Instanton Floer homology, sutures, and Euler characteristics
IINSTANTON FLOER HOMOLOGY, SUTURES, AND EULERCHARACTERISTICS
ZHENKUN LI AND FAN YE
Abstract.
This is a companion paper to an earlier work of the authors. In this paper, we providean axiomatic definition of Floer homology for balanced sutured manifolds and prove that thegraded Euler characteristic of this homology is fully determined by the axioms we proposed. Asa result, we conclude that χ p SHI p M, γ qq “ χ p SF H p M, γ qq for any balanced sutured manifold p M, γ q . In particular, for any link L in S , the Euler characteristic χ p KHI p S , L qq recovers themulti-variable Alexander polynomial of L , which generalizes the knot case. Combined with theauthors’ earlier work, we provide more examples of p , q -knots in lens spaces whose KHI and { HF K have the same dimension. Moreover, for a rationally null-homologous knot in a closedoriented 3-manifold Y , we construct a canonical Z on KHI p Y, K q , the decomposition of I p Y q discussed in the previous paper, and the minus version of instanton knot homology KHI ´ p Y, K q introduced by the first author. Contents
1. Introduction 22. Axioms and formal properties for sutured homology 92.1. Axioms of a Floer-type theory for closed 3-manifolds 92.2. Formal sutured homology of balanced sutured manifold 122.3. Gradings on formal sutured homology 173. Heegaard Floer homology and the graph TQFT 233.1. Heegaard Floer homology for multi-pointed 3-manifolds 233.2. Cobordism maps for restricted graph cobordisms 263.3. Floer’s excision theorem 303.4. Sutured Heegaard Floer homology 374. Equivalence of graded Euler characteristics 394.1. Balanced sutured handlebodies 394.2. Gradings and 2-handle attachments 444.3. General balanced sutured manifolds 465. The canonical mod 2 grading 485.1. The case of the unknot 495.2. Sutured knot complements 515.3. Computations and applications 56References 59 a r X i v : . [ m a t h . G T ] J a n ZHENKUN LI AND FAN YE Introduction
Sutured manifold theory was introduced by Gabai [Gab83] and Floer theory was introducedby Floer [Flo88]. They are both powerful tools in the study of 3-manifolds and knots. The firstcombination of these theories, called sutured Floer homology, was introduced by Juh´asz [Juh06]based on Heegaard Floer theory, with some pioneering work done by Ghiggini [Ghi08] and Ni[Ni07]. Later, Kronheimer and Mrowka made parallel constructions in monopole (Seiberg-Witten)theory and instanton theory [KM10b]. Different versions of Floer theories have different merits. Forexample, Heegaard Floer theory is more computable, while instanton theory is closely related torepresentation varienties of fundamental groups. Hence it is important to understand the relationshipbetween different versions of Floer theories. In this line, Lekili [Lek13], Baldwin and Sivek [BS20b])proved that sutured (Heegaard) Floer homology is isomorphic to sutured monopole Floer homology,though the relation with sutured instanton Floer homology is still open.
Conjecture 1.1 ([KM10b]) . For a balanced sutured manifold p M, γ q , we have SHI p M, γ q –
SF H p M, γ q b C . In particular, for a knot K in a closed oriented 3-manifold Y , there are isomorphisms I p Y q – y HF p Y q b C , and KHI p Y, K q – { HF K p Y, K q b C . Here
SHI is sutured instanton Floer homology [KM10b],
SF H is sutured (Heegaard) Floer homology[Juh06], I is framed instanton Floer homology [KM11], y HF is the hat version of Heegaard Floerhomology [OS04d], KHI is instanton knot homology [KM10b], and { HF K is (Heegaard) knot Floerhomology [OS04b, Ras03].In this paper, instead of studying the full homologies, we study their Euler characteristics undersome gradings, and obtain the following theorem.
Theorem 1.2.
Suppose p M, γ q is a balanced sutured manifold and S , . . . , S n are properly embeddedadmissible surfaces (c.f. Definition 2.21) generating H p M, B M q{ Tors . Then there exist Z n gradingson SHI p M, γ q and SF H p M, γ q induced by these surfaces, respectively. Equivalently, we have SHI p M, γ q “ à p i ,...,i n qP Z n SHI p M, γ, p S , . . . , S n q , p i , . . . , i n qq and the similar result for SF H p M, γ q . Moreover, there exists relative Z gradings SH p M, γ q and SF H p M, γ q , respecting the decompositions. Define (1.1) χ p SHI p M, γ qq : “ ÿ p i ,...,i n qP Z n χ p SHI p M, γ, p S , . . . , S n q , p i , . . . , i n qq ¨ t i ¨ ¨ ¨ t i n n . The graded Euler characteristic χ p SF H p M, γ qq is defined similarly to (1.1). Then we have χ p SHI p M, γ qq „ χ p SF H p M, γ qq , where „ means two polynomials equal up to multiplication by ˘ t j ¨ ¨ ¨ t j n n for some p j , . . . , j n q P Z n ,Remark . Suppose that t , . . . , t n represent generators of H “ H p M q{ Tors – H p M, B M q{ Tors . Then „ means the equality holds for elements in Z r H s{ ˘ H . NSTANTON FLOER HOMOLOGY, SUTURES, AND EULER CHARACTERISTICS 3
The graded Euler characteristic χ p SF H p M, γ qq was studied by Friedl, Juh´asz, and Rasmussen[FJR09]. Applying their results, we can relate the graded Euler characteristics of links with classicalinvariants obtained from fundamental groups.Consider a finitely generated group π “ x x , . . . , x n | r , . . . , r k y . Let H “ H p π q{ Tors be theabelianization of π modulo torsions. For a generator x i and a word w , let B w {B x i be the Foxderivative of w with respect to x i . Equivalently, it satisfies the following conditions.(1) For any word w “ u ¨ v , we have B w B x i “ B u B x ` u ¨ B v B x .(2) B x i B x i “ B x j B x i “ j ‰ i .Consider A “ tB r j {B x i u i,j as a matrix with entries in Z r H s by the projection map Z r π s Ñ Z r H s .Let E p π q be the ideal generated by the minor determinant of A of order n ´
1. Since Z r H s is a uniquefacterization domain, one can consider the greatest common divisor of the elements of E p π q , whichis well-defined up to multiplication by a nonzero element in ˘ H . This is denoted by ∆ p π q and calledthe Alexander polynomial of π ( c.f. [Tur02]). For a 3-manifold M , the Alexander polynomialof M is defined by ∆ p M q : “ ∆ p π p M qq . For an n -component link L in S , we write t , . . . , t n forhomology classes of meridians of components of L and define ∆ L p t , . . . , t n q : “ ∆ p S z int N p L qq asthe multi-variable Alexander polynomial of L . If n “ L “ K is a knot, we can fix theambiguity of ˘ H by assuming ∆ K p t q “ ∆ K p t ´ q and ∆ K p q “
1. In this case, we call it the symmetrized Alexander polynomial of K . Theorem 1.4.
Suppose M is a compact manifold whose boundary consists of tori T , . . . , T n with b p M q ě . Suppose γ “ n ď j “ m j Y p´ m j q consists of two simple closed curves with opposite orientations on each torus. Suppose H “ H p M q{ Tors and r m s , . . . , r m n s are homology classes. Then we have (1.2) χ p SHI p M, γ qq “ ∆ p M q ¨ n ź j “ pr m j s ´ q P Z r H s{ ˘ H. In particular, suppose L Ă S is an n -component link with n ě . Define (1.3) KHI p L q : “ SHI p S z int N p L q , n ď j “ m j Y p´ m j qq , where m , . . . , m n are meridians of components of L . Let p i , . . . , i n q denote the Z n grading on KHI p L q induced by Seifert surfaces of components of L . Then we have χ p KHI p L qq : “ ÿ p i ,...,i n qP Z n χ p KHI p L, p S , . . . , S n q , p i , . . . , i n qqq¨ t i ¨ ¨ ¨ t i n n „ ∆ L p t , . . . , t n q¨ n ź j “ p t j ´ q , where „ means the equality holds for elements in Z r H s{ ˘ H .Remark . The similar result had been proved for link (Heegaard) Floer homology by Ozsv´athand Szab´o [OS08]. For instanton theory, the case of single-variable Alexander polynomial for linkswere understood by Kronheimer and Mrowka [KM10a] and independently by Lim [Lim09], whilethe case of the multi-variable polynomial was unknown before.For knots, the corresponding corollary is the following.
ZHENKUN LI AND FAN YE
Theorem 1.6.
Suppose K is a knot in a closed oriented 3-manifold Y . Suppose Y p K q : “ Y z int N p K q is the knot complement and b p Y p K qq “ . Let r m s P H “ H p Y p K qq{ Tors – Z x t y be the homologyclass of the meridian of K . Define KHI p Y, K q similarly to KHI p L q as in (1.3). Then we have χ p KHI p Y, K qq “ ∆ p Y p K qq ¨ r m s ´ t ´ P Z r H s{ ˘ H. Remark . Analogue results in Heegaard Floer theory can be find in [RR17, Proposition 2.1] and[Ras07, Proposition 3.1].An application of Theorem 1.6 is to compute the instanton knot homology of some special familiesof knots. In [LY20], the authors proved the following.
Theorem 1.8 ([LY20, Theorem 1.4]) . Suppose K Ă Y is a p , q -knot in a lens space (including S ). Then we have dim C KHI p Y, K q ď dim F { HF K p Y, K q . Obviously, a lower bound of dim C KHI p Y, K q can be obtained from χ p KHI p Y, K qq . If this lowerbound coincides with the upper bound from Theorem 1.8, then we figure out the precise dimensionof KHI p Y, K q . This trick applies to p , q -knots in S which are either homologically thin knots orHeegaard Floer L-space knots. In [LY20] we worked with knots in S because prior to the currentpaper, the Euler characteristics of instanton knot homology were only understood in that case. Onthe other hand, in [Ye20], the second author observed some families of p , q -knots in general lensspaces whose dim F { HF K p Y, K q is determined by χ p { HF K p Y, K qq . Hence, with Theorem 1.6 andresults from [Ye20], we conclude the following. Corollary 1.9.
Suppose Y is a lens space, and K Ă Y is a p , q -knot such that(1) either K admits an L -space surgery (c.f. [RR17, Lemma 3.2] and [GLV18, Theorem 2.2]), or K is a constrained knot (c.f. [Ye20, Section 4]),(2) we have H p Y p K qq – Z , where Y p K q is the knot complement of K .Then we have dim C KHI p Y, K q “ dim Z { HF K p Y, K q . Remark . Greene, Lewallen, and Vafaee [GLV18] provided a clear criterion to check if a p , q -knotadmits an L-space surgery. Remark . The condition H p Y p K qq – Z is necessary since terms related Euler characteristicsof torsion spin c structures may cancel out when we consider the map induced by the projection H p Y p K qq Ñ H p Y p K qq{ Tors.Next, we roughly explain the idea to prove Theorem 1.2. For a closed 3-manifold Y , the Eulercharacteristic χ p I p Y qq was understood by Scaduto [Sca15]. The strategy is to build Y from Dehnsurgeries along a link in an integral homology sphere. The behavior of the Euler characteristic underthe surgery exact triangle was fully understood as in [KM07, Section 42.3] and it is known that theFloer homology of any integral homology sphere has Euler characteristic 1. Hence we can calculate χ p I p Y qq inductively.However, things become more complicated when we take into account surfaces inside 3-manifolds.Suppose R Ă Y is a closed homologically essential surface. Then R induces a Z grading I p Y q by µ p R q ( c.f. [KM10b, Section 7]). When trying to understand the graded Euler characteristic, theprevious strategy does not apply directly. The reason is that, the surgery curves may have nontrivialalgebraic intersections with the surface R , so the maps in surgery exact triangles may not behave NSTANTON FLOER HOMOLOGY, SUTURES, AND EULER CHARACTERISTICS 5 well under the grading associated to R . We are faced with the same problem when proving Theorem1.2.Our strategy is the following. Suppose p M, γ q is a balanced sutured manifold and suppose S ,. . . , S n are properly embedded surfaces in M . After attaching 1-handles along B M , we can find a framedlink in the interior of the resulting manifold such that the link is disjoint from all the surfaces.Moreover, surgeries along the link with all slopes chosen between in t , u produce only handlebodies.Since the surgery link is disjoint from the surfaces S , . . . , S n that induce the gradings, the maps insurgery exact triangles preserve the gradings. Hence it suffices to understand the case for suturedhandlebodies. In this case, we can further use bypass exact triangles to reduce sutured handlebodiesto product sutured manifolds. It is known that product sutured manifolds have Floer homologyof dimension one. Since the behavior of Euler characteristics under bypass exact triangles andsurgery exact triangles are the same for both instanton theory and Heegaard Floer theory, we finallyconclude that these two versions of Floer theories must have the same graded Euler characteristic.In the above argument, it is not necessary to treat instanton theory and Heegaard Floer theoryseparately. Instead, we only use some formal properties that are shared by both theories and hencewe can deal with them at the same time. This observation can be made more general. In Kronheimerand Mrowka’s definition of sutured (monopole or instanton) Floer homology, they constructed aclosed 3-manifold called the closure, out of a balanced sutured manifold in a topological manner,and defined the Floer homology for a balanced sutured manifold to be some direct summands ofthe Floer homology of its closure. Then they used the formal properties of monopole theory andinstanton theory to show that the construction is independent of the choice of the closures. In thefollowing series of work [BS15, BS16a, BS16b, Li19, GL19], most arguments were also carried outbased on topological constructions and hence only depend on the formal properties of Floer theories.In this paper, we summarize the necessary properties of Floer theory that are used to build asutured homology for balanced sutured manifolds. In Subsection 2.1, we state three axioms for p ` q -TQFTs (functors from cobordism categories to categories of vector spaces) called ‚ (A1) the adjunction inequality axiom, ‚ (A2) the surgery exact triangle axiom, ‚ (A3) the canonical Z (mod 2) grading axiom.A p ` q -TQFT satisfying these axioms is called a Floer-type theory. For any Floer-type theory H and any balanced sutured manifold p M, γ q , we construct a vector space SH p M, γ q called the formal sutured homology of p M, γ q . More precisely, we have the following theorem. Theorem 1.12.
Suppose H is a p ` q -TQFT and p M, γ q is a balanced sutured manifold. If H satisfies Axioms (A1) and (A2), then there is a vector space SH p M, γ q for any balanced suturedmanifold p M, γ q , well-defined up to multiplication by a nonzero element in the base field F . Suppose S , . . . , S n are properly embedded admissible surfaces generating H p M, B M q{ Tors . Then there exista Z n grading on SH p M, γ q induced by these surfaces. Equivalently, we have (1.4) SH p M, γ q “ à p i ,...,i n qP Z n SH p M, γ, p S , . . . , S n q , p i , . . . , i n qq . Furthermore, if H satisfies Axiom (A3), then there exists a relative Z grading SH p M, γ q ,respecting the decomposition in (1.4). Moreover, the graded Euler characteristic χ p SH p M, γ qq ,defined similarly to (1.1) and determined up to multiplication by a nonzero element in ˘ H p M q{ Tors ,is independent of the choice of the Floer-type theory.
ZHENKUN LI AND FAN YE
Remark . A priori , the definition fo formal sutured homology depends on a large and fixedinteger g . See Convection 2.2. Remark . The construction of SH is essentially due to work of Kronheimer and Mrowka [KM10b].Note that instanton theory, monopole theory and Heegaard Floer theory all satisfy Axioms (A1),(A2), and (A3) with coefficients C , F and F , respectively, up to mild modifications ( c.f. Subsection2.1). Moreover, Axioms (A1), (A2), and (A3) are not limited by the scope of gauge-theoretic theoriesmentioned above and may hold for other more general p ` q -TQFTs.There is one further step to prove Theorem 1.2 from Theorem 1.12. For Heegaard Floer theory, theconstruction coming from Theorem 1.12 is different from the original version of sutured (Heegaard)Floer homology defined by Juh´asz [Juh06]. It had been shown by Lekili [Lek13], Baldwin and Sivek[BS20b] that these two constructions coincide with each other. Although not shown explicitly, theirproofs also imply that the isomorphism between these two constructions respects gradings. Basedon their work, we show the following proposition. Proposition 1.15.
Suppose p M, γ q is a balanced sutured manifold and suppose H “ H p M q{ Tors .Suppose
SHF is the sutured homology for balanced sutured manifolds constructed in Theorem 1.12for Heegaard Floer theory. Then we have χ p SHF p M, γ qq “ χ p SF H p M, γ qq P Z r H s{ ˘ H. Other than Theorem 1.12, there are more results that can be induced from axioms and formalproperties of the formal sutured homology SH . Since the proofs in the authors’ previous papers[BLY20, LY20] are only based on formal properties, all results can be applied to SH without essentialchanges. In particular, the following theorem is just the main theorem of [BLY20], replacing SHI by SH . Theorem 1.16 ([BLY20, Theorem 1.1]) . Suppose H is an admissible Heegaard diagram for abalanced sutured manifold p M, γ q . Let S p H q be the set of generators of SF C p H q . Suppose H is aFloer-type theory with coefficient F . Then we have dim F SH p M, γ q ď | S p H q| . Combining the lower bound from Theorem 1.12 and the upper bound from Theorem 1.16, weobtain the following corollary.
Corollary 1.17.
Formal sutured homology SH is independent of the choice of the p ` q -TQFTsatisfying Axioms (A1), (A2) and (A3) in the following cases (c.f. Definition 5.1): ‚ alternating knots in S (c.f. [OS03]), ‚ p , q -knots satisfying the assumption of Corollary 1.9, including all torus knots in S , ‚ strong L-spaces (including double branched covers of non-split alternating links) and knotsinduced by strong Heegaard diagrams (c.f. [Gre13, GL16]). Next, we deal with the Z grading on formal sutured homology SH p M, γ q . Following [KM10b],to construct SH p M, γ q , we first construct a closure Y from p M, γ q . From a fixed balanced suturedmanifold p M, γ q , we can construct infinitely many different closures (with the same genus), and theFloer homology of each closure has its own Z grading. Although we can construct isomorphismsbetween the Floer homology of different closures, the maps do not necessarily respect the Z grading.See [KM10a, Section 2.6] for a concrete example. Thus, we cannot obtain a canonical Z grading on SH p M, γ q and the Euler characteristic can only be defined up to a sign (since we do know the mapsbetween closures are homogenous with respect to the Z gradings). NSTANTON FLOER HOMOLOGY, SUTURES, AND EULER CHARACTERISTICS 7
However, if we focus on balanced sutured manifolds whose underlying 3-manifolds are knotcomplements, it is possible to obtain a canonical Z grading. The idea is based on the followingobservation. For knot complements, the unknot in S becomes a good reference. We could compareclosures of a general knot with closures of the unknot and then fix the relative Z grading. As anapplication, for a knot K in a closed oriented 3-manifold Y , we obtain a canonical Z grading on KHI p Y, K q . Moreover, this canonical Z grading also carries over to the decomposition of I p Y q introduced by the authors. Theorem 1.18 ([LY20, Theorem 1.10]) . Suppose Y is a closed 3-manifold, and K Ă Y is a null-homologous knot. Suppose p Y is obtained from Y by performing the q { p surgery along K with q ą .Then there is a decomposition up to isomorphism I p p Y q – q ´ à i “ I p p Y , i q , associated to the knot K and the slope q { p . Proposition 1.19.
Under the hypothesis and the statement of Theorem 1.18, there is a well-defined Z grading on I p p Y , i q . For i “ , . . . , q ´ , we have χ p I p p Y , i qq “ χ p I p Y qq . Corollary 1.20.
Suppose K Ă S is a knot and the 3-manifold S r p K q obtained from S by the r -surgery along K is an instanton L-space. Suppose further that r “ q { p with q ą . Then for i “ , . . . , q ´ , we have I p S r p K q , i q – C . Proof.
Applying Proposition 1.19 to Y “ S , we have(1.5) dim C I p S r p K qq ě | χ p I p S r p K qq| “ | q ´ ÿ i “ χ p I p S r p K q , i qq| “ | q ´ ÿ i “ χ p I p S qq| “ q. By assumption, S r p K q is an instanton L -space, i.e. ,dim C I p S r p K qq “ | H p S r p K qq| “ q. Hence the inequality in (1.5) is sharp, which implies dim C I p S r p K q , i q “ (cid:3) The techniques to prove Proposition 1.19 can also be applied to study the minus version ofinstanton knot homology KHI ´ , which was introduced by the first author in [Li19, Section 5]. Proposition 1.21.
Suppose K Ă S is a knot and ∆ K p t q is the symmetrized Alexander polynomialof K . Then there is a Z grading on KHI ´ p S , K q . Furthermore, we have ÿ i P Z χ p KHI ´ p S , K, i qq ¨ t i “ ´ ∆ K p t q ¨ `8 ÿ i “ t ´ i . ZHENKUN LI AND FAN YE
Conventions.
If it is not mentioned, homology groups and cohomology groups are with Z coefficients.A general field is denoted by F and the field with two elements is denoted by F .If it is not mentioned, all manifolds are smooth and oriented. Moreover, all manifolds areconnected unless we indicate disconnected manifolds are also considered. This usually happensWhen we discussing cobordism maps from a p ` q -TQFT.Suppose M is an oriented manifold. Let ´ M denote the same manifold with the reverse orientation,called the mirror manifold of M . If it is not mentioned, we do not consider orientations of knots.Suppose K is a knot in a 3-manifold M . Then p´ M, K q is the mirror knot in the mirror manifold.For a manifold M , let int p M q denote its interior. For a submanifold A in a manifold Y ,let N p A q denote the tubular neighborhood. The knot complement of K in Y is denoted by Y p K q “ Y z int p N p K qq .For a simple closed curve on a surface, we do not distinguish between its homology class anditself. The algebraic intersection number of two curves α and β on a surface is denoted by α ¨ β ,while the number of intersection points between α and β is denoted by | α X β | . A basis p m, l q of H p T ; Z q satisfies m ¨ l “ ´
1. The surgery means the Dehn surgery and the slope q { p in the basis p m, l q corresponds to the curve qm ` pl ..A knot K Ă Y is called null-homologous if it represents the trivial homology class in H p Y ; Z q ,while it is called rationally null-homologous if it represents the trivial homology class in H p Y ; Q q .An argument holds for large enough or sufficiently large n if there exists a fixed N P Z sothat the argument holds for any integer n ą N . Organization.
The paper is organized as follows. In Section 2, we introduce three axioms to defineformal sutured homology for balanced sutured manifolds and prove the first part of Theorem 1.12.Moreover, we state many useful properties for the proof of the second part of Theorem 1.12. In 3, wediscuss the modification of Heegaard Floer theory to make it suitable to formal sutured homologyand prove Proposition 1.15. In Section 4, we prove the second part of Theorem 1.12. In Section 5,we construct a canonical Z grading for balanced sutured manifolds obtained from knots in closed3-manifolds and prove Proposition 1.19 and Proposition 1.21. Acknowledgement.
The second author would like to thank his supervisor Jacob Rasmussenfor patient guidance and helpful comments, and thank his parents for support and constantencouragement. The authors would also like to thank Ciprian Manolescu for helpful comments onthe draft of the paper, and thank John A. Baldwin, Qilong Guo, Joshua Wang, and Yi Xie forhelpful conversations. The second author is also grateful to Yi Liu for inviting him to BICMR,Peking University, Linsheng Wang for valuable discussion on algebra, and Honghuai Fang, ZekunChen, Fei Chen and Shengyu Zou for the company when he was writing this paper.
NSTANTON FLOER HOMOLOGY, SUTURES, AND EULER CHARACTERISTICS 9 Axioms and formal properties for sutured homology
In this section, we construct formal sutured homology and introduce many important properties.2.1.
Axioms of a Floer-type theory for closed 3-manifolds.
In this subsection, we set up axioms to construct formal sutured homology for balanced suturedmanifolds.Let
Cob ` be the cobordism category whose objects are closed oriented (possibly disconnected)3-manifolds, and whose morphisms are oriented 4-dimensional cobordisms between closed oriented3-manifolds. Let Vect F be the category of F -vector spaces, where F is a suitably chosen coefficientfield. A p ` q dimensional topological quantum field theory , or in short p ` q -TQFT , isa monoidal functor H : Cob ` Ñ Vect F . For a closed oriented 3-manifold Y , we write H p Y q for the related vector space, called the Floerhomology of Y . For an oriented cobordism W , we write H p W q for the induced map between Floerhomologies, called the cobordism map associated to W . Note that by definition of the categories,we have H p Y \ Y q “ H p Y q b F H p Y q , and H p´ Y q – Hom F p H p Y q , F q . Note that Floer theories are special cases of p ` q -TQFTs. Summarized from known Floertheories, we propose the following definition. Definition 2.1. A p ` q -TQFT H is called a Floer-type theory if it satisfies the following threeaxioms (A1), (A2), and (A3). A1 . For a closed oriented 3-manifold Y and a second homology class α P H p Y q , there is a Z -grading of H p Y q associated to α , i.e. , we have H p Y q “ à i P Z H p Y, α, i q . This grading satisfies the following properties.
A1-1 . For any odd integer i , we have H p Y, α, i q “ A1-2 . For i P Z zt u , the summand H p Y, α, i q is a finite dimensional vector space over F . A1-3 . For i P Z , we have H p Y, α, i q – H p Y, α, ´ i q . A1-4 . (Adjunction inequality). Suppose Σ is a connected closed oriented surface embedded in Y with g p Σ q ě
1. For | i | ą g p Σ q ´
2, we have H p Y, r Σ s , i q “ . A1-5 . If Y is a surface bundle over S such that the fibre Σ is a connected closed orientedsurface with g p Σ q ě
2, then H p Y, r Σ s , g p Σ q ´ q – F . A1-6 . The gradings coming from multiple homology classes are compatible with each other, i.e. , if we have α , . . . , α n P H p Y q , then there is a Z n -grading on H p Y q , denoted by H p Y q “ à p i ,...,i n qP Z n H p Y, p α , . . . , α n q , p i , . . . , i n qq . Moreover, we have H p Y, α ` ¨ ¨ ¨ ` α n , i q “ à p i ,...,i n qP Z n i `¨¨¨` i n “ i H p Y, p α , . . . , α n q , p i , . . . , i n qq . A1-7 . Suppose W is an oriented cobordism from Y to Y . Suppose α , . . . , α n P H p Y q and β , . . . , β n P H p Y q are homology classes such that for i “ , . . . , n , we have α i “ β i P H p W q . Then the cobordism map H p W q respects the grading associated to those homology classes: H p W q : H p Y , p α , . . . , α n q , p i , . . . , i n qq Ñ H p Y , p β , . . . , β n q , p i , . . . , i n qq . A2 . (Surgery exact triangle) Suppose M is a connected compact oriented 3-manifold with toroidalboundary. Let γ , γ , γ be three connected oriented simple closed curves on B M such that γ ¨ γ “ γ ¨ γ “ γ ¨ γ “ ´ . Let Y , Y , and Y be the Dehn fillings of M along curves γ , γ , and γ , respectively. Then there isan exact triangle(2.1) H p Y q (cid:47) (cid:47) H p Y q (cid:122) (cid:122) H p Y q (cid:100) (cid:100) Moreover, maps in the above triangle are induced by the natural cobordisms associated to differentDehn fillings.
Remark . It is worth mentioning that Axioms (A1) and (A2) are enough for defining formalsutured homology for balanced sutured manifolds. The following Axiom (A3) is only involved whenconsidering Euler characteristics. A3 . For any closed oriented 3-manifold Y , there is a canonical Z grading on H p Y q , denoted by H p Y q “ H p Y q ‘ H p Y q . This grading satisfies the following properties.
A3-1 . The Z grading is compatible with the grading in Axiom (A1). More precisely, if wehave α , . . . , α n P H p Y q , then there is a Z ‘ Z n grading on H p Y q : H p Y q “ à j Pt , u à p i ,...,i n qP Z n H j p Y, p α , . . . , α n q , p i , . . . , i n qq . A3-2 . Suppose Σ g is a connected closed oriented surface of genus g ě
2. Suppose Y “ S ˆ Σ g and Σ “ t u ˆ Σ g . Then we have H p Y, r Σ g s , g ´ q “ H p Y, r Σ g s , g ´ q – F . A3-3 . Suppose W is a cobordism from Y to Y . Then H p W q is homogeneous with respect tothe canonical Z grading. Its degree can be calculated by the following degree formula(2.2) deg p H p W qq ” p χ p W q ` σ p W q ` b p Y q ´ b p Y q ` b p Y q ´ b p Y qq p mod 2 q . Remark . The canonical Z grading is essentially determined by Axioms (A3-2) and (A3-3) ( c.f. [KM07, Section 25.4]). The normalization of the Z grading for the generator of H p Y, r Σ g s , g ´ q is not essential. Assuming H p Y, r Σ g s , g ´ q “ H p Y, r Σ g s , g ´ q shifts the canonical Z gradingfor all 3-manifolds. Based on [LPCS20, Lemma 3.8], we may relax the assumption in Axiom (A3-2)by g “ NSTANTON FLOER HOMOLOGY, SUTURES, AND EULER CHARACTERISTICS 11
The degrees of the maps in Axiom (A2) were described explicitly by Kronheimer and Mrowka[KM07, Section 42.3]. For the convenience of later usage, we present the discussion here.
Proposition 2.4 ([KM07, Section 42.3]) . Suppose δ is a nonzero element in ker p i ˚ q for the map i ˚ : H pB M ; Q q Ñ H p M ; Q q . In the surgery exact triangle (2.1), we can determine the parities of the maps f , f , and f asfollows.(1) If there is an i “ , , so that γ i ¨ δ “ , then f i ´ is odd and the other two are even.(2) If γ i ¨ δ ‰ for any i “ , , , then there is a unique j P t , , u so that γ j ¨ δ and γ j ` ¨ δ areof the same sign. Then the map f j is odd and the other two are even.Here the indices are taken mod 3. With Proposition 2.4, the following lemma is straightforward.
Lemma 2.5.
In the surgery exact triangle (2.1), after arbitrary shifts on the canonical Z -gradingon I ω i p Y i q for all i “ , , , exactly one of the following two cases happens.(1) If all three maps f i are odd, then we have χ p H p Y qq ` χ p H p Y qq ` χ p H p Y qq “ . (2) If there exists i “ , , so that f i is odd and the other two are even, then χ p H p Y i ´ qq “ χ p H p Y i qq ` χ p H p Y i ` qq . Here the indices are taken mod 3.Remark . If there are no shifts, then case (2) in Lemma 2.5 happens due to Proposition 2.4.In this paper we discuss three Floer theories, namely instanton theory, monopole (Seiberg-Witten)theory and Heegaard Floer theory. However, for any of these theories, we need some modificationsas follows. Suppoe Y is an object of Cob ` and W is a morphism of Cob ` . Instanton theory . We should consider the decorated cobordism category
Cob ` ω rather than Cob ` . The objects should be admissible pairs p Y, ω q , where ω Ă Y is a 1-cycle such that anycomponent of Y contains at least one component of ω . The admissible condition means that forany component Y of Y , there exists a closed oriented surface Σ Ă Y such that g p Σ q ě ω ¨ Σ is odd. Morphisms are pairs p W, ν q , where ν is a 2-cyclerestricting to the given 1-cycles on B W . The Floer homology and the cobordism map are I ω p Y q and I p W, ν q ( c.f. [Flo90]), respectively.The coefficient field is F “ C . The decorations ω and ν do not influence Axiom (A1), wherethe Z grading is induced by the generalized eigenspaces of p µ p α q , µ p pt qq actions for α P H p Y q ( c.f. [KM10b, Section 7]).In the original statement of the surgery exact triangle in [Flo90], different 3-manifolds in thesurgery exact triangle may have different choices of ω . However, by the argument in [BS20a, Section2.2], we can assume that, in Axiom (A2), the 1-cycle ω is unchanged in all manifolds involved in thetriangle.The canonical Z grading for instanton theory was discussed by Kronheimer and Mrowka [KM10a,Section 2.6]. Indeed, the degree formula (2.2) is from their discussion. Monopole theory . The Floer homology and the cobordism map are ~ HM ‚ p Y q and ~ HM ‚ p W q ( c.f. [KM07]), respectively. Although we use ~ HM ‚ for monopole theory, p ` q -TQFTs associatedto other versions of monopole Floer homology z HM ‚ and HM ‚ are also implicitly used in the proof of the Floer’s excision theorem ( c.f. [KM10b, Section 3]), which is important to show the suturedhomology for balanced sutured manifolds is well-defined.The Z grading in Axiom (A1) is induced by x c p s q , α y for s P Spin c p Y q and α P H p Y q ( c.f. [KM10b, Section 2.4]).The coefficient field is F “ F . This is because originally the surgery exact triangle is onlyproved in characteristics two ( c.f. [KM07, Section 42]). However, it is worth mentioning thatrecently the surgery exact triangle in characteristic zero was study by Freeman [Fre], so it mightbe possible to extend the discussion to F “ Q or C for monopole theory. Here we do not discussthe construction using local coefficients as in [KM10b, Section 2.2], so we only use closures of fixedgenus ( c.f. Definition 2.8).The canonical Z grading for monopole theory was discussed by Kronheimer and Mrowka [KM07,Section 25.4]. When considering cobordisms of connected 3-manifolds, the degree formula (2.2) isthe same as the formula in [KM07, Definition 25.4.1]. Heegaard Floer theory . The Floer homology and the cobordism map are HF ` p Y q and HF ` p W q ( c.f. [OS04d]). Similar to monopole theory, we will use other versions of Heegaard Floerhomology HF ´ , HF , HF ´ , HF for the proof of the Floer’s excision theorem. See Section 3 fordetails.The coefficient field is F “ F . This is because we have to use the naturality results in [JTZ18],which works only for F . Originally, to obtain a p ` q -TQFT, we should consider the graphcobordism category Cob ` ( c.f. [Zem19]) rather than Cob ` . However, after modifying thenaturality results in Section 3, we can show the Floer homology and and the cobordism map areindependent of the choice of basepoints and graphs.For characteristic zero, the naturality results for closed 3-manifolds were obtained by Gartner in[Gar19]. However, the naturality results for cobordisms are still under working. Hence we choose tofocus on characteristics two.Similar to monopole theory, the Z grading in Axiom (A1) is induced by x c p s q , α y for s P Spin c p Y q and α P H p Y q .There are many ways to fix the Z grading for Heegaard Floer theory. See [OS04c, Section 10.4]and [FJR09, Section 2.4]. However, we can arrange the canonical Z grading to be the same asthose for instanton theory and monopole theory. This is possible because the degree formula (2.2)only depends on algebraic-topological information of cobordisms and 3-manifolds.2.2. Formal sutured homology of balanced sutured manifold.
In [KM10b], Kronheimer and Mrowka constructed sutured monopole Floer homology
SHM andsutured instanton Floer homology
SHI by closures of balanced sutured manifolds. The constructionof the closure is purely topological. Hence we can adapt their construction to construct a formalversion of sutured homology for balanced sutured manifolds. Here we briefly present the constructionfrom [KM10b].
Definition 2.7 ([Juh06, KM10b]) . A balanced sutured manifold p M, γ q consists of a compactoriented 3-manifold M with non-empty boundary together with a closed 1-submanifold γ on B M .Let A p γ q “ r´ , s ˆ γ be an annular neighborhood of γ Ă B M and let R p γ q “ B M z int p A p γ qq . Theysatisfy the following properties.(1) Neither M nor R p γ q has a closed component.(2) If B A p γ q “ ´B R p γ q is oriented in the same way as γ , then we require this orientation of B R p γ q induces one on R p γ q . The induced orientation on R p γ q is called the canonical orientation . NSTANTON FLOER HOMOLOGY, SUTURES, AND EULER CHARACTERISTICS 13 (3) Let R ` p γ q be the part of R p γ q so that the canonical orientation coincides with the inducedorientation on B M , and let R ´ p γ q “ R p γ qz R ` p γ q . We require that χ p R ` p γ qq “ χ p R ´ p γ qq . If γ is clear in the contents, we simply write R ˘ “ R ˘ p γ q . Definition 2.8.
Suppose p M, γ q is a balanced sutured manifold. Let T be a connected compactoriented surface such that the numbers of components of B T and γ are the same. Let the preclosure Ă M of p M, γ q be Ă M : “ M Y γ “´B T r´ , s ˆ T. The boundary of Ă M consists of two components r R ` “ R ` p γ q Y t u ˆ T, and r R ´ “ R ´ p γ q Y t´ u ˆ T. Let h : r R ` – ÝÑ r R ´ be a diffeomorphism which reverses the boundary orientations ( i.e. preservesthe canonical orientations). Let Y be the 3-manifold obtained from Ă M by gluing r R ` to r R ´ by h and let R be the image of r R ` and r R ´ in Y . The pair p Y, R q is called a closure of p M, γ q . Thegenus of R is called the genus of the closure p Y, R q . For a closure p Y, R q with g p R q ě
2, define H p Y | R q : “ H p Y, r R s , g p R q ´ q . Remark . For instanton theory, we should choose a point p on T and choose the diffeomorphism h such that h pt u ˆ p q “ t´ u ˆ p . The image of r´ , s ˆ p in Y becomes a 1-cycle ω and we have ω ¨ R “
1. We should use p Y, R, ω q for the definition of a closure. We do not mention this subtletylater since everything works well under this modification.Suppose p M, γ q is a balanced sutured manifold. In [BS15], Baldwin and Sivek studied thenaturality of SHM p M, γ q and SHI p M, γ q . To relate different closures of p M, γ q , they constructedcanonical maps between corresponding Floer homologies of closures. Here we briefly present theirconstruction in [BS15, Section 5].Suppose p Y , R q and p Y , R q are two closures of p M, γ q of the same genus. We will construct acanonical map Φ : H p Y | R q Ñ H p Y | R q . Note that Y can be thought of as obtained from Y as follows. There exists an orientation preservingdiffeomorphism h : R Ñ R such that if we cut Y open along R and reglue using h , then we obtain a new 3-manifold Y together with the surface R Ă Y . Furthermore, there exists a diffeomorphism φ : Y Ñ Y such that φ | M “ id M and φ p R q “ R . Let X φ be a cobordism from Y to Y induced by φ . It is straightforward to check H p X φ q : H p Y | R q Ñ H p Y | R q is an isomorphism.We can regard h as a composition of Dehn twists along curves on R : h “ D e α ˝ ¨ ¨ ¨ ˝ D e n α n . Here e i P t˘ u , where e “ e “ ´ N “ t i | e i “ ´ u and P “ t i | e i “ u . Note that the resulting 3-manifold of cutting Y open along R and regluing by D e i α i is the same asthe resulting 3-manifold of performing a p´ e i q -surgery along α i Ă R Ă Y . We take a neighborhood N p R q of R Ă Y , and choose an identification N p R q “ r´ , s ˆ R . Pick ´ ă t ă ¨ ¨ ¨ ă t n ă t i ‰ i “ , . . . , n , and isotope α i to the level t t i u ˆ R Ă N p R q Ă Y . Let Y P be the 3-manifold obtained from Y by performing p´ q -surgeriesalong α i for all i P P . There is a natural cobordism X P from Y to Y P by attaching framed4-dimensional 2-handles to the product r , s ˆ Y along α i ˆ t u . Furthermore, the manifold Y P can also be obtained from Y by performing p´ q -surgeries along α i for all i P N . Hence there isa similar cobordism X N from Y to Y P . Since t i ‰
0, the surface R “ t u ˆ R survives in allsurgeries. Let R P Ă Y P be the corresponding surface. Definition 2.10 ([BS15]) . DefineΦ “ H p X φ q ˝ H p X N q ´ ˝ H p X P q : H p Y | R q Ñ H p Y | R q . Proposition 2.11.
The maps H p X P q : H p Y | R q Ñ H p Y P | R N q and H p X N q : H p Y | R q Ñ H p Y P | R P q are both isomorphisms.Remark . Proposition 2.11 restates [BS15, Lemma 4.9]. However, the proof in that paperinvolves a non-vanishing result for minimal Lefschetz fibrations. See [BS15, Proposition B.1]. Yetthis non-vanishing result is not covered by Axioms (A1), (A2), and (A3), so we present an alternativeproof of Proposition 2.11 based on surgery exact triangles from Axiom (A2). Also, it is worthmentioning that Baldwin and Sivek worked with Z coefficients for monopole theory in [BS15], whilewe work with Z coefficients. This makes a difference as the existing proof of the surgery exacttriangle in monopole theory is only proved with Z coefficients. Proof of Proposition 2.11.
The cobordisms X P and X N are constructed similarly, so we only prove X P is an isomorphism. Furthermore, we can assume that P has only one element α . If it has moreelements, then X P is simply the composition of cobordisms associated to single Dehn surgeries.With this assumption, the manifold Y P is obtained from Y by performing a p´ q -surgery along α . Let Y be obtained from Y by performing a 0-surgery along α , and R survives to become R Ă Y . Then we have an exact triangle by Axioms (A1-7) and (A2):(2.3) H p Y | R q H p X P q (cid:47) (cid:47) H p Y P | R P q (cid:120) (cid:120) H p Y | R q (cid:102) (cid:102) To show that H p X P q is an isomorphism, it suffices to show that H p Y | R q “
0. Indeed, since Y isobtained from a 0-surgery along α , and α can be isotoped to be a simple closed curve on R , thesurface R Ă Y is compressible. Hence H p Y | R q “ (cid:3) NSTANTON FLOER HOMOLOGY, SUTURES, AND EULER CHARACTERISTICS 15
With Proposition 2.11 being settled down, the rest of the argument in [BS15] can be applied toour setup verbatim, and we have the following theorem.
Theorem 2.13 ([BS15]) . Suppose p M, γ q is a balanced sutured manifold and p Y , R q and p Y , R q are two closures of it of the same genus. Then the isomorphism Φ : H p Y | R q Ñ H p Y | R q defined in Definition 2.10 satisfies the following properties.(1) The map Φ is well-defined up to multiplication by a non-zero element in F .(2) If p Y , R q “ p Y , R q , then Φ . “ id , where . “ means the equation holds up to multiplication by a non-zero element in F .(3) If there is a third closure p Y , R q of the same genus, then we have Φ . “ Φ ˝ Φ . Remark . In Baldwin and Sivek’s original work, the requirement that the two closures have thesame genus could be dropped, at the cost of involving local coefficient systems. However, up to theauthors’ knowledge, the discussion for the naturality of Heegaard Floer theory has not been carriedout with local coefficients. Since it is enough to work with closures of a large and fixed closure inthe current paper, we choose not to discuss on the local coefficients.
Definition 2.15 ([JTZ18, BS15]) . A projectively transitive system of vector spaces over afield F consists of(1) a set A and collection of vector spaces t V α u α P A over F ,(2) a collection of linear maps t g αβ u α,β P A well-defined up to multiplication by a nonzero element in F such that(a) g αβ is an isomorphism from V α to V β for any α, β P A , called a canonical map ,(b) g αα . “ id V α for any α P A ,(c) g βγ ˝ g αβ . “ g αγ for any α, β, γ P A .A morphism of projectively transitive systems of vector spaces over a field F from p A, t V α u , t g αβ uq to p B, t U γ u , t h γδ uq is a collection of maps t f αγ u α P A,γ P B such that(1) f αγ is a linear map from V α to U γ well-defined up to multiplication by a nonzero element in F for any α P A and γ P B ,(2) f βδ ˝ g αβ . “ h γδ ˝ f αγ for any α, β P A and γ, δ P B .A transitive system of vector spaces over a field F if it is a projectively transitive system andall equations with . “ are replaced by ones with “ . A morphism of transitive systems of vector spacesover a field F is defined similarly.We can replace vector spaces by groups or chain complexs of vector spaces and define theprojectively transitive system and the transitive system similarly. Remark . A transitive system of vector spaces p A, t V α u , t g αβ uq over a field F canonically definesan actual vector space over F V : “ ž α P A V α { „ , where v α „ v β if and only if g αβ p v α q “ v β for any v α P V α and v β P V β . A morphism of transitivesystems of vector spaces canonically defines an linear map between corresponding actual vectorspaces. Convention. If F “ F , a projectively transitive system over F is also a transitive system since F has only one nonzero element. In this case, we do not distinguish the projectively transitivesystem, the transitive system and the corresponding actual vector space. For a general field F , themorphisms between projectively transitive systems are also called maps. Definition 2.17.
Suppose H is a p ` q -TQFT satisfying Axioms (A1) and (A2), and p M, γ q is a balanced sutured manifold, the formal sutured homology SH g p M, γ q is the projectivelytransitive system consisting of(1) the Floer homology H p Y | R q for closures p Y, R q of p M, γ q with a fixed and large enough genus g .(2) the canonical maps Φ between Floer homologies as in Definition 2.10. Convention.
Through out the paper, when discussing formal sutured homology, we will pre-fixa large enough genus. So we omit it from the notation and write simply SH p M, γ q instead of SH g p M, γ q . Remark . When H also satisfies Axiom (A3), since Φ is constructed by the composition ofcobordism maps and their inverses, it is homogeneous with respect to the Z grading from Axiom(A3). Then there exists an induced relative Z grading on SH p M, γ q .In [BS16a, BS18], Baldwin and Sivek proved the bypass exact triangle for sutured monopoleFloer homology and sutured instanton Floer homology. Their proof can be exported to our setup. Theorem 2.19 ([BS16a, Theorem 5.2] and [BS18, Theorem 1.20]) . Suppose p M, γ q , p M, γ q , p M, γ q are three balanced sutured manifold such that the underlying 3-manifold is the same, andthe sutures γ , γ , and γ only differ in a disk as depicted in Figure 1. Then there exists an exacttriangle SH p´ M, ´ γ q ψ (cid:47) (cid:47) SH p´ M, ´ γ q ψ (cid:118) (cid:118) SH p´ M, ´ γ q ψ (cid:104) (cid:104) Moreover, the maps ψ i are induced by cobordisms, hence is homogeneous with respect to therelative Z grading on SH p M, γ i q . We un-package the proof of Theorem 2.19 for later convenience.
Proposition 2.20 ([BS16a, Section 5] and [BS18, Section 4]) . Consider p M, γ i q for i “ , , inTheorem 2.19, there is a closure p Y , R q of p´ M, ´ γ q with the following significance.(1) The genus g p R q is large enough.(2) There are pairwise disjoint curves ζ , ζ , ζ Ă Y so that the following is true.(a) For i “ , , , we have ζ i X int p M q “ H and ζ i can be isotoped to be disjoint from R .(b) If we perform a suitable Dehn surgery along ζ , then we obtain a closure p Y , R q of p´ M, ´ γ q . If we perform a suitable Dehn surgery in Y along ζ , then we obtain aclosure p Y , R q of p´ M, ´ γ q . If we perform a suitable Dehn surgery in Y along ζ , thenwe obtain the closure p Y , R q of p´ M, ´ γ q again.(c) The maps ψ , ψ , and ψ are induced the cobordism associated to Dehn surgeries along ζ , ζ , and ζ , respectively.(3) There are two curves η and η on R , so that if we perform p´ q -surgeries on both of them,with respect to the surface framings from R , then the surgeries along ζ , ζ , and ζ form anexact triangle as in Axiom (A2). NSTANTON FLOER HOMOLOGY, SUTURES, AND EULER CHARACTERISTICS 17 γ γ γ Figure 1.
The bypass triangle.2.3.
Gradings on formal sutured homology.
Suppose p M, γ q is a balanced sutured manifold and S Ă p
M, γ q is a properly embedded surface in M . If S satisfies some admissible conditions, the first author constructed a grading on SHM p M, γ q and SHI p M, γ q in [Li19]. In this subsection, we adapt the construction to formal sutured homology SH p M, γ q . Definition 2.21 ([GL19]) . Suppose p M, γ q is a balanced sutured manifold and S Ă M is a properlyembedded surface. The surface S is called an admissible surface if the followings hold.(1) Every boundary component of S intersects γ transversely and non-trivially.(2) | S X γ | ´ χ p S q is an even integer.Recall the construction of a closure of p M, γ q in Definition 2.8. Let T be a connected compactoriented surface of large enough genus and B T – ´ γ. Then we take Ă M “ M Y r´ , s ˆ T, with B Ă M “ r R ` \ p´ r R ´ q . Suppose n “ |B S X γ | and B S X γ “ t p , . . . , p n u . Definition 2.22 ([Li19]) . A pairing P of size n is a collection of n couples P “ tp i , j q , . . . , p i n , j n qu such that the followings hold.(1) t , . . . , n u “ t i , j , . . . , i n , j n u .(2) For any k P t , . . . , n u , the points p i k and p j k are positive and negative, respectively, asintersection points of B S and γ on B M .Given a pairing P of size n , and assuming that g p T q is large enough, we can extend S to aproperly embedded surface in Ă M as follows. Let α ,. . . , α n be pairwise disjoint properly embeddedarcs on T such that the followings hold.(1) The arcs α , . . . , α n represent linearly independent homology classes in H p T, B T q .(2) For any k P t , . . . , n u , we have B α i “ t p i k , p j k u . Givn α , . . . , α n , we construct a surface properly embedded in Ă M r S P : “ S Y r´ , s ˆ p α Y ¨ ¨ ¨ Y α n q . Definition 2.23.
A pairing P is called balanced if r S P X r R ` and r S P X r R ´ have the same numberof components.For any balanced pairing P , we can pick an orientation preserving diffeomorphism h : r R ` – ÝÑ r R ´ so that h p r S P X r R ` q “ r S P X r R ´ . Thus, we obtain a closure p Y, R q of p M, γ q as well as a closed oriented surface ¯ S P Ă Y . Then wedefine SH p M, γ, S, i q : “ H p Y, pr R s , r ¯ S P sq , p g p R q ´ , i qq . Theorem 2.24.
Given an admissible surface S in a balanced sutured manifold p M, γ q , the decom-position SH p M, γ q “ à i P Z SH p M, γ, S, i q is independent of all the choices made in the construction and hence is well-defined.Proof. The decomposition follows from Axioms (A1-1) and (A1-7). This gives a Z grading on SH p M, γ q . To show that this grading is well-defined, we need to show that it is independent of thefollowing three types of choices:(1) the choice of the balanced pairing P ,(2) the choice of arcs α ,. . . , α n with fixed endpoints,(3) the choice of the diffeomorphism h .In [Li19], the grading had been shown to be independent of the choices of type (2) and (3). Theproof involves only Axioms (A1) and (A2) and hence can be applied to our current setup. However,the original argument for choices of type (1) in [Li19] involves closures of different genus, whichis beyond the scope of our current paper as mentioned in Remark 2.14. Hence, we provide analternative proof here. For the moment let us write the grading as SH p M, γ, S, P , i q to emphasize that the grading a priori depends on the choice of the balanced pairing. Theorem 2.24then follows from the following proposition. (cid:3) Proposition 2.25.
Suppose P and P are two balanced pairings, then for any i P Z , we have SH p M, γ, S, P , i q “ SH p M, γ, S, P , i q . To relate two different pairings, in [Li19], the author introduced the following operation.
Definition 2.26.
Suppose P is a pairing of size n and α , . . . , α n are related arcs. Suppose k, l P t , . . . , n u are two indices so that the followings hold.(1) The arcs t u ˆ α i k and t u ˆ α i l belong to different components of r S P X R ` .(2) The arcs t´ u ˆ α i k and t´ u ˆ α i l belong to different components of r S P X R ´ . NSTANTON FLOER HOMOLOGY, SUTURES, AND EULER CHARACTERISTICS 19
Then we can construct another pairing P “ p P ztp i k , j k q , p i l , j l quq Y tp i k , j l q , p i l , j k qu . The operation of replacing P by P is called a cut and glue operation . Theorem 2.27 ([Kav19]) . Balanced pairings always exist. Moreover, any two balanced pairings arerelated by a finite sequence of cut and glue operations and their inverses.
Lemma 2.28.
Suppose P and P are two balanced pairings that are related by a cut and glueoperation, then for any i P Z , we have SH p M, γ, S, P , i q “ SH p M, γ, S, P , i q . Proof.
Suppose k and l are the indices involved in the operation. From the first part of the proof ofTheorem 2.24, we can freely make choices of type (2) and (3). Hence we can assume that there is adisk D Ă int p T q so that α k and α l intersects D in two arcs as depicted in Figure 2. Suppose D ` “ t u ˆ D Ă R ` , and D ´ “ t´ u ˆ D Ă R ´ . We can choose an orientation preserving diffeomorphism h : r R ` – ÝÑ r R ´ such that h p r S P X r R ` q “ r S P X r R ´ and h p D ` q “ D ´ . Let p Y, R q be the corresponding closure of p M, γ q and s S P be the closed surface defining the grading SH p M, γ, S, P q . Let β k “ α k X D and β l “ α l X D . It is obvious that if we remove the two arcs β k and β l from D Ă T , and glue back two new arcs β k and β l as shown in the middle subfigure ofFigure 2, then we obtain two new properly embedded arcs α k and α l on T so that B α k “ t p i k , p j l u and B α l “ t p i l , p j k u . Hence we change from P to P . Inside Y , if we remove S ˆ p β k Y β l q Ă S ˆ D Ă Y and glueback S ˆ p β k Y β l q Ă S ˆ D , then we obtain the surface s S P Ă Y that gives rise to the grading SH p M, γ, S, P q . The lemma then follows from the fact r s S P s “ r s S P s P H p Y q .p i k p j k p j l p j k β l β k β k β l UD D D ˆ r , s Figure 2.
The disk D , the arcs β k , β l , β k , β l , and the surface U . This fact (2.3) can be proved by constructing an explicit cobordism in Y ˆr , s from s S P Ă Y ˆt u to s S P Ă Y ˆ t u : in the product p Y ˆ r , s , s S P ˆ r , sq , we can remove S ˆ p β Y β q ˆ r , s Ă S ˆ D ˆ r , s Ă Y ˆ r , s and glue back S ˆ U Ă D ˆ r , s , where U Ă D ˆ r , s is the surface shown in the right subfigureof Figure 2. (cid:3) Proof of Proposition 2.25.
It follows immediately from Theorem 2.27 and Lemma 2.28. (cid:3)
Having constructed the grading, the rest of the arguments in [Li19] can be applied to our currentsetup verbatim. Hence we have the following.
Theorem 2.29 ([Li19]) . Suppose p M, γ q is a balanced sutured manifold and S Ă p
M, γ q is anadmissible surface. Then there is a Z grading on SH p M, γ q induced by S , which we write as SH p M, γ q “ à i P Z SH p M, γ, S, i q . This decomposition satisfies the following properties.(1) Suppose n “ |B S X γ | . If | i | ą p n ´ χ p S qq , then SH p M, γ, S, i q “ . (2) If there is a sutured manifold decomposition p M, γ q S (cid:32) p M , γ q in the sense of Gabai [Gab83],then we have SH p M, γ, S, p n ´ χ p S qqq – SH p M , γ q . (3) For any i P Z , we have SH p M, γ, S, i q “ SH p M, γ, ´ S, ´ i q . (4) For any i P Z , we have SH p M, γ, S, i q – SH p M, γ, S, ´ i q – SH p M, ´ γ, S, i q – SH p M, γ, S, ´ i q . (5) For any i P Z , we have SH p´ M, γ, S, i q –
Hom F p SH p M, γ, S, ´ i q , F q . Proof.
Term (1) comes from the adjunction inequality in (A1-4). Term (2) is a restatement of[KM10b, Proposition 7.11]. Term (3) is straighforward from the definition. Term (4) is from Axiom(A1-3). Term (5) is from the pairing ( c.f. [Li18]): x¨ , ¨y : SH p M, γ q ˆ SH p´ M, γ q Ñ F . (cid:3) Based on term (2) in Theorem 2.29, we can show formal sutured homology detects the tautnessand the productness of balanced sutured manifolds.
Definition 2.30 ([Juh06]) . A sutured manifold p M, γ q is called taut if M is irreducible and R ` p γ q and R ´ p γ q are both incompressible and Thurston norm-minimizing in the homology class that theyrepresent in H p M, γ q . Theorem 2.31 ([Juh06, Juh08, KM10b]) . Suppose p M, γ q is a balanced sutured manifold so that M is irreducible. Then p M, γ q is taut if and only if SH p M, γ q ‰ . NSTANTON FLOER HOMOLOGY, SUTURES, AND EULER CHARACTERISTICS 21
Definition 2.32.
Suppose p M, γ q is a balanced sutured manifold. It is called a homology product if H p M, R ` p γ qq “ H p M, R ´ p γ qq “
0. It is called a product sutured manifold if p M, γ q – pr´ , s ˆ Σ , t u ˆ B Σ q , where Σ is a compact surface with boundary. Theorem 2.33 ([Ni07, Juh08, KM10b]) . Suppose p M, γ q is a sutured manifold and a homologyproduct. Then p M, γ q is a product sutured manifold if and only if SH p M, γ q – F . If S Ă p
M, γ q is not admissible, then we can perform an isotopy on S to make it admissible. Definition 2.34.
Suppose p M, γ q is a balanced sutured manifold, and S is a properly embeddedsurface. A stabilization of S is an isotopy of S to a surface S in the following sense. This isotopycreates a new pair of intersection points B S X γ “ pB S X γ q Y t p ` , p ´ u . We require that there are arcs α Ă B S and β Ă γ , oriented in the same way as B S and γ , respectively,so that the followings hold.(1) We have B α “ B β “ t p ` , p ´ u .(2) The curves α and β cobound a disk D so that int p D q X p γ Y B S q “ H .The stabilization is called negative if D can be oriented so that B D “ α Y β as oriented curves. Itis called positive if B D “ p´ α q Y β . See Figure 3. We denote by S ˘ k the result of performing k many positive or negative stabilizations of S . B Sγγ (cid:8)(cid:8)(cid:8)(cid:8)(cid:42)(cid:72)(cid:72)(cid:72)(cid:72)(cid:106) positivenegative
D α β Dαβ
Figure 3.
The positive and negative stabilizations of S .The following lemma is straightforward. Lemma 2.35.
Suppose p M, γ q is a balanced sutured manifold, and S is a properly embedded orientedsurface. Suppose S ` and S ´ are the results of performing a positive and a negative stabilization on S , respectively. Then we have the following.(1) If we decompose p M, γ q along S or S ` , then the resulting two balanced sutured manifolds arediffeomorphic. (2) If we decompose p M, γ q along S ´ , then the resulting balanced sutured manifold p M , γ q is nottaut, as R ˘ p γ q would both become compressible.Remark . Note that the definition of stabilizations of a surface depends on the orientationsof the suture and the surface. If the orientation of the suture reverses, then positive and negativestabilizations switch between each other. Similarly, if we switch the orientation of the surface, thenthe positive and negative stabilizations also switch between each other.One can also relate the gradings associated to different stabilizations of a fixed surface. The prooffor
SHM and
SHI in [Li19] and [Wan20] can be adapted to our setup as well.
Theorem 2.37 ([Li19, Proposition 4.3] and [Wan20, Proposition 4.17]) . Suppose p M, γ q is abalanced sutured manifold and S is a properly embedded surface in M , which intersects the suture γ transversely. Suppose S has a distinguished boundary component so that all the stabilizationsmentioned below are performed on this boundary component. Then, for any p, k, l P Z so that thestabilized surfaces S p and S p ` k are both admissible, we have SH p M, γ, S p , l q “ SH p M, γ, S p ` k , l ` k q . Note S p is a stabilization of S as introduced in Definition 2.34, and, in particular, S “ S . If we have multiple admissible surfaces, then they together induce a multi-grading. This is provedfor
SHM and
SHI by Ghosh and the first author in [GL19]. The proof can be adapted to our casewithout essential changes.
Theorem 2.38 ([GL19]) . Suppose p M, γ q is a balanced sutured manifold and S , . . . , S n are admis-sible surfaces in p M, γ q . Then there exists a Z n grading on SH p M, γ q induced by S , . . . , S n , whichwe write as SH p M, γ q “ à p i ,...,i n qP Z n SH p M, γ, p S , . . . , S n q , p i , . . . , i n qq . Theorem 2.39 ([GL19]) . Suppose p M, γ q is a balanced sutured manifold and α P H p M, B M q is anontrivial homology class. Suppose S and S are two admissible surfaces in p M, γ q such that rB S , B S s “ rB S , B S s “ α P H p M, B M q . Then, there exist a constant C so that SH p M, γ, S , l q “ SH p M, γ, S , l ` C q . Based on the relative Z grading from Remark 2.18 and the Z n grading from Theorem 2.38, wecan define graded Euler characteristic of formal sutured homology. Definition 2.40.
Suppose p M, γ q is a balanced sutured manifold and S , . . . , S n are admissiblesurfaces in p M, γ q . Consider the canonical isomorphisms H p M, B M q – H p M q – H p M q{ Tors . Let ρ , . . . , ρ n P H “ H p M q{ Tors be images of rp S , B S qs , . . . , rp S n , B S n qs respectively. The graded Euler characteristic of SH p M, γ q is χ p SH p M, γ qq : “ ÿ p i ,...,i n qP Z n χ p SH p M, γ, p S , . . . , S n q , p i , . . . , i n qqq ¨ p ρ i ¨ ¨ ¨ ρ i n n q P Z r H s{ ˘ H. Remark . Theorem 2.39 shows the definition of graded Euler characteristic is independent of thechoice of S , . . . , S n if we regard it as an element in Z r H s{ ˘ H . If the admissible surfaces S , . . . , S n and a particular closure of p M, γ q is fixed, then the ambiguity of ˘ H can be removed. NSTANTON FLOER HOMOLOGY, SUTURES, AND EULER CHARACTERISTICS 23
At last, from Theorem 2.19, Proposition 2.20, and Axiom (A1-7), the following proposition isstraightforward.
Proposition 2.42.
Suppose p M, γ q is a balanced sutured manifold and S Ă p
M, γ q is an admissiblesurface. Suppose we perform the bypass change in a disk in Figure 1 that is disjoint from B S . Let γ and γ be the resulting two sutures. Then all the maps in the bypass exact triangle (2.19) aregrading preserving, i.e., for any i P Z , we have an exact triangle SH p´ M, ´ γ , S, i q ψ ,i (cid:47) (cid:47) SH p´ M, ´ γ , S, i q ψ ,i (cid:117) (cid:117) SH p´ M, ´ γ , S, i q ψ ,i (cid:105) (cid:105) where ψ k,i are the restriction of ψ k in (2.19). Heegaard Floer homology and the graph TQFT
In this section, we discuss the modification of Heegaard Floer theory to make it suitable to formalsutured homology.3.1.
Heegaard Floer homology for multi-pointed 3-manifolds.
In this subsection and the next subsection, we provide an overview of the graph TQFT forHeegaard Floer theory, constructed by Zemke [Zem19] (See also [HMZ18, Zem18]), and list someproperties which are relevant to proofs in the third subsection about the Floer’s excision theorem.
Definition 3.1. A multi-pointed 3-manifold is a pair p Y, w q consisting of a closed, oriented3-manifold Y (not necessarily connected), together with a finite collection of basepoints w Ă Y ,such that each component of Y contains at least one basepoint.Given two multi-pointed 3-manifolds p Y , w q and p Y , w q , a ribbon graph cobordism from p Y , w q to p Y , w q is a pair p W, Γ q satisfying the following:(1) W is a cobordism from Y to Y .(2) Γ is an embedded graph in W such that Γ X Y i “ w i for i “ ,
1. Furthermore, each point of w i has valence 1 in Γ.(3) Γ has finitely many edges and vertices, and no vertices of valence 0.(4) The embedding of Γ is smooth on each edge.(5) Γ is decorated with a formal ribbon structure, i.e. , a formal choice of cyclic ordering of theedges adjacent to each vertex. Definition 3.2. A restricted graph is a graph whose vertices have valence either 1 or 2. A ribbongraph cobordism p W, Γ q from p Y , w q to p Y , w q is called a restricted graph cobordism if Γ isrestricted (so the cyclic ordering is unique) and any component of Γ does not connect two basepointsof the same manifold Y i for i “ , Definition 3.3 ([OS08]) . Suppose p Y, w q is a connected multi-pointed 3-manifold. A multi-pointed Heegaard diagram H “ p Σ , α, β, w q for p Y, w q is a tuple satisfying the following:(1) Σ is a closed, oriented surface, embedded in Y , such that w Ă Σ zp α Y β q . Furthermore, Σsplits Y into two handlebodies U α and U β , oriented so that Σ “ B U α “ ´ U β .(2) α “ t α , . . . , α n u is a collection of n “ g p Σ q ` | w | ´ U α . Each component of Σ z α is planar andcontains a single basepoint. (3) β “ t β , . . . , β n u is a collection of pairwise disjoint, simple, closed curves on Σ bounding pairwisedisjoint compressing disks in U β . Each component of Σ z β is planar and contains a singlebasepoint.Suppose w “ t w , . . . , w m u . Let the polynomial ring associated to w be F r U w s : “ F r U w , . . . , U w m s . Let F r U w , U ´ w s be the ring obtained by formally inverting each of the variables.If k “ p k , . . . , k m q is an m -tuple, let U kw : “ U k w ¨ ¨ ¨ U k m w m . Suppose H “ p Σ , α, β, w q is a multi-pointed Heegaard diagram of a connected multi-pointed3-manifold p Y, w q . Suppose n “ g p Σ q ` | w | ´
1. Consider two tori T α : “ α ˆ ¨ ¨ ¨ ˆ α n and T β : “ β ˆ ¨ ¨ ¨ ˆ β n in the symmetric product Sym n Σ : “ p n ź i “ Σ q{ S n . The chain complex CF ´ p H q is a free F r U w s -module generated by intersection points x P T α X T β .Define CF p H q : “ CF ´ p H q b F r U w s F r U w , U ´ w s , and CF ` p H q : “ CF p H q{ CF ´ p H q . To construct a differential on CF ´ p H q , suppose H satisfies some extra admissibility conditionsif b p Y q ą c.f. [Zem19, Section 4.7]). Let p J s q s Pr , s be an auxiliary path of almost complexstructures on Sym n Σ and let π p x , y q be the set of homology classes of Whitney disks connectingintersection points x and y ( c.f. [OS08, Section 3.4]). For φ P π p x , y q , let M J s p φ q be the modulispace of J s -holomorphic maps u : r , s ˆ R Ñ Sym n Σ which represent φ . The moduli space M J s p φ q has a natural action of R , corresponding to reparametrization of the source. We write x M J s p φ q : “ M J s p ψ q{ R . For φ P π p x , y q , let µ p φ q be the expected dimension of M J s p φ q for generic J s and let n w i p φ q bethe algebraic intersection number of t w i u ˆ Sym n ´ Σ and any representative of φ . Define n w p φ q : “ p n w p φ q , . . . , n w m p φ qq . For a generic path J s , define the differential on CF ´ p H q by B J s p x q “ ÿ y P T α X T β ÿ φ P π p x , y q µ p φ q“ x M J s p φ q U n w p φ q w ¨ y , extended linearly over F r U w s . The differential B J s can be extended on CF p H q and CF ` H q bytensoring with the identity map. Lemma 3.4 ([OS08, Lemma 4.3]) . For a generic path J s , the map B J s on CF ˝ p H q , where ˝ Pt8 , ` , ´u , satisfies B J s ˝ B J s “ . NSTANTON FLOER HOMOLOGY, SUTURES, AND EULER CHARACTERISTICS 25
For a disconnected multi-pointed 3-manifold p Y, w q “ p Y , w q \ p Y , w q , where Y i is connectedfor i “ ,
2, suppose H i is an admissible multi-pointed Heegaard diagram of Y i and suppose J s i are corresponding generic paths of almost complex structures. Let the chain complex associated to p Y, w q be(3.1) p CF ˝ p H \ H q , B J s q : “ p CF ˝ p H q , B J s q b F p CF ˝ p H q , B J s q , where ˝ P t8 , ` , ´u . Remark . In Zemke’s original construction [Zem19], one should choose colors of basepoints andgraphs to achieve the functoriality of TQFT. Here we implicitly choose different colors for allbasepoints and we have the relation on the homology level(3.2) H p CF ˝ p H \ H q , B J s q “ H p CF ˝ p H q , B J s q b F H p CF ˝ p H q , B J s q . Since we will only consider restricted graph cobordisms, the choice of colors satisfies the functoriality.Note that in the construction of [HMZ18, Zem18], the colors of all basepoints are the same and all U w i are identified as U , so (3.1) and (3.2) are not true in general.The chain homotopy type of p CF ˝ p H q , B J s q is independent of the choice of the admissible diagram H and the generic path J s . Indeed, we have the following theorem about naturality. Theorem 3.6 ([Zem19, Proposition 4.6], see also [OS04d, JTZ18]) . Suppose that p Y, w q is amulti-pointed 3-manifold. To each (admissible) pair p H , J q and p H , J q , there is a well-defined map Ψ p H ,J qÑp H ,J q : p CF ´ p H q , B J q Ñ CF ´ p H q , B J q , which is well defined up to F r U w s -equivariant chain homotopy. Furthermore, the following aresatisfied:(1) If p H , J q , p H , J q and p H , J q are three (admissible) diagrams with (generic) paths of almostcomplex structures, then Ψ p H ,J qÑp H ,J q » Ψ p H ,J qÑp H ,J q ˝ Ψ p H ,J qÑp H ,J q . (2) Ψ p H ,J qÑp H ,J q » id p CF ´ p H q , B J q . Moreover, similar results hold for CF and CF ` . Since all chain complexes discussed above can be decomposed by spin c structures ( c.f. [OS04d,Section 2.6]), we have the following definition. Definition 3.7.
Suppose p Y, w q is a multi-pointed 3-manifold and s P Spin c p Y q . For ˝ P t8 , ` , ´u ,define CF ˝ p Y, w , s q to be the transitive system of chain complexes with canonical maps fromTheorem 3.6, with respect to s , and define HF ˝ p Y, w , s q to be the induced transitive system ofhomology groups.For further using, we also define the completions of the chain complexes. Definition 3.8.
Let F rr U w ss be the ring of formal power series of U w . For ˝ P t8 , ` , ´u , define CF ˝ p Y, w , s q : “ CF ˝ p Y, w , s q b F r U w s F rr U w ss . Let HF ˝ p Y, w , s q be the induced homology groups. Convention.
When omitting the module structure, we have CF ` p Y, w , s q “ CF ` p Y, w , s q . Hencewe do not distinguish these groups.The advantage of the completions is that we have the following proposition. Proposition 3.9 ([MO17, Section 2], see also [OS04a, Lemma 2.3]) . If p Y, w q is a multi-pointed3-manifold and s P Spin c p Y q on any component is nontorsion, then HF p Y, w , s q “ . Then the boundary map in the following long exact sequence induces a canonical isomorphismbetween HF ´ p Y, w , s q and HF ` p Y, w , s q for any nontorsion spin c structure s . Proposition 3.10.
From the short exact sequence Ñ CF ´ p Y, w , s q Ñ CF p Y, w , s q Ñ CF ` p Y, w , s q Ñ , we have a long exact sequence ¨ ¨ ¨ Ñ HF ´ p Y, w , s q Ñ HF p Y, w , s q Ñ HF ` p Y, w , s q Ñ ¨ ¨ ¨ Definition 3.11.
Suppose p Y, w q is a multi-pointed 3-manifold and s P Spin c p Y q is a nontorsionspin c structure. We write HF p Y, w , s q “ HF red p Y, w , s q : “ HF ` p Y, w , s q – HF ´ p Y, w , s q . Cobordism maps for restricted graph cobordisms.Theorem 3.12 ([Zem19, Theorem A]) . Suppose p W, Γ q : p Y , w q Ñ p Y , w q is a ribbon graphcobordism and s P Spin c p W q . Then there are two chain maps F AW, Γ , s , F BW, Γ , s : CF ´ p Y , w , s | Y q Ñ CF ´ p Y , w , s | Y q , which is a diffeomorphism invariant of p W, Γ q , up to F r U w s -equivariant chain homotopy. Proposition 3.13 ([Zem19, Theorem C]) . Suppose that p W, Γ q is ribbon graph cobordism whichdecomposes as a composition p W, Γ q “ p W , Γ q Y p W , Γ q . If s and s are spin c structures on W and W , respectively, then F AW , Γ , s ˝ F AW , Γ , s “ ÿ s P Spin c p W q s | W “ s s | W “ s F AW, Γ , s . The similar relation holds for F BW, Γ , s . Convention.
Since we will only consider restricted graph cobordisms, the map F AW, Γ , s is chainhomotopic to F BW, Γ , s . Hence we write CF ´ p W, Γ , s q for the chain map and HF ´ p W, Γ , s q for theinduced map on the homology group. If Γ and s are specified, we write CF ´ p W q and HF ´ p W q forsimplicity, respectively. The chain maps on CF , CF ` , CF ´ , CF are obtained by tensoring withthe identity maps, respectively. We use the similar notation for these chain maps and the inducedmaps on homology groups. All maps are called cobordism maps .For CF ´ , the cobordism map is defined by the composition of the following maps. ‚ For 4-dimensional 1-, 2-, and 3-handle attachments away from the basepoints, we use themaps defined by Ozsv´ath and Szab´o [OS06]. ‚ For 4-dimensional 0- and 4-handle attachments, or equivalently adding and removing acopy of S with a single basepoint, respectively, we use the maps defined by the canonicalisomorphism from the tensor product with CF ´ p S , w q – F r U s . ‚ For a ribbon graph cobordism p Y ˆ r , s , Γ q , we project the graph into Y and use the graphaction map defined in [Zem19, Section 7]. NSTANTON FLOER HOMOLOGY, SUTURES, AND EULER CHARACTERISTICS 27
Remark . For 4-dimensional 1-, 2-, and 3-handle attachments, Ozsv´ath and Szab´o’s originalconstruction was for connected cobordisms of connected 3-manifolds. Zemke [Zem19, Section 8]extended the construction to cobordisms between disjoint 3-manifolds. For 4-dimensional 0- and 4-handle attachments, considering the coloring chain complex in [Zem19, Section 4.3], the isomorphismis indeed CF ´ p Y \ S , w Y t w uq – CF ´ p Y, w q b F CF ´ p S , w q – CF ´ p Y, w q b F F r U s . The graph action map is obtained by the composition of maps associated to elementary graphs. Theconstruction involves free-stabilization maps S ˘ w [Zem19, Section 6] and relative homologymaps A λ [Zem19, Section 5], where S ˘ w correspond to adding or removing a basepoint w and A λ correspond to a path λ between two basepoints. When considering restricted graph cobordisms, weonly need maps associated to 1-, 2-, 3-handle attachments and free-stabilizations. Definition 3.15.
Suppose H “ p Σ , α, β, w q is a multi-pointed Heegaard diagram for a multi-pointed3-manifold p Y, w q . Let D Ă Σ zp α Y β q be a small disk containg a new basepoint w P Σ zp α Y β q .Let α and β be two simple closed curves on Σ bounding a disk containing w and | α X β | “ θ ` and θ ´ are the higher and the lower graded intersection points, respectively. Considerthe Heegaard diagram H “ p Σ , α Y t α u , β Y t β u , w Y t w uq . See Figure 4. Figure 4.
Free-stabilization in a small disk D .For appropriately chosen almost complex structures, define the free-stabilization maps S ˘ w by S ` w p x q “ x ˆ θ ` ,S ´ w p x ˆ θ ´ q “ x , , and S ´ w p x ˆ θ ` q “ . Remark . If we collapse B D to a point p , we obtain a doubly-pointed diagram on S with twocurves. Hence H can be considered as the connected sum of H and p S , α , β , t w , p uq . Proposition 3.17 ([Zem19, Section 6 and Lemma 8.13]) . The maps S ˘ w in Definition 3.15 determinewell-defined chain maps on the level of transitive systems of chain complexes S ` w : CF ´ p Y, w q Ñ CF ´ p Y, w Y t w uq ,S ´ w : CF ´ p Y, w Y t w uq Ñ CF ´ p Y, w q . Moreover, they satisfy the following results.(1) The maps S ˘ w commute with maps associated to 1-, 2-, and 3-handle attachments.(2) For ˝ , ˝ P t` , ´u , we have S ˝ w S ˝ w » S ˝ w S ˝ w . Remark . The free-stabilization maps can be regarded as restricted graph cobordisms with W “ Y ˆr , s . The graphs are shown in Figure 5. Alternatively, we can regard them as compositionsof maps associated to handle attachements. The map S ` w is obtained by first attaching a 0-handlewith an arc whose one endpoint is on the boundary and the other is in the interior, and thenattaching a 1-handle away from basepoints. See Figure 5. The map S ´ w is obtained by first attaching3-handle and then 4-handle with an arc similarly. Figure 5.
Restricted graph cobordisms related to free-stabilization maps.
Convention.
All illustrations of cobordisms are from the top to the bottom.We can calculate the effect of free-stabilization maps on the homology explicitly.
Proposition 3.19 ([OS08, Proposition 6.5]) . Consider the construction in Definition 3.15. Forsuitable choices of almost complex structures, the chain complex CF ´ p H q is identified with themapping cone of the following map CF ´ p H q b F F r U s θ ´ U ´ U ÝÝÝÝÝÑ CF ´ p H q b F F r U s θ ` . Corollary 3.20.
The map S ` w induces isomorphisms on HF ´ and HF ´ and induces the zero mapon HF ` . The map S ´ w induces an isomorphism on HF ` and induces zero maps on HF ´ and HF ´ .Proof. The arguments for HF ´ and HF ´ follows directly from Definition 3.15 and Proposition3.19. For HF ` , consider the duality CF ´ p´ Y, w , ¯ s q – Hom F p CF ` p Y, w , s q , F q from [OS04d, Section 5.1] (cid:3) The following proposition implies the choice of the basepoints is not important.
NSTANTON FLOER HOMOLOGY, SUTURES, AND EULER CHARACTERISTICS 29
Proposition 3.21 ([Zem19, Corollary 14.19 and Corollary F]) . Suppose p Y, w q is a multi-pointed3-manifold and w P w . Then the π p Y, w q action on HF ´ p Y, w q is always the identity map.Suppose p Y , w q and p Y , w q are two multi-pointed 3-manifolds with | w | “ | w | . Suppose W is a cobordism from Y to Y and Γ Ă W is a collection of paths connecting w and w . Thenthe cobordism map HF ´ p W, Γ q is independent of the choice of Γ . Moreover, if Y “ Y “ Y , then HF ´ p W, Γ q is an isomorphism.The similar results hold for HF , HF ` , HF ´ , HF . From Corollary 3.20 and Proposition 3.21, we can define a transitive system of groups based ondifferent choices of basepoints.
Definition 3.22.
Suppose Y is a closed, oriented 3-manifold and w , w Ă Y are two collectionsof basepoints in Y . Let w “ w z w and w “ w z w . Define transition maps associated to p w , w q as Ψ ´ w Ñ w : “ ź w P w p S ` w q ´ ˝ ź w P w S ` w for HF ´ and HF ´ Ψ ` w Ñ w : “ ź w P w p S ´ w q ´ ˝ ź w P w S ´ w for HF ` ,where the products mean compositions. Lemma 3.23.
Suppose Y is a closed, oriented 3-manifold and w , w , w Ă Y are three collectionsof basepoints in Y . Suppose w is a basepoint in Y that is not in w i for i “ , . Then the followingshold for transition maps.(1) Ψ ˘ w i Ñ w j is well-defined for i, j P t , , u , i.e., the composition is independent of the order ofmaps.(2) Ψ ˘ w i Ñ w j is an isomorphism for i, j P t , , u .(3) Ψ ˘ w i Ñ w i “ id for i “ , , . (4) Ψ ˘ w Ñ w ˝ Ψ ˘ w Ñ w “ Ψ ˘ w Ñ w .(5) Ψ ˘ w Yt w uÑ w Yt w u ˝ S ` w “ S ` w ˝ Ψ ˘ w Ñ w .(6) Ψ ˘ w Ñ w ˝ S ´ w “ S ´ w ˝ Ψ ˘ w Yt w uÑ w Yt w u .Proof. Terms (1), (4), (5) and (6) follow from term (2) of Proposition 3.17. Note that maps in terms(5) and (6) can be either isomorphisms or zero maps. Term (2) is trivial from the definition. Term(3) follows from Corollary 3.20. (cid:3)
Lemma 3.24.
Suppose Y and Y are closed, oriented 3-manifolds and w , w Ă Y , w , w Ă Y are collections of basepoints. Suppose W is a cobordism from Y to Y that is induced by a compositionof 1-, 2-, 3-handle attachments away from all basepoints. Let Γ and Γ be induced graphs in W with Γ X Y “ w , Γ X Y “ w , Γ X Y “ w , and Γ X Y “ w . Then we have a commutative diagram HF ´ p Y , w q HF ´ p W, Γ q (cid:47) (cid:47) Ψ ´ w Ñ w (cid:15) (cid:15) HF ´ p Y , w q Ψ ´ w Ñ w (cid:15) (cid:15) HF ´ p Y , w q HF ´ p W, Γ q (cid:47) (cid:47) HF ´ p Y , w q The similar commutative diagrams hold for HF ´ and HF ` . Proof.
This follows from term (1) of Proposition 3.17. (cid:3)
Theorem 3.25.
Suppose Y is a closed, oriented 3-manifold. Then groups HF ´ p Y, w q for all w Ă Y and transition maps Ψ ´ w Ñ w for all w , w Ă Y form a transitive system, which is denotedby HF ´ p Y q . Moreover, suppose p W, Γ q is a restricted graph cobordism from p Y , w q to p Y , w q .Then HF ´ p W, Γ q induces a well-defined map from HF ´ p Y q to HF ´ p Y q , which is independent ofthe choice of the restricted graph Γ and denoted by HF ´ p W q .The similar arguments hold for HF ´ and HF ` .Proof. The well-definedness of HF ´ p Y q and HF ´ p W, Γ q follow from Lemma 3.23 and Lemma 3.24.The restricted graph cobordism is a composition of maps associated to 1-, 2-, 3-handle attachmentsand free-stabilizations. Note that free-stabilizations maps can be isomorphisms or zero maps onhomology groups. Then the independence of Γ follows from Proposition 3.21 and above lemmas.The proofs for HF ´ and HF ` are similar. (cid:3) Remark . Groups and maps in Theorem 3.25 also split into spin c structures. Suppose s P Spin c p W q is a nontorsion spin c structure which restricts to nontorsion spin c structure s i on Y i for i “ ,
2. Then HF ´ p Y i , s i q and HF ` p Y i , s i q are canonically identified by the boundary mapin Proposition 3.10. Moreover, the maps HF ´ p W, s q and HF ` p W, s q are the same under thisidentification. We write the map as HF p W, s q . Floer’s excision theorem.
In this subsection, we follow Kronheimer and Mrowka’s idea in [KM10b, Section 3] to prove anexcision theorem in Heegaard Floer theory.Let Y be a closed, oriented 3-manifold, of either one or two components. In the latter case, let Y and Y be two components of Y . Let Σ and Σ be two closed, connected, oriented surfaces in Y with g p Σ q “ g p Σ q . If Y has two components, suppose Σ i is a non-separating surface in Y i for i “ ,
2. If Y is connected, suppose Σ and Σ represent independent homology classes. In eithercase, let F “ Σ Y Σ . Let h be an orientation-preserving diffeomorphism from Σ to Σ .We construct a new manifold r Y as follows. Let Y be obtained from Y by cutting along Σ. Wehave B Y “ Σ Y p´ Σ q Y Σ Y p´ Σ q . If Y has two components, then we have Y “ Y Y Y , where Y i is obtained from Y i by cuttingalong Σ i for i “ ,
2. Let r Y be obtained by gluing the boundary component Σ to the boundarycomponent ´ Σ and gluing Σ to ´ Σ , using the diffeomorphism of h both times. See Figure 6 forthe case that Y has two components.In either case, r Y is connected. Let r Σ i be the image of Σ i in r Y for i “ , r F “ r Σ Y r Σ . Definition 3.27.
Suppose Y is a closed, oriented 3-manifold and F Ă Y is a closed, orientedsurface. Let F i for i “ , . . . , m be components of F . Suppose further that g p F i q ě Y contains at least one component of F . Let Spin c p Y | F q denote the set of spin c structures s P Spin c p Y q satisfying(3.3) x c p s q , F i y “ g p F i q ´ ą F i . Define HF p Y | F q : “ à s P Spin c p Y | F q HF p Y, s q . NSTANTON FLOER HOMOLOGY, SUTURES, AND EULER CHARACTERISTICS 31
Figure 6.
Construction of r Y .Suppose p W, Γ q is a restricted graph cobordism and G Ă W is a closed, oriented surface. Let G i for i “ , . . . , n be components of G . Suppose further that g p G q ě W contains at least one component of G . Let Spin c p W | G q denote the set of spin c structures s P Spin c p W q satisfying similar conditions in (3.3) by replacing F i by G i . Define HF ´ p W, Γ | G q : “ ÿ s P Spin c p W | G q HF ´ p W, Γ , s q . Let HF ` p W, Γ | G q , HF ´ p W, Γ | G q and HF p W, Γ | G q be defined similarly. Remark . Note all spin c structures in Spin c p Y | F q are nontorsion, so HF p Y, s q is well-defined.The following is the main theorem of this subsection. Theorem 3.29 (Floer’s excision theorem) . Consider Y and r Y constructed as above. If g p Σ q “ g p Σ q ě , then there is an isomorphism HF p Y | F q – HF p r Y | r F q . Moreover, this isomorphism and its inverse is induced by restricted graph cobordisms.
Before proving the main theorem, we introduce some lemmas parallel to results in monopoletheory ( c.f. [KM10b, Lemma 2.2, Proposition 2.5 and Lemma 4.7])
Lemma 3.30 ([Lek13, Corollary 17]) . Let Y Ñ S be a fibered 3-manifold whose fiber F is a closed,connected, oriented surface with g “ g p F q ě . Then there is a unique s P Spin c p Y | R q so that HF p Y, s q ‰ . Moreover, we have HF p Y | F q “ HF p Y, s q – F . Lemma 3.31.
Suppose Y “ Σ ˆ S such that Σ “ Σ ˆ t u Ă Y is a closed, connected, orientedsurface with g p Σ q ě . Suppose w P S and w P Y are basepoints. Let W be obtained from Σ ˆ D by removing a 4-ball, considered as a cobordism from S to Y . Let Γ Ă W be any path connecting w and w . Then the map HF ´ p W, Γ | Σ q : F rr U ss – HF ´ p S , w q Ñ HF p Y | Σ q – F is nontrivial. Consider ´ W as a cobordism from Σ ˆ S to S . Then the map HF ` p´ W, Γ | Σ q : F – HF p Y | Σ q Ñ HF ` p S , w q – F r U , U ´ s{ F r U s is also nontrivial. Figure 7.
Nontrivial cobordism maps from compositions.
Proof.
Suppose P is 2-dimensional pair of pants as shown in the left subfigure of Figure 7. Consider W “ Σ ˆ P as a cobordism from Y \ Y to Y , where Y i – Y for i “ , ,
3. Suppose w is anotherbasepoint in Y . Let w i and w i be images of w and w in Y i for i “ , ,
3. Let Γ Ă W be acollection of two paths γ and γ , where γ connects w to w and γ connects w to w .Let p W , Γ q “ p Y ˆ I, w ˆ I q be the product cobordism. Suppose Σ i Ă Y i is the image of Σ Ă Y for i “ , ,
3. Consider the composition of the cobordism maps HF ´ p W , Γ | Σ Y Σ Y Σ q ˝ HF ´ p W \ W | Σ Y Σ q : HF p Y | Σ q b F HF ´ p S , w q Ñ HF p Y | Σ q . After filling the S component by a 4-ball, or equivalently composing it with the map associated toa 0-handle attachment, we obtain the free-stabilization map S ` w ( c.f. Remark 3.18). By Corollary3.20, the result map is an isomorphism HF p Y | Σ q – HF p Y | Σ q . Since HF ´ p W \ W | Σ Y Σ q “ HF ´ p W | Σ q b F HF ´ p W | Σ q , we know HF ´ p W | Σ q is a nontrivial map.The proof about HF ` p´ W, Γ | Σ q is similar. We replace W by ´ W and S ` w by S ´ w in the aboveproof. See the right subfigure of Figure 7 for the illustration of the composition of the cobordisms. (cid:3) Corollary 3.32.
Consider Y “ Σ ˆ S and p W, Γ q in Lemma 3.31. After filling two S boundariesby 4-balls, we have HF ´ p´ W, Γ | Σ q ˝ HF ` p W, Γ | Σ q “ id HF p Y | Σ q . NSTANTON FLOER HOMOLOGY, SUTURES, AND EULER CHARACTERISTICS 33
Proof.
By Lemma 3.30, we have HF p Y | Σ q – F , so any nontrivial map induces the identity map.By Lemma 3.31, the composition is a nontrivial map. (cid:3) Figure 8.
Restricted graph cobordisms inducing the same map.
Remark . The proof of [KM10b, Proposition 2.5] involves some arguments about the trace map,while the setting about trace cobordism in [Zem18] is different from what we use here ( c.f.
Remark3.5). Corollary 3.32 implies that we can replace the cobordism in the left subfigure of Figure 8 bythe right subfigure. This is not true for a general coloring in [Zem19]. To apply this replacement,we should decompose cobordisms in [KM10b, Section 3.2] into compositions of cobordisms.Now we start to prove the main theorem of this subsection. Many constructions are similar tothose in [KM10b, Section 3.2].
Proof of Theorem 3.29.
Step 1 . We construct a cobordism W from r Y to Y and a cobordism ¯ W from Y to r Y .Recall that Y is obtained from Y by cutting along Σ and Σ and we have B Y “ Σ Y p´ Σ q Y Σ Y p´ Σ q . Suppose P is a saddle surface, which can be regarded as a submanifold of a pair of pants withone boundary component on the top and two boundary components at the bottom. See Figure 9.Suppose B P “ λ Y λ Y µ Y µ Y η , Y η , Y η , Y η , , where λ and λ are two arcs in the top boundary component of the pair of pants, µ and µ aretwo arcs in the bottom boundary components of the pair of pants, and η i,j is the arc connecting λ i and µ j for i, j P t , u .Suppose Σ – Σ – Σ . Remember that we have fixed a diffeomorphism h from Σ to Σ . Suppose h is an orientation-preserving diffeomorphism from Σ to Σ . Let W be the union P ˆ Σ Y Y ˆ I, where η , ˆ Σ is glued to Σ ˆ I , η , ˆ Σ is glued to ´ Σ ˆ I , η , ˆ Σ is glued to Σ ˆ I , and η , ˆ Σ is glued to ´ Σ ˆ I , using h and h ˝ h , respectively. Figure 9 illustrates the case that Y has two components Y and Y . By the construction of r Y , the result manifold W is a cobordismfrom r Y to Y .The cobordism ¯ W is constructed similarly. Let P be another saddle surface and let ¯ W beobtained by gluing P ˆ Σ and Y ˆ I as shown in Figure 9. Figure 9.
Compositions of W and ¯ W . Step 2 . For some restricted graph Γ and some surface G in W A “ ¯ W Y r Y W , we show thecobordism map HF p W A , Γ | G q : “ HF ` p W A , Γ | G q “ HF ´ p W A , Γ | G q induce the identity map on HF p Y | F q : “ HF ` p Y | F q – HF ´ p Y | F q . We prove for the case that Y has two components Y and Y . The proof for the case that Y isconnected is similar. The cobordism W A is shown in the right subfigure of Figure 9. Let w and w be two basepoints on Y and let w and w be two basepoints on Y . Moreover, we can suppose w i P Y i for i “ ,
2. For i “ ,
2, let γ i be the union of images of w i ˆ I Ă Y i ˆ I in ¯ W and W . Let γ Ă W A be a path in the union of images of P ˆ Σ and P ˆ Σ connecting w to w , as shown inFigure 10. Let Γ A “ γ Y γ Y γ . It is straighforward to check by definition that Γ A is restricted. Let w be a basepoint on Σ ˆ S .We decompose the restricted graph cobordism p W A , Γ q into 4 pieces W i as shown in Figure 10,where W and W are product cobordisms for p Y \ Σ ˆ S \ Y , t w , w, w uqq . Let G “ F Y Σ ˆ t u “ Σ Y Σ Y Σ ˆ t u . NSTANTON FLOER HOMOLOGY, SUTURES, AND EULER CHARACTERISTICS 35
By Remark 3.26, we know HF ´ p W A , Γ A | G q and HF ` p W A , Γ A | G q induce the same map, so HF ´ p W i , Γ A X W i | G q and HF ` p W A , Γ A X W i | G q also induce the same map. We assign HF ´ to W and W and HF ` for W and W . Figure 10.
Decomposition of the cobordism W A and the graph Γ.Note that we have HF p Y \ Σ ˆ S \ Y | G qq – HF p Y | Σ q b F HF p Σ ˆ S | Σ ˆ t uq b F HF p Y | Σ q . By Corollary 3.32, we can replace the components of W and W corresponding to Σ ˆ S by thecobordism in the left subfigure of Figure 8. After filling two S boundaries by 4-balls, the resultinggraph cobordism p W A , Γ A q is the composition of the graph cobordisms associated to S ` w and S ´ w .By Corollary 3.20 and our assignments of the cobordism maps, we know that HF p W A , Γ A | G q “ HF p W A , Γ A | G q is an isomorphism. By construction in Definition 3.22, we know that HF p W A , Γ A | G q induces theidentity map on HF p Y | F q . Step 3.
For some restricted graph Γ and some surface G in W B “ W Y Y ¯ W , we show thecobordism map HF p W B , Γ | G q : “ HF ` p W B , Γ | G q “ HF ´ p W B , Γ | G q induce the identity map on HF p r Y | r F q : “ HF ` p r Y | r F q – HF ´ p r Y | r F q . We prove for the case that Y has two components Y and Y . The proof for the case that Y isconnected is similar. The cobordism W B is shown in the left subfigure of Figure 9. The proof isessentially the same as that in Step 2. However, the decomposition of W B is not obvious as before. Let w , w , w and w be basepoints on r Y . Let γ and γ be defined similarly to γ and γ ,respectively. Let γ Ă W B be a path in the union of images of P ˆ Σ and P ˆ Σ connecting w to w , as shown in Figure 11. Let Γ B “ γ Y γ Y γ . It is straighforward to check by definition that Γ B is restricted. Figure 11.
The cobordism W B and the graph Γ .Similar to the proof for W A , we decompose W B into 4 pieces W i , where W and W are productcobordisms for p r Y \ Σ ˆ S , t w , w , w uqq . In Figure 11 we only draw W and W . Let G “ r F Y Σ ˆ t u “ r Σ Y r Σ Y Σ ˆ t u . Similarly, We assign HF ´ to W and W and HF ` for W and W . By similar argument to that inthe proof for W A , we know that HF p W B , Γ B | G q induces the identity map on HF p r Y | r F q . Step 4 . The results in Step 2 and Step 3 implydim F HF p Y | F q ě dim F HF p r Y | r F q ě dim F HF p Y | F q . Then we know restricted graph cobordisms p W, Γ A X W q , p ¯ W , Γ A X ¯ W q , p W, Γ B X W q , and p ¯ W , Γ B X ¯ W q with respect to G and G induce isomorphisms HF p W q and HF p ¯ W q between HF p Y | F q and HF p r Y | r F q , respectively. (cid:3) NSTANTON FLOER HOMOLOGY, SUTURES, AND EULER CHARACTERISTICS 37
Sutured Heegaard Floer homology.
In this subsection, we introduce two equvalent definitions of sutured Heegaard Floer homology.The first one, is due to Juh´asz [Juh06], based on balanced diagrams of balanced sutured manifolds.The other follows from the construction in Section 2.2, which is essentially due to Kronheimer andMrowka [KM10b]. These definitions are denoted by
SF H and
SHF , respectively. The equivalenceof these definitions was shown by Lekili [Lek13], Baldwin and Sivek [BS20b]. We will focus on theequality for graded Euler characteristics of two homologies.
Definition 3.34 ([Juh06]) . A balanced diagram H “ p Σ , α, β q is a tuple satisfying the following.(1) Σ is a compact, oriented surface with boundary.(2) α “ t α , . . . , α n u and β “ t β , . . . , β n u are two sets of pairwise disjoint simple closed curves inthe interior of Σ.(3) The maps π pB Σ q Ñ π p Σ z α q and π pB Σ q Ñ π p Σ z β q are surjective.For such triple, let N be the 3-manifold obtained from Σ ˆ r´ , s by attaching 3–dimensional2–handles along α i ˆ t´ u and β i ˆ t u for i “ , . . . , n and let ν “ B Σ ˆ t u . A balanced diagram p Σ , α, β q is called compatible with a balanced sutured manifold p M, γ q if the sutured manifold p N, ν q is diffeomorphic to p M, γ q .Suppose H “ p Σ , α, β q is a balanced diagram with g “ g p Σ q and n “ | α | “ | β | . Consider two tori T α : “ α ˆ ¨ ¨ ¨ ˆ α n and T β : “ β ˆ ¨ ¨ ¨ ˆ β n in the symmetric product Sym n Σ : “ p n ź i “ Σ q{ S n . The chain complex
SF C p H q is a free F -module generated by intersection points x P T α X T β .Similar to the construction of CF ´ , for a generic path of almost complex structures J s on Sym n Σ,define the differential on
SF C p H q by B J s p x q “ ÿ y P T α X T β ÿ φ P π p x , y q µ p φ q“ x M J s p φ q ¨ y , Theorem 3.35 ([Juh06, JTZ18]) . Suppose p M, γ q is a balanced sutured manifold. Then there is abalanced diagram H compatible with p M, γ q . The groups H p SF C p H q , B J s q for different choices of H and J s , together with some canonical maps, form a transitive system. Let SF H p M, γ q denote thistransitive system and also the associated actual group. Moreover, there is a decomposition SF H p M, γ q “ à s P Spin c p M, B M q SF H p M, γ, s q . Remark . The group
SF H p M, γ q generalizes Heegaard Floer homology [OS04d] and knot Florhomology [OS04b, Ras03]. Suppose Y is a closed 3-manifold and K Ă Y is a knot. Let Y p q beobtained from Y by removing a 3-ball and let δ be a simple closed curve on B Y p q . Let γ consist oftwo meridians of K . Then there are isomorphisms SF H p Y p q , δ q – y HF p Y q , and SF H p Y p K q , γ q – { HF K p Y, K q . Definition 3.37.
For a balanced sutured manifold p M, γ q , let the Z grading of SF H p M, γ q beinduced by the sign of intersection points of T α and T β for some compatible diagram H “ p Σ , α, β q ( c.f. [FJR09, Section 3.4]). Suppose H “ H p M, B M q and choose s P Spin c p M, B M q . The gradedEuler characteristic of SF H p M, γ q is χ p SF H p M, γ qq : “ ÿ s P Spin c p M, B M q s ´ s “ h P H χ p SF H p M, γ, s qq ¨ h P Z r H s{ ˘ H. Theorem 3.38 ([FJR09]) . Suppose p M, γ q is a balanced sutured manifold. Then χ p SF H p M, γ qq “ τ p M, γ q , where τ p M, γ q is a torsion element computed from the map π p R ´ p γ q , pt q Ñ π p M, pt q by Fox calculus. Convention.
The group H p M, B M q may have torsions. To compare the graded Euler characteristicin Definition 3.37 with the one induced by admissible surfaces in Definition 2.40, we also regard χ p SF H p M, γ qq as an element in Z r H s{ ˘ H , which is induced by the composition of maps H “ H p M, B M q – H p M q Ñ H p M q{ Tors “ H . Then we define the second version of sutured Heegaard Floer homology.
Definition 3.39.
Suppose p M, γ q is a balanced sutured manifold and p Y, R q is a closure of p M, γ q as in Definition 2.8. Define SHF p M, γ q : “ HF p Y | R q “ à s P Spin c p Y | R q HF ` p Y, s q . Remark . By work of Kutluhan, Lee, and Taubes [KLT10], for any s P Spin c p Y q , there is anisomorphism HF ` p Y, s q – ~ HM ˚ p Y, s q “ ~ HM ‚ p Y, s q . The last group is used to define
SHM in [KM10b].Following the discussion in Section 2.2, we can prove the naturality of
SHF p M, γ q based on theFloer’s excision theorem. Let SHF p M, γ q be the transitive system corresponding to SHF p M, γ q . Theorem 3.41 ([Lek13, Theorem 24], see also [BS20b, Theorem 3.26]) . Suppose p M, γ q is abalanced sutured manifold and p Y, R q is a closure of p M, γ q . Then there exists a balanced diagram H “ p Σ , α, β q compatible with p M, γ q and a singly-pointed Heegaard diagram H “ p Σ , α , β q sothat the followings hold.(1) Σ is a submanifold of Σ .(2) α and β are subsets of α and β , respectively.(3) Suppose α “ α Y α and β “ β Y β . There exists an intersection point x P T α X T β sothat the map f : SF C p H q Ñ HF ` p H q c ÞÑ c ˆ x is a quasi-isomorphism. Corollary 3.42 (Proposition 1.15) . Suppose p M, γ q is a balanced sutured manifold and H “ H p M, B M q{ Tors . Then we have χ p SF H p M, γ qq “ χ p SHF p M, γ qq P Z r H s{ ˘ H, where χ p SHF p M, γ qq is defined as in Definition 2.40. NSTANTON FLOER HOMOLOGY, SUTURES, AND EULER CHARACTERISTICS 39
Proof.
It suffices to show the quasi-isomorphism in Theorem 3.41 respects spin c structures and Z gradings.Consider the Z gradings at first. Suppose c and c are two generators of SF C p H q . Note thatthe Z grading of c i is defined by the sign of the corresponding intersection point in T α X T β for i “ ,
2. For c i ˆ x , the Z grading is defined by mod 2 Maslov grading, which coincides with thesign of the corresponding intersection point in T α X T β . Thus, we havegr p c q ´ gr p c q “ gr p c ˆ x q ´ gr p c ˆ x q , where gr is the Z grading.Then we consider spin c structures. Consider c i for i “ , γ c ´ γ c such that s p c q ´ s p c q “ PD pr γ c ´ γ c sq , where s p¨q : T α X T β Ñ Spin c p M, B M q is defined in [Juh06, Definition 4.5], and PD : H p M q Ñ H p M, B M q is the Poincar´e duality map.From [OS04d, Lemma 2.19], we have s z p c ˆ x q ´ s z p c ˆ x q “ PD p i ˚ pr γ c ´ γ c sqq , where s z p¨q : T α X T β Ñ Spin c p Y q is defined in [OS04d, Section 2.6] and PD : H p Y q Ñ H p Y q isthe Poincar´e duality map, and i ˚ : H p M q Ñ H p Y q is the map induced by inclusion i : M Ñ Y .Hence we have c p s z p c ˆ x qq ´ c p s z p c ˆ x qq “ p i ˚ pr γ c ´ γ c sqq . Finally, this corollary follows from definitions of graded Euler characteristics. (cid:3) Equivalence of graded Euler characteristics
In this section, we prove the graded Euler characteristic of formal sutured homology is independentof the choice of the Floer-type theory. Throughout the section, we assume that H is a Floer-typetheory, i.e. , it satisfies all three axioms (A1), (A2), and (A3). For simplicity, we say ‘a propertyis independent of H ’ if a property is independent of the choice of the Floer-type theory. Suppose p M, γ q is a balanced sutured manifold and H “ H p M q{ Tors. If the admissible surfaces and theclosure of p M, γ q are fixed, then the graded Euler characteristic χ p SH p M, γ qq in Definition 2.40 isconsidered as a well-defined element in Z r H s , rather than Z r H s{ ˘ H . See Remark 2.41.4.1. Balanced sutured handlebodies.
In this subsection, we deal with Z n gradings for a balanced sutured handlebody. We start withthe following lemma. Lemma 4.1.
Suppose p M, γ q is a balanced sutured manifold, S Ă p
M, γ q is an admissible surface.Suppose p Y , R q and p Y , R q are two closures of p M, γ q of the same genus so that S extends toclosed surfaces ¯ S and ¯ S as in Subsection 2.3. If the graded Euler characteristic of H p Y | R q hasbeen known, then the graded Euler characteristic of H p Y | R q is determined from that of H p Y | R q and the topological data of p Y , R q and p Y , R q .Proof. In Subsection 2.2, we construct a canonical mapΦ : H p Y | R q Ñ H p Y | R q . From the proof of Theorem 2.24, the canonical map Φ necessarily preserves the grading inducedby S , since that is the reason why the grading associated to S on SH p M, γ q is well-defined. From the construction of Φ in Subsection 2.2, the canonical map is a composition of a few cobordismmaps (or the inverse). Then the lemma follows from Axiom (A3-3). (cid:3) Next, we consider gradings associated to embedded disks.
Proposition 4.2.
Suppose H is a genus g ą handlebody and γ Ă B H is a closed oriented 1-submanifold so that p H, γ q is a balanced sutured manifold. Pick D , . . . , D g a set of pairwise disjointmeridian disks in H so that r D s , . . . , r D g s generate H p H, B H q . Then for any fixed multi-grading i “ p i , . . . , i g q P Z g associated to D , . . . , D g , the Euler characteristic χ p SH p´ H, ´ γ, i qq P Z {t˘ u depends only on p H, γ q , D , . . . , D g and i P Z g , and is independent of H . Furthermore, if a particularclosure of p´ H, ´ γ q is fixed, then the sign ambiguity can be resolved.Proof. We can think of H and D , . . . , D g Ă H as fixed while the suture γ may vary. For any suture γ , define I p γ q “ min γ is isotopic to γ g ÿ j “ | D j X γ | , where | ¨ | denotes the number of points. We prove the proposition by induction on I p γ q . Since r γ s “ P H pB H q , we know | D j X γ | is always even for j “ , . . . , g .Note that an isotopy of γ can be understood as combinations of positive and negative stabilizationsin the sense of Definition 2.34, and the grading shifting behavior under such isotopy is described byProposition 2.37, which is determined purely by topological data and is independent of H . Hencewe always assume that the suture γ has already realized I p γ q .First, if I p γ q ă g , then there exists a meridian disk D j with D j X γ “ H . Then it follows fromTheorem 2.31 that SH p´ H, ´ γ q “ ´ H is irreducible while p´ H, ´ γ q is not taut. Hence forany multi-grading i P Z g , we have χ p SH p´ H, ´ γ, i qq “ . If I p γ q “ g , then either there exists some integer j so that D j X γ “ H or for j “ , . . . , g , we have | D j X γ | “ . In the former case, we know that SH p´ H, ´ γ q “ χ p SH p´ H, ´ γ, i qq “ i P Z g . In the later case, we know that p´ H, ´ γ q is a product sutured manifold.It follows from Proposition 2.33 and Proposition 2.29 that SH p´ H, ´ γ q “ SH p´ H, ´ γ, q – F . Hence χ p SH p´ H, ´ γ, i qq “ ˘ i “ “ p , . . . , q i P Z g zt u Note the ambiguity ˘ Y of p´ H, ´ γ q , then the Euler characteristic has no sign ambiguity. Since p H, γ q is a product suturedmanifold, there is a ‘standard’ closure p S ˆ Σ , t u ˆ Σ q as in [KM10b]. By Axiom (A3-2), we have χ p H p S ˆ Σ |t u ˆ Σ qq “ ´ . NSTANTON FLOER HOMOLOGY, SUTURES, AND EULER CHARACTERISTICS 41
Then for any other closure p Y, R q , we can apply Lemma 4.1 and hence SH p Y, R q has a fixed Eulercharacteristic with no sign ambiguity.Now suppose we have proved that, for all γ so that I p γ q ă n , the Euler characteristic of SH p´ H, ´ γ, i q , viewed as an element in Z {t˘ u , is independent of H , and that when we look atany fixed closure of p´ H, ´ γ q , the sign ambiguity can be resolved. Next we deal with the case when I p γ q “ n .Note we have dealt with the base case I p γ q ď g , so we can assume that n ě g `
1. Hence,without loss of generality, we can assume that | D X γ | ě
4. Within a neighborhood of B D , thesuture γ can be depicted as in Figure 12. We can pick the bypass α as shown in the same figure.From Proposition 2.42, for any multi-grading i P Z g , we have an exact triangle(4.1) SH p´ H, ´ γ, i q (cid:40) (cid:40) SH p´ H, ´ γ , i q (cid:54) (cid:54) SH p´ H, ´ γ , i q (cid:111) (cid:111) α B D γ γ γ Figure 12.
The bypass arc α that reduces the intersection function I .Note that the suture γ and γ are determined by the original suture γ and the bypass arc α ,which are all topological data. From Figure 12, it is clear that I p γ q ď I p γ q ´ I p γ q ď I p γ q ´ . Hence the inductive hypothesis applies, and we know that the Euler characteristics of SH p´ H, ´ γ , i q and SH p´ H, ´ γ , i q can be fixed independently of H . Note the maps in the bypass exact triangle (4.1)are described by Proposition 2.20. Hence we conclude that the Euler characteristic of SH p´ H, ´ γ, i q is also independent of H . Thus, we finish the proof by induction. (cid:3) Next, we deal with gradings associated to general admissible surfaces.
Proposition 4.3.
Suppose H is a genus g handlebody, and S is a properly embedded surface in H .Suppose γ Ă B H is a suture so that p H, γ q is a balanced sutured manifold and S is an admissiblesurface. Then the Euler characteristic χ p SH p´ H, ´ γ, S, j qq P Z {t˘ u depends only on p H, γ q , S , and j P Z and is independent of H . Furthermore, if we fix a particularclosure of p´ H, ´ γ q , then the sign ambiguity can also be removed. Before proving the theorem, we need the following lemma.
Lemma 4.4.
Suppose p M, γ q is a balanced sutured manifold and S Ă p
M, γ q is a properly embeddedadmissible surface. Suppose α is a boundary component of S so that α bounds a disk D Ă B M and | α X γ | “ . Let S be the surface obtained by taking the union S Y D and then push D into theinterior of M . Then for any i P Z , we have SH p M, γ, S, i q “ SH p M, γ, S , i q . Proof.
Push the interior of D into the interior of M and make D X S “ H . It is clear that r S s “ r S Y D s P H p M, B M q and B S “ Bp S Y D q . In Subsection 2.3, when constructing the grading associated to S Y D , we can pick a closure p Y, R q of p M, γ q , so that S and D extend to closed surfaces ¯ S and ¯ D in Y , respectively. Since |B D X γ | “ D is a torus. Since B S “ Bp S Y D q , we know that S also extends to a closed surface¯ S and from the fact that r S s “ r S Y D s we know that r ¯ S s “ r ¯ S Y ¯ D s “ r ¯ S s ` r ¯ D s . Since ¯ D is a torus, from Axioms (A1-4) and (A1-6), we know that the decompositions of H p Y | R q with respect to ¯ S and ¯ S are the same. Thus it follows that SH p M, γ, S, i q “ SH p M, γ, S , i q . (cid:3) Proof of Proposition 4.3.
It is a basic fact that the map B ˚ : H p H, B H q Ñ H pB H q is injective, and H p H, B H q is generated by g meridian disks, which we fix as D , . . . , D g . Hence weassume that r S s “ a r D s ` ¨ ¨ ¨ ` a g r D g s P H p H, B H q . Case 1 . B S consists of only of B D i , i.e. , B S “ g ď i “ pY a i B D i q , where Y a i B D i means the union of a i parallel copies of B D i .Then it follows immediately from the construction of the grading and Axiom (A1-6) that SH p´ H, ´ γ, S, j q “ SH p´ H, ´ γ, g ď i “ pY a i D i q , j q“ à j `¨¨¨` j g “ j SH p´ H, ´ γ, p D , . . . , D g q , p j , . . . , j g qq . Hence this case follows from Proposition 4.2.
Case 2 . B S contains some component that is not parallel to B D i for j “ , . . . , g . Step 1 . We modify S and show that it suffices to deal with the case when S X D j “ H for j “ , . . . , g .Note that im pB ˚ q Ă H pB H q is generated by rB D s , . . . , rB D g s , so we have B S ¨ B D i “ j “ , . . . , g . Here ¨ denotes the algebraic intersection number of two oriented curves on B H . Thismeans that for j “ , . . . , g , the intersection points of B D i with B S can be divided into pairs. Supposetwo intersection points of B D with B S of opposite signs are adjacent to each other on B D , as NSTANTON FLOER HOMOLOGY, SUTURES, AND EULER CHARACTERISTICS 43 depicted in Figure 13. We can perform a cut and paste surgery along D and S to obtain a newsurface S . From the same figure, it is clear that after isotopy, we can make |B D X B S | ď |B D X B S | ´ .D S cut andpaste S Figure 13.
The cut and paste surgery on D and S . ´ D S cut andpaste S θ Figure 14.
The cut and paste surgery on ´ D and S .Note also that if we perform a cut and paste surgery along S and ´ D , we obtain anothersurface S . From Figure 14 it is clear that B S “ B S Y θ, where θ is the union of some null-homotopic closed curves on B H . We can isotope S to make eachcomponent of θ intersects the suture twice. Let S be the result of such an isotopy and S be thesurface obtained from S by capping off every component of θ . Note we have r S s “ r S s P H p H, B H q and B S “ B S . Then from Lemma 4.4 we know that SH p´ H, ´ γ, S, j q “ SH p´ H, γ, S , j q“ SH p´ H, γ, S , j q“ SH p´ H, ´ γ, S , j ` j p S , S qq“ à j ` j “ j ` j p S ,S q SH p´ H, ´ γ, p D , S q , p j , j qq Note the shift j p S , S q depends on the isotopy from S to S and by Proposition 2.37, it isdetermined by the topological data and is independent of H . Hence we reduce the problem tounderstanding the Euler characteristic of SH p´ H, ´ γ q with multi-grading associated to p D , S q ,with |B D X B S | ď |B D X B S | ´ . Repeating this argument, we finally reduce to the problem of understanding the Euler characteristicof H p´ H, ´ γ q with multi-grading associated to p D , . . . , D g , S g q , with B D i X B S g “ H for j “ , . . . , g . Step 2 . We modify S further to reduce to Case 1.If every component of B S g is homotopically trivial, then we know that r S g s “ P H p H, B H q since the map H p H, B H q Ñ H pB H q is injective. We isotope each component of B S g by stabilizationto make it intersect the suture γ twice and then cap it off by a disk. The resulting surface S g ` is ahomologically trivial closed surface in H , so SH is totally supported at grading 0 with respect to S g ` . The grading shift between S g and S g ` can then be understood by Proposition 2.37, and isindependent of H .Note that B H zpB D Y ¨ ¨ ¨ Y B D g q is a 2 g -punctured sphere, so B S is homotopically trivial whenremoving punctures on the sphere. If some component C of B S g is not null-homotopic, then C isobtained from some B D j by performing handle slides (or equivalently, band sums) over B D , . . . , B D g for some times.If we isotope C to make it intersect some B D i twice and then apply the cut and paste surgery,the result curve is isotopic to the one obtained by performing a handle slide over B D i . Explicitly,in Figure 13, suppose two right endpoints of arcs in B S (the green arcs) are connected, then theright part of B S is a trivial circle and the left part of B S is obtained from B S by performing ahandle slide over B D . Thus, we can apply the cut and paste surgery for many times, which isequivalent to performing handle slides over B D , . . . , B D g for some times. Finally, we reduce C tothe curve isotopic to B D j . Then we reduce the problem to understanding the Euler characteristic of SH p´ H, ´ γ q with multi-grading associated to p D , . . . , D g , S g ` q , where S g ` is a surface so thateach component of B S g ` is parallel to ˘B D i for some i . Case 1 applies to S g ` and we finish theproof. (cid:3) Corollary 4.5.
Suppose H is a handlebody and γ is a suture on B H so that p H, γ q is a balancedsutured manifold. Suppose S , . . . , S n are properly embedded admissible surfaces in p H, γ q . Then theEuler characteristic χ p SH p´ H, ´ γ, p S , . . . , S n q , p i , . . . , i n qqq P Z {t˘ u depends only on p H, γ q , S , . . . , S n , and p i , . . . , i n q P Z n , and is independent of H . Furthermore, ifwe fix a particular closure of p´ H, ´ γ q , then the sign ambiguity can also be removed.Proof. The proof is similar to that for Proposition 4.3. (cid:3)
Gradings and 2-handle attachments.
Suppose M is a compact oriented 3-manifold with boundary, and S Ă M is a properly embeddedsurface. Suppose α Ă M is a properly embedded arc that intersects S transversely and B α X B S “ H . NSTANTON FLOER HOMOLOGY, SUTURES, AND EULER CHARACTERISTICS 45
Let N “ M z int p N p α qq , S N “ S X N , and µ Ă B N be a meridian of α that is disjoint from S N . Let γ N be a suture on B N satisfies the following properties.(1) p N, γ N q is balanced, S is admissible, and | γ N X µ | “ µ , then we obtain a balanced sutured manifold p M, γ M q .From [BS16a, Section 4.2], there is a map C µ : SHI p´ N, ´ γ N q Ñ SHI p´ M, ´ γ M q constructed as follows.Push µ into the interior of N to become µ . Suppose p N , γ N, q is the manifold obtained from p N, γ N q by a 0-surgery along µ with respect to the framing from B N . Equivalently, p N , γ N, q can be obtained from p M, γ M q by attaching a 1-handle. Since µ Ă int p N q , the construction ofthe closure of p N, γ N q does not affect µ . Thus, we can construct a cobordism between closures of p N, γ N q and p N , γ N, q by attaching a 4-dimensional 2-handle associated to the surgery on µ . Thiscobordism induces a cobordism map C µ : SH p´ N, ´ γ N q Ñ SH p´ N , ´ γ N, q . It is shown in [BS16a, Section 4.2] (or also [KM10b]) that attaching a 1-handle does not change theclosure, so there is an identification ι : SH p´ M, ´ γ M q “ ÝÑ SH p´ N , ´ γ N, q . Thus, we define C µ “ ι ´ ˝ C µ . In this subsection, we prove the following proposition.
Proposition 4.6.
For any i P Z , we have C µ p SH p´ N, ´ γ N , S N , i qq Ă SH p´ M, ´ γ M , S, i q . Proof.
Step 1 . We consider how the map C µ changes the gradings associated to S N and S M .Since µ is disjoint from S , so we can also make µ disjoint from S N “ S X N . As a result,the surface S N survives in p N , γ N, q . From Axiom (A1-7), the cobordism map associated to the0-surgery along µ preserves the grading associated to S N C µ p SH p´ N, ´ γ N , S N , i qq Ă SH p´ N , ´ γ N, , S N , i q . Step 2 . We show ι : SH p´ M, ´ γ M , S, i q “ ÝÑ SH p´ N , ´ γ N, , S, i q . As discussed above, p N , γ N, q is obtained from p M, γ M q by a 1-handle attachment. This one handlecan be described explicitly as follows. In p N , γ N, q , there is an annulus A bounded by µ and itspush-off µ . We can cap off µ by the disk coming from the 0-surgery, and hence obtain a disk D with B D “ µ . By assumption, we know that |B D X γ N, | “ | µ X γ N | “
2. Hence D is a compressingdisk that intersects the suture twice. If we perform a sutured manifold decomposition on p N , γ N, q along D , it is straightforward to check the resulting balanced sutured manifold is p M, γ M q . However,in [Juh16], it is shown that decomposing along such a disk is the inverse operation of attachinga 1-handle, and the disk is precisely the co-core of the 1-handle. From this description, we canconsider the 1-handle attached to p M, γ M q as along two endpoints of α . Since B α X B S “ H , thesurface S naturally becomes a properly embedded surface in p N , γ N, q . From Axiom (A1-7), weknow that the map ι preserves the gradings induced by S . Step 3 . We show that SH p´ N , ´ γ N, , S, i q “ SH p´ N , ´ γ N, , S N , i q . If S X α “ H , then S “ S N “ S X N and the above argument is trivial. If S X α ‰ H , then S N is obtained from S by removing disks containing intersection points in α X S . Then B S N zB S consists of a few copies of meridians of α . For simplicity, we assume that there is only one copy ofthe meridian of α , i.e. , B S N zB S “ µ . The general case is similar to prove.After performing the 0-surgery along µ , we know that the surface S N Ă N is compressible.Indeed, we can pick µ Ă int p S N q parallel to µ Ă B S N . Then there is an annulus A bounded by µ and µ , and we obtain a disk D by capping µ off by the disk coming from the 0-surgery. Performinga compression along the disk D , we know that S N becomes the disjoint union of a disk D and thesurface S Ă N . Note B D is parallel to the disk D discussed above. Since Bp D Y S q “ B S N and r D Y S s “ r S N s P H p N , B N q , From (A1-6), we know that SH p´ N , ´ γ N, , S N , i q “ SH p´ N , ´ γ N, , S Y D , i q“ ÿ i ` i “ i SH p´ N , ´ γ N, , p S, D q , p i , i qq . Since the disk D intersects γ N twice, from term (2) of Proposition 2.29, we know that SH p´ N , ´ γ N, q “ SH p´ N , ´ γ N, , D , q . Hence we conclude that SH p´ N , ´ γ N, , S N , i q “ ÿ i ` i “ i SH p´ N , ´ γ N, , p S, D q , p i , i qq“ SH p´ N , ´ γ N, , S, i q . (cid:3) Remark . Theorem 4.6 is a generalization of [BLY20, Lemma 2.2], where α is a tangle and S N isan annulus.4.3. General balanced sutured manifolds.
In this subsection, we prove our main theorem, which is a restatement of the second part ofTheorem 1.12.
Theorem 4.8.
Suppose p M, γ q is a balanced sutured manifold and t S , . . . , S n u is a collection ofproperly embedded admissible surfaces. Then the Euler characteristic χ p SH p´ M, ´ γ, p S , . . . , S g q , p i , . . . , i n qqq depends only on p M, γ q , S , . . . , S n , and p i , . . . , i n q P Z n , and is independent of H . Corollary 4.9.
Suppose p M, γ q is a balanced sutured manifold and suppose H “ H p M q{ Tors .Then the graded Euler characteristic χ p SH p M, γ qq “ χ p SH g p M, γ qq P Z r H s{ ˘ H is independent of the choice of the fixed genus g of closures. NSTANTON FLOER HOMOLOGY, SUTURES, AND EULER CHARACTERISTICS 47
Proof.
From Corollary 3.42 and Theorem 4.8, we know χ p SH g p M, γ qq “ χ p SF H p M, γ qq P Z r H s{ ˘ H, where the right hand side is independent of the choice of the fixed genus g of closures. (cid:3) Proof of Theorem 4.8.
First we can attach product 1-handles disjoint from S , . . . , S n . From [BS16a,Section 4.2], attaching a 1-handle does not change the closure and hence does not make any differenceto the multi-grading associated to p S , . . . , S n q . Hence we can assume that γ is connected from nowon. From [LY20, Section 3.1], we can pick a disjoint union of properly embedded arcs α “ α Y ¨ ¨ ¨ Y α m so that(1) for k “ , . . . , m , we have B α k X R ` p γ q ‰ H and B α k X R ´ p γ q ‰ H , (2) M z int p N p α qq is a handlebody.Then we apply the arguments involved in [LY20, Section 3.2]: since γ is connected, we can pickpairwise disjoint arcs ζ , . . . , ζ m so that for any k “ , . . . , m, we have B ζ k “ B α k and | ζ k X γ | “ . For any k “ , . . . , g , let β k Ă ζ k be a neighborhood of the intersection point ζ k X γ and let ζ k z β k “ ζ k, ` Y ζ k, ´ , where ζ k, ˘ Ă R ˘ p γ q . Push the interior of β k into the interior of M to make it a properly embeddedarc, which we still call β k . Let β “ β Y ¨ ¨ ¨ Y β m . Let N “ M z int p N p β qq , and let γ N be the disjoint union of γ and a meridian for each componentof β . It is explained in [LY20, Section 3.2] that p N, γ N q can be obtained from p M, γ q by attaching1-handles disjoint from S ,. . . , S m , so there is a canonical identification SH p´ M, ´ γ, p S , . . . , S n q , p i , . . . , i n qq “ SH p´ N, ´ γ N , p S , . . . , S n q , p i , . . . , i n qq Let H “ M z int p N p α Y β qq . It is straightforward to check that H is a handlebody. Let Γ µ be thedisjoint union of γ and a meridian for each component of α Y β . Let the suture Γ be obtainedfrom Γ µ by performing band sums along ζ k, ` and ζ k, ´ for k “ , . . . , m . See Figure 15. It isstraightforward to check that p N, γ N q can be obtained from p H, Γ q by attaching 2-handles alongthe meridians of all components of α .We prove the theorem in the case when m “
1, while the general case follows from a straightforwardinduction. If m “
1, then α is connected. Suppose µ is the meridian of α . As explained in Subsection4.2, attching a 2-handle along µ is the same as performing a 0-surgery along a push-off µ of µ .There is an exact triangle associated to the surgeries along µ that is discussed in [LY20, Section3.2] (see also [GLW19, Section 3.1]):(4.2) SH p´ N, ´ γ N q (cid:40) (cid:40) SH p´ H, ´ Γ q C µ (cid:54) (cid:54) SH p´ H, ´ Γ q (cid:111) (cid:111) The map C µ is the map associated to the 2-handle attachment as discussed in Subsection 4.2. Thesuture Γ is obtained from Γ by twisting along ´ µ once. for j “ , . . . , n , take S j,H “ S j X H . ζ ´ ζ ` N p β q N p α q µ Γ Figure 15.
The suture Γ .Since µ is disjoint from S j,H for j “ , . . . , n , the proof of Proposition 4.6 implies there is a gradedversion of the exact triangle (4.2):(4.3) SH p´ N, ´ γ N , p S , . . . , S n q , p i , . . . , i n qq (cid:15) (cid:15) SH p´ H, ´ Γ , p S ,H , . . . , S n,H q , pp i , . . . , i n qqq C µ (cid:50) (cid:50) SH p´ H, ´ Γ , p S ,H , . . . , S n,H q , pp i , . . . , i n qqq (cid:108) (cid:108) Then Theorem 4.8 follows from Proposition 2.4 and Corollary 4.5. (cid:3) The canonical mod 2 grading
Throughout this section, we focus on special cases of balanced sutured manifolds obtained fromconnected closed 3-manifolds and knots in them ( c.f.
Remark 3.36).
Definition 5.1.
Suppose that Y is a closed 3-manifold and z P Y is a basepoint. Let Y p q beobtained from Y by removing a 3-ball containing z and let δ be a simple closed curve on B Y p q – S .Suppose that K Ă Y is a knot and w is a basepoint on K . Let Y p K q be the knot complement of K and let γ “ m Y p´ m q consist of two meridians with opposite orientations of K near w . Then p Y p q , δ q and p Y p K q , γ q are balanced sutured manifolds. Define r H p Y, z q : “ SH p Y p q , δ q , and KH p Y, K, w q : “ SH p Y p K q , γ q . Convention.
Different choices of the basepoints give isomorphism vector spaces. Since we onlycare about the isomorphism class of the vector spaces, we omit the basepoints and simply write r H p Y q and KH p Y, K q instead. NSTANTON FLOER HOMOLOGY, SUTURES, AND EULER CHARACTERISTICS 49
To be more specific and consistent with [LY20], in this section, we focus on instanton theory.Based on the discussion in Subsection 2.1, we specify the Floer homology H p Y q and the cobordismmap H p W q to be I ω p Y q and I p W, ν q . For a connected closed 3-manifold, the framed instanton Floerhomology I p Y q defined in [KM11] is isomorphic to r H p Y q when H is instanton theory. Hence wereplace r H p Y q by I p Y q throughout this section. Also we replace SH and KH by SHI and
KHI ,respectively. Recall that the definitions of
SHI and
KHI a priori depend on the choice of a fixedand large genus g of closures. We write SHI g and KHI g explicitly in this section. However, for instanton theory, closures of different genus induce isomorphicgroups and we can use closures of genus one to define sutured instanton Floer homology ( c.f. [KM10b,Section 7]).In this section, we discuss the canonical Z grading on KHI g and the decomposition of I inTheorem 1.18. For other Floer-type theory, the construction in [LY20, Section 4.3] can be adaptedwithout essential changes and we have a decomposition for r H p Y q similar to that in Theorem 1.18.The results in this section also apply without essential changes.5.1. The case of the unknot.
In this subsection, we study the model case: the unknot U in S . Suppose µ U and λ U are themeridian and the longitude of U , respectively. The knot complement is identified with a solid torus S ˆ D :(5.1) ρ : S p U q – ÝÑ S ˆ D , where ρ p µ U q “ S ˆ t u , and ρ p λ U q “ t u ˆ B D . For co-prime integers x and y , let γ p x,y q “ γ xλ U ` yµ U Ă B S p U q be the suture consists of two disjoint simple closed curves representing ˘p xλ U ` yµ U q . Convention.
Note γ p x,y q “ γ p´ x, ´ y q . From term (4) in Proposition 2.29, the orientation of thesuture does not influence the isomorphism type of formal sutured homology. Hence we do not careabout the orientation of the suture and we always assume y ě p S p U q , γ p x,y q q as follows.Let Σ be a connected closed surface of genus g ě
1. Suppose Y Σ “ S ˆ Σ and Σ “ t u ˆ Σ. Picka non-separating simple closed curve α Ă Σ and suppose its complement is Y Σ p α q “ Y Σ z int p N p α qq . There is a framing on B Y Σ p α q induced by the surface Σ. Let µ α and λ α be the correspondingmeridian and longitude, respectively. Also, suppose p P Σ is a point disjoint from α . According tothe discussion in Section 2, we can form a closure p ¯ Y , R, ω q of p S p U q , γ p x,y q q as follows:(5.2) ¯ Y “ S p U q Y φ Y Σ p α q , R “ Σ , and ω “ S ˆ t pt u , where φ : B S p U q – ÝÑ B Y p α q is an orientation reversing diffeomorphism such that(5.3) φ p xλ U ` yµ U q “ λ α . Note different choices of the preimage of µ α lead to different closures of p S p U q , γ p x,y q q . From (5.3),we know that φ p λ U q “ zλ α ` yµ α , where z “ x if y “ z “ y is arbitrary if y “
1, and zx ” p mod y q in other cases. Again differentchoices of z lead to different closures. From now on, we fix the value of z as follows: z “ x of y “ z “ y “
1, and z is the minimal positive integer so that y |p xz ´ q . Now, composing φ with theinverse of the map ρ in (5.1), suppose p Y “ Y p α q Y φ ˝ ρ ´ S ˆ D , where φ ˝ ρ ´ : Bp S ˆ D q Ñ B Y p α q is a diffeomorphism such that φ ˝ ρ ´ pt u ˆ B D q “ zλ α ` yµ α . Hence, ¯ Y is obtained from Y by performing a y { z surgery and we also write ¯ Y “ p Y y { z . Lemma 5.2.
For any suture γ p x,y q on B S p U q , we have χ p SHI g p S p U q , γ p x,y q qq “ ˘ y. Proof.
First, we can focus on the closure p ¯ Y “ p Y y { z , R “ Σ , ω q as in (5.2). We need to compute theEuler characteristic of SHI g p S p U q , γ p x,y q q : “ I ω p p Y y { z | Σ q . If y “
0, then x “ ˘
1, but p S p U q , γ p˘ , q q are both irreducible and non-taut. By Theorem 2.31,we know that SHI g p S p U q , γ p , q q “ . If y “
1, then z “ p Y { “ S ˆ Σ . By Axiom (A1-5), we have χ p I ω p p Y { | Σ qq “ ´ . If y ą
1, we have y ą z ě
1. If z “
1, then we have an exact triangle from Axiom (A2) I ω p p Y y ´ | Σ q (cid:47) (cid:47) I ω p p Y y | Σ q f (cid:120) (cid:120) I ω p p Y { | Σ q (cid:102) (cid:102) where the map f is odd and the rest two are even by Proposition 2.4. Hence we conclude byinduction that χ p I ω p p Y y | Σ qq “ ´ y. Finally, when y ą z ą
1, suppose the continued fraction of ´ y { z is ´ yz “ r a , . . . , a n s “ a ´ a ´ ¨¨¨´ an , where a n ď ´
2. Define ´ y z “ r a , . . . , a n ´ s , and ´ y z “ r a , . . . , a n ´ ` s , where y , y ě
0. From a basic property of continued fraction, we have y “ y ` y , and z “ z ` z . NSTANTON FLOER HOMOLOGY, SUTURES, AND EULER CHARACTERISTICS 51
From Axiom (A2), there exists an exact triangle I ω p p Y y { z | Σ q (cid:47) (cid:47) I ω p p Y y { z | Σ q (cid:120) (cid:120) I ω p p Y y { z | Σ q f (cid:103) (cid:103) where the map f is odd and the other two are even by Proposition 2.4. (cid:3) Remark . It is worth mentioning that different papers have different normalization for thecanonical Z grading. Our choice of normalization in Axiom (A3) is the same as in [KM10a]. InLidman, Pinz´on-Caicedo, and Scaduto’s setup [LPCS20], they adapted another normalization andproved χ p I ω p S ˆ Σ | Σ qq “ Corollary 5.4.
Suppose p Y , R , ω q is a closure of p S p U q , γ p x,y q q , then χ p I p Y | R qq “ ˘ y. Proof.
This corollary follows directly from the fact that canonical maps from I ω p p Y | Σ q to I ω p Y | R q is a composition of cobordism maps and hence is homogeneous. (cid:3) Sutured knot complements.
Suppose Y is a closed 3-manifold and K Ă Y is a null-homologous knot. Any Seifert surface S of K gives rise to a framing on B Y p K q : the meridian µ can be picked as the meridian of the solidtorus N p K q , and the longitude λ can be picked as S X B Y p K q . The ‘half lives and half dies’ fact for3-manifolds implies that the following map has a 1-dimensional image: B ˚ : H p Y p K q , B Y p K q ; Q q Ñ H pB Y p K q ; Q q . Hence any two Seifert surfaces lead to the same framing on B Y p K q . Definition 5.5.
The framing p λ, ´ µ q defined as above is called the canonical framing of p Y, K q .With this canonical framing, let γ p x,y q “ γ xλ ` yµ Ă B Y p K q be the suture consists of two disjoint simple closed curves representing ˘p xλ ` yµ q .Our goal in this subsection is to define a canonical Z grading on SHI g p Y p K q , γ p x,y q q for anyfixed large enough g . Recall SHI g p M, γ q is the projective transitive system formed by closures of p M, γ q of a fixed genus g . We first assign a Z grading for any closure of p Y p K q , γ p x,y q q .Suppose p ¯ Y , R, ω q is a closure of p Y p K q , γ p x,y q q . Then we can form a closure p ¯ Y U , R, ω q of p S p U q , γ p x,y q q by taking(5.4) ¯ Y U “ ¯ Y zp int p Y p K qqq Y id S p U q . Here id is the diffeomorphism between toroidal boundaries which respect the canonical framings onboth boundaries.
Definition 5.6.
The modified Z grading on I ω p ¯ Y | R q is defined as follows. ‚ If χ p I ω p ¯ Y U | R qq “ is negative, then it is the same as the canonical Z grading on I ω p ¯ Y | R q . ‚ If χ p I ω p ¯ Y U | R qq is positive, then it is the same as the canonical Z grading on I ω p ¯ Y | R q witheven and odd part switched. Suppose p ¯ Y , R, ω q and p ¯ Y , R, ω q are two closures of p Y p K q , γ p x,y q q so that ¯ Y is obtained from ¯ Y by a Dehn surgery along a curve β Ă ¯ Y , which is disjoint from int p M q , R and ω . Then there is amap F : I ω p ¯ Y | R q Ñ I ω p ¯ Y | R q associated to the Dehn surgery along β Ă ¯ Y . Let p ¯ Y U , R, ω q and p ¯ Y U , R, ω q be the closures of p S p U q , γ p x,y q q constructed as in (5.4). There is also a map F U : I ω p ¯ Y U | R q Ñ I ω p ¯ Y U | R q associated to the same Dehn surgery along β Ă ¯ Y U . Then we have the following. Lemma 5.7.
The maps F and F U have the same parity with respect to the canonical Z gradingson corresponding instanton Floer homologies.Proof. Note that H p S p U q ; Q q – Q x µ U y and the map i U ˚ : H pB S p U q ; Q q Ñ H p S p U q ; Q q induced by the inclusion has a 1-dimensional kernel generated by λ U . For a null-homologous knot K Ă Y , we know that the map i ˚ : H pB Y p K q ; Q q Ñ H p Y p K q ; Q q induced by the inclusion has a 1-dimensional kernel generated by the longitude λ of K and has a1-dimensional image generated by the meridian µ of K . Hence, from the Mayer-Vietoris sequence,we know that there is an injective map j : H p ¯ Y U ; Q q ã Ñ H p ¯ Y ; Q q , that sends r µ U s to r µ s and sends every homology class in ¯ Y U z S p U q “ ¯ Y z Y p K q using the naturalmap i : H p ¯ Y z Y p K q ; Q q Ñ H p ¯ Y ; Q q . Similarly, since β X int p Y p K qq “ H , we know that there is an injective map j β : H p ¯ Y U p β q ; Q q ã Ñ H p ¯ Y p β q ; Q q , which fits into the following commutative diagram H pB ¯ Y U p β q ; Q q ι U ˚ (cid:47) (cid:47) “ (cid:15) (cid:15) H p ¯ Y U p β q ; Q q j β (cid:15) (cid:15) H pB ¯ Y p β q ; Q q ι ˚ (cid:47) (cid:47) H p ¯ Y p β q ; Q q where ι ˚ and ι U ˚ are induced by natural inclusions. Hence we conclude, under the identification H pB ¯ Y U p β q ; Q q “ H pB ¯ Y p β q ; Q q , two kernels are also identified:ker p ι U ˚ q “ ker p ι ˚ q . Since F and F U are associated to Dehn surgeries along β of the same slopes, we conclude from 2.4that their parity must be the same. (cid:3) NSTANTON FLOER HOMOLOGY, SUTURES, AND EULER CHARACTERISTICS 53
Lemma 5.8.
Suppose p ¯ Y , R, ω q and p ¯ Y , R , ω q are two different closures of p Y p K q , γ p x,y q q so that g p R q “ g p R q . There is a canonical isomorphism Φ : I ω p ¯ Y | R q – ÝÑ I ω p ¯ Y | R q as in Definition 2.10. Then with respect to the modified Z grading as in Definition 5.6, the canonicalmap Φ is homogeneous of degree zero.Proof. From the definition of Φ and Axiom (A3-3), we know that Φ is always homogeneous. Fortwo closures p ¯ Y , R, ω q and p ¯ Y , R , ω q of p Y p K q , γ p x,y q q , we can form two corresponding closures p ¯ Y U , R, ω q and p ¯ Y U , R , ω q of p S p U q , γ p x,y q q as in (5.2), respectively. There is a canonical mapΦ U : I ω p ¯ Y U | R q – ÝÑ I ω p ¯ Y U | R q . From Definition 5.6, the modified Z gradings on I ω p ¯ Y | R q and I ω p ¯ Y | R q coincide if and only ifthe canonical Z gradings on I ω p ¯ Y U | R q and I ω p ¯ Y U | R q coincide. The definition of the modified Z grading automatically makes Φ U even under the modified Z grading. Hence to show that Φ is even,it suffices to show that Φ and Φ U have the same parity under the modified Z grading.From the construction of the canonical maps, there is a sequence of simple closed curves β , . . . , β n on R , such that the map Φ is the composition of cobordism maps induced by a diffeomorphismand the sequence of Dehn surgeries. Similarly, the map Φ U is the composition of the maps inducedanother diffeomorphism and the same sequence of Dehn surgeries on ¯ Y U . Since the surgery curves areall on R and disjoint from int p Y p K qq , cobordism maps induced by diffeomorphisms are always witheven degrees. For Dehn surgeries along β i , we can apply Lemma 5.7 and then finish the proof. (cid:3) Corollary 5.9.
Suppose p Y p K q , γ p x,y q q is the balanced sutured manifold constructed as beforeand g is a fixed large enough genus of closures, there is a well-defined absolute Z grading on SHI g p Y p K q , γ p x,y q q . It is called the canonical Z grading. In particular, we can define thecanonical Z grading on KHI g p Y, K q .Proof. If we fix a genus g , then the fact that SHI g and KHI g admit well-defined Z gradings followsfrom above discussions. Now we want to compare the Z grading for closures of different genuses.First we deal with the case of unknot. As in Subsection 5.1, we can constructed a standard closure p ¯ Y y { z , Σ q for p Y p K q , γ p x,y q q . Here, the genus of Σ can be arbitrary. To specify the genus, in theproof of this corollary, we temporarily write p ¯ Y y { z , Σ q as p ¯ Y gy { z , Σ g q . In [BS15], the canonical mapΦ g,g ` : I ω p ¯ Y gy { z | Σ g q Ñ I ω p ¯ Y g ` y { z | Σ g ` q is constructed as follows: Recall that we have α Ă Σ g being a non-separating simple closed curve,and ¯ Y gy { z “ Y p K q Y p S ˆ Σ g qz N p α q . Now pick β Ă Σ g be another simple closed curve so that α X β “ H . Take a curve θ Ă Σ benon-separating simple closed curve as well. Then we can form p ¯ Y g ` y { z , Σ g ` q from p ¯ Y gy { z , Σ g q and p S ˆ Σ , Σ q by cutting them open along S ˆ β and S ˆ θ respectively, and then glue the twopieces together along toroidal boundaries by the identifications S “ S and β “ θ . Then, as in[KM10b], there is a cobordism W g,g ` from p ¯ Y gy { z , Σ g q and p S ˆ Σ , Σ q to p ¯ Y g ` y { z , Σ g ` q , knownas the Floer excision cobordism. In the proof of [LPCS20, Proposition 3.6] and [KM10a, Lemma3.3], the degree of the cobordism induced by cobordisms constructed in the same way as W g,g ` has been computed explicitly. By a similar computation, we see that the degree of canonical mapΦ g,g ` is 0 with respect to the modified Z grading. (Note by the above argument, the modified Z
24 ZHENKUN LI AND FAN YE grading for p ¯ Y gy { z , Σ g q is the same as the canonical grading.) For any two closures of p Y p K q , γ p x,y q q ,as in [BS15], the canonical maps are constructed by composing the maps induced by some W g,g ` and canonical maps for closures of the same genuses. Since both types have been shown to be ofdegree 0, we know that any canonical map is of degree 0 and hence we are done. (cid:3) In the rest of this subsection, we deal with bypass exact triangles. Suppose y { x is a surgeryslope with y ě
0. According to Honda [Hon00], there are two basic bypasses on the balancedsutured manifold p Y p K q , γ p x ,y q q , whose arcs are depicted as in Figure 16. The sutures involved inthe bypass triangles were described explicitly in Honda [Hon00]. Figure 16.
Bypass arcs on γ p , ´ q . Definition 5.10.
For a surgery slope y { x with y ě
0, suppose its continued fraction is y x “ r a , a , . . . , a n s “ a ´ a ´ ¨¨¨´ an , where a i ď ´
2. If y ą ´ x ą
0, let y x “ r a , . . . , a n ´ s and y x “ r a , . . . , a n ` s . If ´ x ą y ą
0, we do the same thing for x {p´ y q . If y ą x ą
0, we do the same thing for y {p´ x q . If x ą y ą
0, we do the same thing for x {p´ y q . If y { x “ {
0, then y { x “ { y { x “ {p´ q . If y { x “ {
1, let y { x “ {p´ q and y { x “ {
1. We always require that y ě y ě Remark . It is straightforward to use induction to verify that for y ą ´ x ą x “ x ` x , and y “ y ` y . Then the bypass exact triangle in Theorem 2.19 becomes the following.
Proposition 5.12.
Suppose K Ă Y is a null-homologous knot, and suppose the surgery slopes y i { x i for i “ , , are defined as in Definition 5.10. Suppose ψ ˚` , ˚ and ψ ˚´ , ˚ are from two differentbypasses, where ˚ means the corresponding slope. Then there are two exact triangles about ψ ˚` , ˚ and NSTANTON FLOER HOMOLOGY, SUTURES, AND EULER CHARACTERISTICS 55 ψ ˚´ , ˚ , respectively. SHI g p´ Y p K q , ´ γ p x ,y q q ψ y { x ˘ ,y { x (cid:47) (cid:47) SHI g p´ Y p K q , ´ γ p x ,y q q ψ y { x ˘ ,y { x (cid:116) (cid:116) SHI g p´ Y p K q , ´ γ p x ,y q q ψ y { x ˘ ,y { x (cid:106) (cid:106) As stated in Proposition 2.20, the bypass maps ψ , ψ , and ψ are induced by some cobordismmaps. Then we have the following. Lemma 5.13.
Suppose g is a large enough integer and suppose y i { x i for i “ , , is from Definition5.10. Suppose further x “ x ` x , and y “ y ` y . With respect to the well-defined Z grading on SHI g in Corollary 5.9, the map ψ is odd and therest two are even. As a consequence, χ p SHI g p´ Y p K q , ´ γ p x ,y q qq “ χ p SHI g p´ Y p K q , ´ γ p x ,y q qq ` χ p SHI g p´ Y p K q , ´ γ p x ,y q qq . Proof.
As in Proposition 2.20, we can fix a large enough g so that for i “ , ,
3, there are closures p ¯ Y i , R, ω q for p´ Y p K q , ´ γ p x i ,y i q q of genus g , and the bypass maps ψ , ψ , and ψ have the same Z degrees as the maps induced by Dehn surgeries along three curves ζ , ζ , and ζ in correspondingclosures ¯ Y i z int p Y p K qq . Since we only care about the Z degrees of maps, in a slight abuse of notation,we do not distinguish the bypass map and the map induced by Dehn surgery.For i “ , ,
3, we can form corresponding closures p ¯ Y Ui , R, ω q as in (5.2) so that the curves ζ , ζ , and ζ remains and suitable surgeries along them induces an exact triangle SHI g p´ S p U q , ´ γ p x ,y q q ψ U (cid:47) (cid:47) SHI g p´ S p U q , ´ γ p x ,y q q ψ U (cid:116) (cid:116) SHI g p´ S p U q , ´ γ p x ,y q q ψ U (cid:106) (cid:106) As in the proof of Lemma 5.8, with the help of Lemma 5.7, it suffices to check that the maps ψ U , ψ U , and ψ U are odd or even as desired, with respect to the Z grading on SHI g from Corollary 5.9.For the case of the unknot, the argument becomes straightforward: from Definition 5.6 andLemma 5.2, we know that for any y ą χ p SHI g p´ S p U q , ´ γ p x,y q qq “ ´ y Then the equation y “ y ` y implies that χ p SHI g p´ S p U q , ´ γ p x ,y q qq “ χ p SHI g p´ S p U q , ´ γ p x ,y q qq ` χ p SHI g p´ S p U q , ´ γ p x ,y q qq . Note the maps ψ Ui for i “ , , SHI g could possibly be shifted due to the normalization in Definition5.6 and the surgery along curves η and η as in Proposition 2.20. Hence they still satisfies thehypothesis of Lemma 2.5. Thus, we conclude that ψ U is odd and the other two are even. Similarly, ψ is odd and the other two are even, and we have χ p SHI g p´ Y p K q , ´ γ p x ,y q qq “ χ p SHI g p´ Y p K q , ´ γ p x ,y q qq ` χ p SHI g p´ Y p K q , ´ γ p x ,y q qq . (cid:3) Computations and applications.
Let Y be a closed 3-manifold and let K Ă Y be a null-homologous knot. Suppose S is a minimalgenus Seifert surface of K . Its genus is always denoted by g p S q , which is distinguished with g , thefixed genus of closures. We refer [LY20, Section 4] for the definitions of sutures Γ n , Γ n p y { x q , theadmissible surface with stablization S τ , the bypass maps ψ ˚` , ˚ , ψ ˚´ , ˚ , and numbers i ˚ max , i ˚ min . Tosimplify our notation, we write(5.5) χ gy { x p´ Y, K, i q “ χ p SHI g p´ Y p K q , ´ γ p x,y q , S τ , i qq , where the Euler characteristic is with respect to the canonical Z grading on SHI g as in Corollary5.9. We write(5.6) χ gy { x p´ Y, K q “ ÿ i P Z χ gy { x p´ Y, K, i q When | x | “
1, we write y { x as an integer. Also, we write χ gµ p´ Y, K, i q “ χ g { p´ Y, K, i q to specify the meridional suture. Lemma 5.14.
Suppose Y is a closed oriented 3-manifold, and K Ă Y is a null-homologous knot.For g P Z large enough and any i P Z , we have χ g p´ Y, K, i q “ χ gµ p´ Y, K, i q , and χ g p´ Y, K, i q “ . Proof.
From [LY20, Proposition 4.16], we have the following two bypass exact triangles:
SHI g p´ Y p K q , ´ Γ , S τ , i q ψ ´ , (cid:47) (cid:47) SHI g p´ Y p K q , ´ Γ , S τ , i q ψ ´ ,µ (cid:115) (cid:115) SHI g p´ Y p K q , ´ Γ , S τ , i q ψ µ ´ , (cid:79) (cid:79) and SHI g p´ Y p K q , ´ Γ , S τ , i ` q ψ ` , (cid:47) (cid:47) SHI g p´ Y p K q , ´ Γ , S τ , i q ψ ` ,µ (cid:115) (cid:115) SHI g p´ Y p K q , ´ Γ , S τ , i q ψ µ ` , (cid:79) (cid:79) Hence we obtain the following two equations from Lemma 5.13 χ g p´ Y, K, i q “ χ g p´ Y, K, i q ` χ gµ p´ Y, K, i q χ g p´ Y, K, i q “ χ g p´ Y, K, i ` q ` χ gµ p´ Y, K, i q By Axiom (A1-4), for i ą g p S q , we have SHI g p´ Y p K q , ´ Γ , S τ , i q “ . Hence we conclude by (5.5) and (5.6) that χ g p´ Y, K, g p S qq “ χ gµ p´ Y, K, g p S qq , and χ g p´ Y, K, g p S qq “ . The lemma follows from the induction on the grading i . (cid:3) NSTANTON FLOER HOMOLOGY, SUTURES, AND EULER CHARACTERISTICS 57
Lemma 5.15.
Suppose Y is a closed oriented 3-manifold, and K Ă Y is a null-homologous knot.For the suture γ p x,y q with y ą , we know that χ gy { x p´ Y, K, i q “ y ´ ÿ j “ χ gµ p´ Y, K, i ´ i ymax ` i µmax ` j q . Proof.
We only prove the case when x ă
0. The other case is similar. First, if x “
1, then we havea bypass exact triangle (in this case we write y “ n ) SHI g p´ Y p K q , ´ Γ n , S τ , i q ψ n ´ ,n ` (cid:47) (cid:47) SHI g p´ Y p K q , ´ Γ n ` , S τ , i q ψ n ` ´ ,µ (cid:114) (cid:114) SHI g p´ Y p K q , ´ Γ µ , S τ , i ` q ψ µ ´ ,n (cid:79) (cid:79) Hence, we can apply Lemma 5.13, Lemma 5.14, and the induction to conclude that χ gn p´ Y, K, i q “ n ´ ÿ j “ χ gµ p´ Y, K, i ´ i nmax ` i µmax ` j q . If x ą
1, we can consult to the continued fraction description of y { x and apply an induction inthe same spirit as in the proof of Lemma 5.2. (cid:3) Corollary 5.16.
Suppose Y is a closed oriented 3-manifold, and K Ă Y is a null-homologous knot.For the suture γ p x,y q where y ą g p S q , and for any i P Z so that i ymax ´ g p S q ě i ě i ymin ` g p S q , we know that χ gy { x p´ Y, K, i q “ χ gµ p´ Y, K q . Proof.
The corollary follows immediately from Lemma 5.15 and the fact that there are only p g p S q` q gradings with nontrivial elements for SHI g p´ Y p K q , ´ Γ µ , S τ q . (cid:3) Lemma 5.17.
Suppose Y is a closed oriented 3-manifold, and K Ă Y is a null-homologous knot.Then we have χ gµ p´ Y, K q “ ˘ χ p I p´ Y qq . Proof.
From Lemma 5.15, we know that for any n P Z ě , we have χ gn p´ Y, K q “ n ¨ χ gµ p´ Y, K q . From [LY20, Lemma 4.11], we know that there is an exact triangle
SHI g p´ Y p K q , ´ Γ n q (cid:47) (cid:47) SHI g p´ Y p K q , ´ Γ n ` q (cid:118) (cid:118) I p´ Y q (cid:104) (cid:104) Hence, by Lemma 2.5, we know that there is a proper sign assignment for all n so that ˘ χ gn p´ Y, K q ˘ χ gn ` p´ Y, K q ˘ χ p I p Y qq “ . Hence the only possibilities are χ p I p Y qq “ ˘ χ gµ p´ Y, K q . (cid:3) Proof of Proposition 1.19.
It is an immediate corollary following Corollary 5.16, Lemma 5.17 andthe definition of the decomposition from [LY20, Section 4.3]. (cid:3)
For a knot K in S , we can actually fix the sign ambiguity coming from different choices of thefixed genus of the closures. Lemma 5.18.
For any knot K Ă S and any positive integer g , we have χ gµ p´ S , K q “ ´ . Proof.
Since we adapt the normalization from Kronheimer and Mrowka [KM10a, Section 2.6], wecan directly apply the results from them. In particular, for any knot K , in [KM10a, Section 2.4], apreferred closure p ¯ Y , Σ , ω q of p´ S p K q , ´ Γ µ q with g p Σ q “ χ p I ω p ¯ Y | Σ qq “ ´ ∆ K p q “ ´ . Note this coincide with our choice of modified Z grading: when the Euler characteristic is negative,we do not perform any shifts.The case g “ p ¯ Y , Σ , ω q with g p Σ q “ p´ S p K q , ´ Γ µ q is constructed, and there is a cobordism W from ¯ Y Y S ˆ Σ to ¯ Y coming from the Floer’s excision theorem. The canonical generatorof I ω p S ˆ Σ | Σ q is proved to be at the odd grading ( c.f. [LPCS20, Lemma 3.8], though thenormalization of the canonical Z grading is different). Moreover, the degree of the cobordism map W is odd ( c.f. [LPCS20, Proposition 3.6]).For general g , it is straightforward to generalize the above construction for ¯ Y and ¯ Y to ¯ Y g and¯ Y g ` . There is a similar cobordism W g from ¯ Y g Y S ˆ Σ to ¯ Y g ` , the degree of which can becomputed easily to be odd. Hence by induction we conclude that for all g , χ gµ p´ S , K q “ ´ . (cid:3) By Lemma 5.18, we can identify χ gµ p´ S , K q for all large enough g , we simply write χ µ p´ S , K q “ χ gµ p´ S , K q instead. Applying Lemma 5.15, we know that for any g large enough and y ą χ gy { x p´ S , K q “ ´ y. Similarly, we simply write χ y { x p´ S , K q instead.Finally, we consider the projectively system SHI p M, γ q for a balanced sutured manifold p M, γ q defined in [BS15], which is independent of the choice of the genus of the closures. The isomorphismclass of SHI p M, γ q and SHI g p M, γ q are the same. Similar to SHI g p M, γ q , it has a decompositionassociated to an admissible surface S Ă p
M, γ q . Definition 5.19.
Suppose p M, γ q is a balanced sutured manifold and S is an admissible surface in p M, γ q . For any i, j P Z , defineSHI p M, γ, S, i qr j s “ SHI p M, γ, S, i ´ j q . In [Li19, Section 5], the first author constructed a minus version of the instanton knot homologyvia the direct system(5.7) ¨ ¨ ¨ Ñ
SHI p´ S p K q , Γ n , S τ qr g p K q ´ i nmax s ψ n ´ ,n ` ÝÝÝÝÝÑ
SHI p´ S p K q , Γ n ` , S τ qr g p K q ´ i n ` max s Ñ ¨ ¨ ¨ NSTANTON FLOER HOMOLOGY, SUTURES, AND EULER CHARACTERISTICS 59 and define KHI ´ p´ S , K q to be the direct limit of (5.7). All ψ n ´ ,n ` are grading preserving aftershifting, so there is a well-defined Z -grading (the Alexander grading) on KHI ´ p´ S , K q , which wewrite as KHI ´ p´ S , K, i q . By [Li19, Corollary 2.20], there is a commutative diagramSHI p´ S p K q , Γ n , S τ qr g p K q ´ i nmax s ψ n ´ ,n ` (cid:47) (cid:47) ψ n ` ,n ` (cid:15) (cid:15) SHI p´ S p K q , Γ n ` , S τ qr g p K q ´ i n ` max s ψ n ` ` ,n ` (cid:15) (cid:15) SHI p´ S p K q , Γ n ` , S τ qr g p K q ´ i n ` max s ψ n ` ´ ,n ` (cid:47) (cid:47) SHI p´ S p K q , Γ n ` , S τ qr g p K q ´ i n ` max s Hence the maps t ψ n ` ,n ` u induces an map U on KHI ´ p´ S , K q . Proof of Proposition 1.21.
From Lemma 5.13, the maps ψ n ´ ,n ` are all even, hence there is a well-defined Z grading on KHI ´ p´ S , K q . Again from Lemma 5.13, we know that U is even, i.e. ,preserving the Z grading on KHI ´ p´ S , K q . Finally, we can apply Lemma 5.15 and the fact χ p KHI g p´ S , K qq “ ´ ∆ K p t q to conclude the desired formula. Note that by our the normalization, the sign is negative. (cid:3) References [BLY20] John A. Baldwin, Zhenkun Li, and Fan Ye. Sutured instanton homology and Heegaarddiagrams.
ArXiv: 2011.09424, v1 , 2020.[BS15] John A. Baldwin and Steven Sivek. Naturality in sutured monopole and instantonhomology.
J. Differ. Geom. , 100(3):395–480, 2015.[BS16a] John A. Baldwin and Steven Sivek. A contact invariant in sutured monopole homology.
Forum Math. Sigma , 4:e12, 82, 2016.[BS16b] John A. Baldwin and Steven Sivek. Instanton Floer homology and contact structures.
Selecta Math. (N.S.) , 22(2):939–978, 2016.[BS18] John A. Baldwin and Steven Sivek. Khovanov homology detects the trefoils.
ArXiv:1801.07634, v1 , 2018.[BS20a] John A. Baldwin and Steven Sivek. Framed instanton homology and concordance.
ArXiv:2004.08699, v2 , 2020.[BS20b] John A. Baldwin and Steven Sivek. On the equivalence of contact invariants in suturedFloer homology theories.
ArXiv:1601.04973, v3 , 2020.[FJR09] Stefan Friedl, Andr´as Juh´asz, and Jacob Rasmussen. The decategorification of suturedFloer homology.
J. Topol. , 4(2):431–478, 2009.[Flo88] Andreas Floer. An instanton-invariant for 3-manifolds.
Comm. Math. Phys. , 118(2):215–240, 1988.[Flo90] Andreas Floer. Instanton homology, surgery, and knots. In
Geometry of low-dimensionalmanifolds, 1 (Durham, 1989) , volume 150 of
London Math. Soc. Lecture Note Ser. , pages97–114. Cambridge Univ. Press, Cambridge, 1990.[Fre] Jesse Freeman. The surgery exact triangle in monopole Floer homology with Z r i s coefficients and applications. In preparation.[Gab83] David Gabai. Foliations and the topology of 3-manifolds. J. Differ. Geom. , 18(3):445–503,1983. [Gar19] Mike Gartner. Projective naturality in Heegaard Floer homology.
ArXiv: 1908.06237, v1 ,2019.[Ghi08] Paolo Ghiggini. Knot Floer homology detects genus-one fibred knots.
Amer. J. Math. ,130(5):1151–1169, 2008.[GL16] Joshua Evan Greene and Adam Simon Levine. Strong Heegaard diagrams and strongL–spaces.
Algebr. Geom. Topol. , 16(6):3167–3208, 2016.[GL19] Sudipta Ghosh and Zhenkun Li. Decomposing sutured monopole and instanton Floerhomologies.
ArXiv:1910.10842, v2 , 2019.[GLV18] Joshua Evan Greene, Sam Lewallen, and Faramarz Vafaee. (1,1) L-space knots.
Compos.Math. , 154(5):918–933, 2018.[GLW19] Sudipta Ghosh, Zhenkun Li, and C.-M. Michael Wong. Tau invariants in monopole andinstanton theories.
ArXiv:1910.01758, v3 , 2019.[Gre13] Joshua Evan Greene. A spanning tree model for the Heegaard Floer homology of abranched double-cover.
J. Topol. , 6(2):525–567, 2013.[HMZ18] Kristen Hendricks, Ciprian Manolescu, and Ian Zemke. A connected sum formula forinvolutive Heegaard Floer homology.
Sel. Math. New Ser. , 24(2):1183–1245, 2018.[Hon00] Ko Honda. On the classification of tight contact structures I.
Geom. Topol. , 4:309–368,2000.[JTZ18] Andr´as Juh´asz, Dylan P. Thurston, and Ian Zemke. Naturality and mapping class groupsin Heegaard Floer homology.
ArXiv:1210.4996, v4 , 2018.[Juh06] Andr´as Juh´asz. Holomorphic discs and sutured manifolds.
Algebr. Geom. Topol. , 6:1429–1457, 2006.[Juh08] Andr´as Juh´asz. Floer homology and surface decompositions.
Geom. Topol. , 12(1):299–350,2008.[Juh16] Andr´as Juh´asz. Cobordisms of sutured manifolds and the functoriality of link Floerhomology.
Adv. Math. , 299:940–1038, 2016.[Kav19] Nithin Kavi. Cutting and gluing surfaces.
ArXiv:1910.11954, v1 , 2019.[KLT10] C¸ a˘gatay Kutluhan, Yi-Jen Lee, and Clifford Henry Taubes. HF=HM I: Heegaard Floerhomology and Seiberg-Witten Floer homology.
ArXiv:1007.1979, v1 , 2010.[KM07] Peter B. Kronheimer and Tomasz S. Mrowka.
Monopoles and three-manifolds , volume 10of
New Mathematical Monographs . Cambridge University Press, Cambridge, 2007.[KM10a] Peter B. Kronheimer and Tomasz S. Mrowka. Instanton Floer homology and the Alexanderpolynomial.
Algebr. Geom. Topol. , 10(3):1715–1738, 2010.[KM10b] Peter B. Kronheimer and Tomasz S. Mrowka. Knots, sutures, and excision.
J. Differ.Geom. , 84(2):301–364, 2010.[KM11] Peter B. Kronheimer and Tomasz S. Mrowka. Khovanov homology is an unknot-detector.
Publ. Math. Inst. Hautes ´Etudes Sci. , 113:97–208, 2011.[Lek13] Yankı Lekili. Heegaard-Floer homology of broken fibrations over the circle.
Adv. Math. ,244:268–302, 2013.[Li18] Zhenkun Li. Gluing maps and cobordism maps for sutured monopole Floer homology.
ArXiv:1810.13071, v3 , 2018.[Li19] Zhenkun Li. Knot homologies in monopole and instanton theories via sutures.
ArXiv:1901.06679, v6 , 2019.[Lim09] Yuhan Lim. Instanton homology and the Alexander polynomial.
ArXiv:0907.4185, v2 ,2009.
NSTANTON FLOER HOMOLOGY, SUTURES, AND EULER CHARACTERISTICS 61 [LPCS20] Tye Lidman, Juanita Pinz´on-Caicedo, and Christopher Scaduto. Framed instantonhomology of surgeries on L-space knots.
ArXiv:2003.03329, v1 , 2020.[LY20] Zhenkun Li and Fan Ye. Instanton Floer homology, sutures, and Heegaard diagrams.
ArXiv: 2010.07836, v2 , 2020.[MO17] Ciprian Manolescu and Peter S. Ozsvath. Heegaard Floer homology and integer surgerieson links.
ArXiv: 1011.1317, v4 , 2017.[Ni07] Yi Ni. Knot Floer homology detects fibred knots.
Invent. Math. , 170(3):577–608, 2007.[OS03] Peter S. Ozsv´ath and Zolt´an Szab´o. Heegaard Floer homology and alternating knots.
Geom. Topol. , 7(1):225–254, 2003.[OS04a] Peter Ozsv´ath and Zolt´an Szab´o. Holomorphic triangle invariants and the topology ofsymplectic four-manifolds.
Duke Math. J. , 121(1):1–34, 2004.[OS04b] Peter S. Ozsv´ath and Zolt´an Szab´o. Holomorphic disks and knot invariants.
Adv. Math. ,186(1):58–116, 2004.[OS04c] Peter S. Ozsv´ath and Zolt´an Szab´o. Holomorphic disks and three-manifold invariants:Properties and applications.
Ann. of Math. , 159:1159–1245, 2004.[OS04d] Peter S. Ozsv´ath and Zolt´an Szab´o. Holomorphic disks and topological invariants forclosed three-manifolds.
Ann. of Math. (2) , 159(3):1027–1158, 2004.[OS06] Peter S. Ozsv´ath and Zolt´an Szab´o. Holomorphic triangles and invariants for smoothfour-manifolds.
Adv. Math. , 202:326–400, 2006.[OS08] Peter S. Ozsv´ath and Zolt´an Szab´o. Holomorphic disks, link invariants and the multi-variable Alexander polynomial.
Algebr. Geom. Topol. , 8(2):615–692, 2008.[Ras03] Jacob Rasmussen. Floer homology and knot complements.
ArXiv:math/0306378, v1 ,2003.[Ras07] Jacob Rasmussen. Lens space surgeries and L-space homology spheres.
ArXiv:0710.2531,v1 , 2007.[RR17] Jacob Rasmussen and Sarah Dean Rasmussen. Floer simple manifolds and L-spaceintervals.
Adv. Math. , 322:738–805, 2017.[Sca15] Christopher Scaduto. Instantons and odd Khovanov homology.
J. Topol. , 8(3):744–810,2015.[Tur02] Vladimir Turaev.
Torsions of 3-dimensional manifolds . Birkh¨auser Basel, 2002.[Wan20] Joshua Wang. The cosmetic crossing conjecture for split links.
ArXiv:2006.01070, v1 ,2020.[Ye20] Fan Ye. Constrained knots in lens spaces.
ArXiv:2007.04237, v1 , 2020.[Zem18] Ian Zemke. Duality and mapping tori in Heegaard Floer homology.
ArXiv: 1801.09270,v1 , 2018.[Zem19] Ian Zemke. Graph cobordisms and Heegaard Floer homology.
ArXiv: 1512.01184, v3 ,2019.
Department of Mathematics, Stanford University
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