\cap F_{i}$ , and identify $D_{1}$ with $D_{2}$ by an orientation-reversing map so that the sutures $s(\gamma)\cap D_{1}$ and $s(\gamma)\cap D_{2}$ are identified. Do this identification in such a way that the orientations of the sutures are reversed. The resulting manifold naturally inherits an orientation and sutures for which there is at least one fewer boundary component $F$ with $\chi(F\cap R_{+}(\gamma))\neq\chi(F\cap R_{-}(\gamma))$ . This reverse of this procedure is a product disc decomposition along $D_{1}=D_{2}$ .

Let $(M,\gamma)$ be a balanced sutured manifold, and let $\zeta\in H_{1}(M,\partial M)$ . Construct a strongly-balanced sutured manifold $(M^{\prime},\gamma^{\prime})$ from $(M,\gamma)$ as explained. Let $z\subset M$ be a properly-embedded oriented $1$ -manifold representing $\zeta$ which is disjoint from the discs on the boundary used in the construction of $(M^{\prime},\gamma^{\prime})$ . Then $z$ represents a class $\zeta^{\prime}\in H_{1}(M^{\prime},\partial M^{\prime})$ for which $i_{*}(\zeta^{\prime})=\zeta$ under the sequence of product disc decompositions from $(M^{\prime},\gamma^{\prime})$ to $(M,\gamma)$ . Note that $\operatorname{SFH}(M,\gamma)$ and $\operatorname{SFH}(M^{\prime},\gamma^{\prime})$ are isomorphic as $\mathbf{F}_{2}[X]/X^{2}$ -modules with respect to the actions of $\zeta$ and $\zeta^{\prime}$ , respectively, by Lemma 2.6.

If there is an embedded $2$ -sphere $S$ in $M$ with $S\cdot\zeta$ odd, then the same $2$ -sphere viewed in $M^{\prime}$ also has $S\cdot\zeta^{\prime}$ odd. Thus $\operatorname{SFH}(M^{\prime},\gamma^{\prime})$ and $\operatorname{SFH}(M,\gamma)$ are free. Conversely, if $\operatorname{SFH}(M,\gamma)$ is free, then $\operatorname{SFH}(M^{\prime},\gamma^{\prime})$ is free as well, so there is an embedded $2$ -sphere $S^{\prime}$ in $(M^{\prime},\gamma^{\prime})$ which intersects $z$ transversely in an odd number of points. By using an innermost argument, we may compress $S^{\prime}$ along discs in the product discs of the sequence of decompositions from $(M^{\prime},\gamma^{\prime})$ to $(M,\gamma)$ to obtain a collection of embedded $2$ -spheres $S_{i}$ in $M$ for which $\sum_{i}S_{i}\cdot\zeta=S^{\prime}\cdot\zeta^{\prime}$ is odd. Thus there is at least one $S_{i}$ for which $S_{i}\cdot\zeta$ is odd. ∎

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