Contact Structures with Distinct Heegaard Floer Invariants

HF=HM I : Heegaard Floer homology and Seiberg--Witten Floer homology

Let M be a closed, connected and oriented 3-manifold. This article is the first of a five part series that constructs an isomorphism between the Heegaard Floer homology groups of M and the corresponding Seiberg-Witten Floer homology groups of M.

Perturbed Floer Homology of some fibered three manifolds

In this paper, we write down a special Heegaard diagram for a given product three manifold $Σ_g\times S^1$. We use the diagram to compute its perturbed Heegaard Floer homology.

Grid diagrams in Heegaard Floer theory

We review the use of grid diagrams in the development of Heegaard Floer theory. We describe the construction of the combinatorial link Floer complex, and the resulting algorithm for unknot detection. We also explain how grid diagrams can be used to show that the Heegaard Floer invariants of 3-manifolds and 4-manifolds are algorithmically computable (mod 2).

A note on combinatorial splicing formulas for Heegaard Floer homology

We give a precise description of splicing formulas from a previous paper in terms of knot Floer complex associated with a knot in homology sphere.

Heegaard Floer 理论中的典范定向

Canonical orientations in Heegaard Floer theory

本文在整数环上建立 Heegaard Floer 理论，利用典范定向证明其自然性，并给出手术正合三角的新证明及对合 Heegaard Floer 同调的定义。

Algebraic torsion via Heegaard Floer homology

We outline Hutchings's prescription that produces an ECH analog of Latschev and Wendl's algebraic $k$-torsion in the context of $ech$, a variant of ECH used in a proof of the isomorphism between Heegaard Floer and Seiberg-Witten Floer homologies; and we explain how it translates into Heegaard Floer homology.

Heegaard Floer 理论中 Maslov 指数公式的组合证明

Combinatorial proof of Maslov index formula in Heegaard Floer Theory

本文通过 Heegaard 图的组合学方法，证明了 Heegaard Floer 同调中 Lipshitz 的 Maslov 指数公式。

A topological grading on bordered Heegaard Floer homology

In this paper, we construct a canonical grading on bordered Heegaard Floer homology by homotopy classes of nonvanishing vector fields. This grading is a generalization of our construction of an absolute grading on Heegaard Floer homology and it extends the well-known grading with values in a noncommutative group defined by Lipshitz-Ozsváth-Thurston.

Introduction to the basics of Heegaard Floer homology

This paper provides an introduction to the basics of Heegaard Floer homology with some emphasis on the hat theory and to the contact geometric invariants in the theory. The exposition is designed to be comprehensible to people without any prior knowledge of the subject.

A note on the rank of Heegaard Floer homology

We show that if K is a non-trivial knot inside a homology sphere Y, then the rank of knot Floer homology associated with K is strictly bigger than the rank of Heegaard Floer homology of Y.

A combinatorial description of the Heegaard Floer contact invariant

In this short note, we observe that the Heegaard Floer contact invariant is combinatorial by applying the algorithm of Sarkar--Wang to the description of the contact invariant due to Honda--Kazez--Matic. We include an example of this combinatorial calculation.

循环3-轨形的Heegaard Floer不变量

Heegaard Floer invariants for cyclic 3-orbifolds

本文为具有任意循环奇点的三维轨形定义了Heegaard Floer同调，推广了Biji Wong近期假设奇点轨迹连通的工作。

高维Heegaard Floer同调的一个例子

An example of higher-dimensional Heegaard Floer homology

本文通过计数二维圆盘不交余切纤维的高维Heegaard Floer同调中的伪全纯曲线，证明所得代数同构于对称群对应的Hecke代数。

Fukaya categories and bordered Heegaard-Floer homology

We outline an interpretation of Heegaard-Floer homology of 3-manifolds (closed or with boundary) in terms of the symplectic topology of symmetric products of Riemann surfaces, as suggested by recent work of Tim Perutz and Yanki Lekili. In particular we discuss the connection between the Fukaya category of the symmetric product and the bordered algebra introduced by Robert Lipshitz, Peter Ozsvath and Dylan Thurston, and recast bordered Heegaard-Floer homology in this language.

HF=HM II: Reeb orbits and holomorphic curves for the ech/Heegaard-Floer correspondence

This is the second of five papers that construct an isomorphism between the Seiberg-Witten Floer homology and the Heegaard Floer homology of a given compact, oriented 3-manifold. The isomorphism is given as a composition of three isomorphisms; the first of these relates a version of embedded contact homology on an an auxillary manifold to the Heegaard Floer homology on the original. This paper describes this auxilliary manifold, its geometry, and the relationship between the generators of the embedded contact homology chain complex and those of the Heegaard Floer chain complex. The pseudoholom…

The equivalence of Heegaard Floer homology and embedded contact homology via open book decompositions II

This paper is the sequel to "The equivalence of Heegaard Floer homology and embedded contact homology via open book decompositions I" and is devoted to proving some of the technical parts of the HF=ECH isomorphism.

Fukaya categories of symmetric products and bordered Heegaard-Floer homology

The main goal of this paper is to discuss a symplectic interpretation of Lipshitz, Ozsvath and Thurston's bordered Heegaard-Floer homology in terms of Fukaya categories of symmetric products and Lagrangian correspondences. More specifically, we give a description of the algebra A(F) which appears in the work of Lipshitz, Ozsvath and Thurston in terms of (partially wrapped) Floer homology for product Lagrangians in the symmetric product, and outline how bordered Heegaard-Floer homology itself can conjecturally be understood in this language.

An exposition of the equivalence of Heegaard Floer homology and embedded contact homology

This article is a survey on the authors' proof of the isomorphism between Heegaard Floer homology and embedded contact homology appeared in This article is a survey on the authors' proof of the isomorphism between Heegaard Floer homology and embedded contact homology appeared in arXiv:1208.1074, arXiv:1208.1077 and arXiv:1208.1526.

Filtration of Heegaard Floer homology and gluing formulas

We introduce an extra filtration of $\CFK(Y,K)$ and use it in order to obtain formulas for Floer homology of $(Y,K)$, which is obtained from $(Y_i,K_i), i=1,2$ by gluing the knot complements on the framed torus boundaries.

Heegaard Floer homology of broken fibrations over the circle

We extend Perutz's Lagrangian matching invariants to 3-manifolds which are not necessarily fibred using the technology of holomorphic quilts. We prove an isomorphism of these invariants with Ozsvath-Szabo's Heegaard Floer invariants for certain extremal spin^c structures. As applications, we give new calculations of Heegaard Floer homology of certain classes of 3-manifolds, and a characterization of Juhasz's sutured Floer homology.

A cylindrical reformulation of Heegaard Floer homology

We reformulate Heegaard Floer homology in terms of holomorphic curves in the cylindrical manifold Sigma x [0,1] x R, where Sigma is the Heegaard surface, instead of Sym^g(Sigma). We then show that the entire invariance proof can be carried out in our setting. In the process, we derive a new formula for the index of the dbar-operator in Heegaard Floer homology, and shorten several proofs. After proving invariance, we show that our construction is equivalent to the original construction of Ozsvath-Szabo. We conclude with a discussion of elaborations of Heegaard Floer homology suggested by our co…

HF=HM III: Holomorphic curves and the differential for the ech/Heegaard Floer correspondence

This is the third of five papers that construct an isomorphism between the Seiberg-Witten Floer homology and the Heegaard Floer homology of a given compact, oriented 3-manifold. The isomorphism is given as a composition of three isomorphisms; the first of these relates a version of embedded contact homology on an an auxillary manifold to the Heegaard Floer homology on the original. This paper describes the relationship between the differential on the embedded contact homology chain complex and the differential on the Heegaard Floer chain complex. The paper also describes the relationship betwe…

Combinatorial cobordism maps in hat Heegaard Floer theory

In a previous paper, Sarkar and the third author gave a combinatorial description of the hat version of Heegaard Floer homology for three-manifolds. Given a cobordism between two connected three-manifolds, there is an induced map between their Heegaard Floer homologies. Assume that the first homology group of each boundary component surjects onto the first homology group of the cobordism (modulo torsion). Under this assumption, we present a procedure for finding the rank of the induced Heegaard Floer map combinatorially, in the hat version.

Errata to 'A cylindrical reformulation of Heegaard Floer homology'

This note corrects one serious mistake and several smaller mistakes from arXiv:math/0502404. The main results of that paper are unchanged.

Heegaard Floer homology of Some Mazur Type Manifolds

We show that an infinite family of contractible 4-manifolds have the same boundary as a special type of plumbing. Consequently their Ozsvath--Szabo invariants can be calculated algorithmically. We run this algorithm for the first few members of the family and list the resulting Heegaard--Floer homologies. We also show that the rank of the Heegaard--Floer homology can get arbitrary large values in this family by using its relation with the Casson invariant. For comparison, we list the ranks of Floer homologies of all the examples of Briekorn spheres that are known to bound contractible manifold…

Absolute gradings on ECH and Heegaard Floer homology

In joint work with Yang Huang, we defined a canonical absolute grading on Heegaard Floer homology by homotopy classes of oriented 2-plane fields. A similar grading was defined on embedded contact homology by Michael Hutchings. In this paper we show that the isomorphism between these homology theories defined by Colin-Ghiggini-Honda preserves this grading.

On the Heegaard Floer homology of branched double-covers

Let $L\subset S^3$ be a link. We study the Heegaard Floer homology of the branched double-cover $Σ(L)$ of $S^3$, branched along $L$. When $L$ is an alternating link, $\HFa$ of its branched double-cover has a particularly simple form, determined entirely by the determinant of the link. For the general case, we derive a spectral sequence whose $E^2$ term is a suitable variant of Khovanov's homology for the link $L$, converging to the Heegaard Floer homology of $Σ(L)$.

Slicing planar grid diagrams: a gentle introduction to bordered Heegaard Floer homology

We describe some of the algebra underlying the decomposition of planar grid diagrams. This provides a useful toy model for an extension of Heegaard Floer homology to 3-manifolds with parametrized boundary. This paper is meant to serve as a gentle introduction to the subject, and does not itself have immediate topological applications.

分支覆盖的Heegaard Floer同调的秩不等式

Rank inequalities for the Heegaard Floer homology of branched covers

本文研究沿零同调链环分支的二重覆盖，建立了Heegaard Floer同调维数的不等式，并探讨其与L-空间猜想的关系及拓扑应用。

Heegaard Floer homologies and contact structures

Given a contact structure on a closed, oriented three-manifold $Y$, we describe an invariant which takes values in the three-manifold's Floer homology $\HFa$. This invariant vanishes for overtwisted contact structures and is non-zero for Stein fillable ones. The construction uses of Giroux's interpretation of contact structures in terms of open book decompositions, and the knot Floer homologies introduced in math.GT/0209056.

Knots with unknotting number one and Heegaard Floer homology

We use Heegaard Floer homology to give obstructions to unknotting a knot with a single crossing change. These restrictions are particularly useful in the case where the knot in question is alternating. As an example, we use them to classify all knots with crossing number less than or equal to nine and unknotting number equal to one. We also classify alternating knots with ten crossings and unknotting number equal to one.

Hamiltonian handleslides for Heegaard Floer homology

A $g$-tuple of disjoint, linearly independent circles in a Riemann surface of genus $g$ determines a `Heegaard torus' in its $g$-fold symmetric product. Changing the circles by a handleslide produces a new torus. It is proved that, for symplectic forms with certain properties, these two tori are Hamiltonian-isotopic Lagrangian submanifolds. This provides an alternative route to the handleslide-invariance of Ozsvath-Szabo's Heegaard Floer homology.

Grid diagrams and Heegaard Floer invariants

We give combinatorial descriptions of the Heegaard Floer homology groups for arbitrary three-manifolds (with coefficients in Z/2). The descriptions are based on presenting the three-manifold as an integer surgery on a link in the three-sphere, and then using a grid diagram for the link. We also give combinatorial descriptions of the mod 2 Ozsvath-Szabo mixed invariants of closed four-manifolds, in terms of grid diagrams.

Heegaard diagrams and Floer homology

We review the construction of Heegaard Floer homology for closed three-manifolds and also for knots and links in the three-sphere. We also discuss three applications of this invariant to knot theory: studying the Thurston norm of a link complement, the slice genus of a knot, and the unknotting number of a knot. We emphasize the application to the Thurston norm, and illustrate the theory in the case of the Conway link.

The equivalence of Heegaard Floer homology and embedded contact homology via open book decompositions I

Given an open book decomposition (S,h) adapted to a closed, oriented 3-manifold M, we define a chain map from a certain Heegaard Floer chain complex associated to (S,h) to a certain embedded contact homology chain complex associated to (S,h), as defined in arXiv:1008.2734, and prove that it induces an isomorphism on the level of homology. This implies the isomorphism between the hat version of Heegaard Floer homology of -M and the hat version of embedded contact homology of M.

The equivalence of Heegaard Floer homology and embedded contact homology III: from hat to plus

Given a closed oriented 3-manifold M, we establish an isomorphism between the Heegaard Floer homology group HF^+(-M) and the embedded contact homology group ECH(M). Starting from an open book decomposition (S,h) of M, we construct a chain map Φ^+ from a Heegaard Floer chain complex associated to (S,h) to an embedded contact homology chain complex for a contact form supported by (S,h). The chain map Φ^+ commutes up to homotopy with the U-maps defined on both sides and reduces to the quasi-isomorphism Φfrom "The equivalence of Heegaard Floer homology and embedded contact homology I, II" on subco…

实Heegaard Floer同调

Real Heegaard Floer Homology

本文提出了一种实Heegaard Floer同调，用于研究具有非空固定点集的三流形，其欧拉特征与Miyazawa不变量相关，并能对所有结进行组合计算。

Dehn Twists in Heegaard Floer Homology

We derive a new exact sequence in the hat-version of Heegaard Floer homology. As a consequence we see a functorial connection between the invariant of Legendrian knots and the contact element. As an application we derive two vanishing results of the contact element making it possible to easily read off its vanishing out of a surgery presentation in suitable situations.

Knot concordance and Heegaard Floer homology invariants in branched covers

By studying the Heegaard Floer homology of the preimage of a knot K in S^3 inside its double branched cover, we develop simple obstructions to K having finite order in the classical smooth concordance group. As an application, we prove that all 2-bridge knots of crossing number at most 12 for which the smooth concordance order was previously unknown have infinite smooth concordance order.

Heegaard Floer Invariants and Tight Contact Three--Manifolds

Let Y(r) be the closed, oriented three-manifold obtained by performing rational r-surgery on the right-handed trefoil knot in the three-sphere. Using contact surgery and the Heegaard Floer contact invariants we construct positive, tight contact structures on Y(r) for every r not equal to 1. This implies, in particular, that the oriented boundaries of the positive E6 and E7 plumbings carry positive, tight contact structures, solving a well-known open problem.

Covering spaces and Q-gradings on Heegaard Floer homology

Heegaard Floer homology, first introduced by P. Ozsvath and Z. Szabo, associates to a 3-manifold Y a family of relatively graded Abelian groups HF(Y,t), indexed by Spin^c structures t on Y. In the case that Y is a rational homology sphere, Ozsvath and Szabo lift the relative Z-grading to an absolute Q-grading. This induces a relative Q-grading on \bigoplus_{t\in Spin^c(Y)} HF(Y,t). In this paper we describe an alternate construction of this relative Q-grading by studying the Heegaard Floer homology of covering spaces.

An absolute grading on Heegaard Floer homology by homotopy classes of oriented 2-plane fields

For a closed oriented 3-manifold Y, we define an absolute grading on the Heegaard Floer homology groups of Y by homotopy classes of oriented 2-plane fields. We show that this absolute grading refines the relative one and that it is compatible with the maps induced by cobordisms. We also prove that if ξ is a contact structure on Y, then the grading of the contact invariant c(ξ) is the homotopy class of ξ.

关于实Heegaard Floer同调与局部化的注记

A note on real Heegaard Floer homology and localization

本文证明了Guth和Manolescu最近构造的实Heegaard Floer同调的帽子变体存在局部化谱序列，并将其应用于分支双覆盖和强可逆纽结。

On Heegaard Floer homology and Seifert fibered surgeries

We explore certain restrictions on knots in the three-sphere which admit non-trivial Seifert fibered surgeries. These restrictions stem from the Heegaard Floer homology for Seifert fibered spaces, and hence they have consequences for both the Alexander polynomial of such knots, and also their knot Floer homology. In particular, we show that certain polynomials are never the Alexander polynomials of knots which admit homology three-sphere Seifert fibered surgeries. The knot Floer homology restrictions, on the other hand, apply also in cases where the Alexander polynomial gives no information, s…

Noncommutative Hodge-to-de Rham spectral sequence and the Heegaard Floer homology of double covers

Let A be a dg algebra over F_2 and let M be a dg A-bimodule. We show that under certain technical hypotheses on A, a noncommutative analog of the Hodge-to-de Rham spectral sequence starts at the Hochschild homology of the derived tensor product of M with itself and converges to the Hochschild homology of M. We apply this result to bordered Heegaard Floer theory, giving spectral sequences associated to Heegaard Floer homology groups of certain branched and unbranched double covers.

Heegaard Floer invariants of contact structures on links of surface singularities

Let a contact 3-manifold $(Y, ξ_0)$ be the link of a normal surface singularity equipped with its canonical contact structure $ξ_0$. We prove a special property of such contact 3-manifolds of "algebraic" origin: the Heegaard Floer invariant $c^+(ξ_0)\in HF^+(-Y)$ cannot lie in the image of the $U$-action on $HF^+(-Y)$. It follows that Karakurt's "height of $U$-tower" invariants are always 0 for canonical contact structures on singularity links, which contrasts the fact that the height of $U$-tower can be arbitrary for general fillable contact structures. Our proof uses the interplay between th…

Heegaard Floer homology and genus one, one boundary component open books

We compute the Heegaard Floer homology of any rational homology 3-sphere with an open book decomposition of the form (T,ϕ), where T is a genus one surface with one boundary component. In addition, we compute the Heegaard Floer homology of any T^2-bundle over S^1 with first Betti number equal to one, and we compare our results with those of Lebow on the embedded contact homology of such torus bundles. We use these computations to place restrictions on Stein-filllings of the contact structures compatible with such open books, to narrow down somewhat the class of 3-braid knots with finite concord…

Heegaard Floer homology as morphism spaces

In this paper we prove another pairing theorem for bordered Floer homology. Unlike the original pairing theorem, this one is stated in terms of homomorphisms, not tensor products. The present formulation is closer in spirit to the usual TQFT framework, and allows a more direct comparison with Fukaya-categorical constructions. The result also leads to various dualities in bordered Floer homology.

Transverse knots, branched double covers and Heegaard Floer contact invariants

Given a transverse link in the standard contact 3-sphere, we study the contact manifold that arises as a branched double cover of the sphere. We give a contact surgery description of such manifolds, which allows to determine the Heegaard Floer contact invariants for some of them. By example of the knots of Birman--Menasco, we show that these contact manifolds may fail to distinguish between non-isotopic transverse knots. We also investigate the relation between the Heegaard Floer contact invariants of the branched double covers and the Khovanov homology, in particular, the transverse link inva…

Heegaard Floer invariants of Legendrian knots in contact three--manifolds

We define invariants of null--homologous Legendrian and transverse knots in contact 3--manifolds. The invariants are determined by elements of the knot Floer homology of the underlying smooth knot. We compute these invariants, and show that they do not vanish for certain non--loose knots in overtwisted 3--spheres. Moreover, we apply the invariants to find transversely non--simple knot types in many overtwisted contact 3--manifolds.

On the naturality of the spectral sequence from Khovanov homology to Heegaard Floer homology

Ozsvath and Szabo have established an algebraic relationship, in the form of a spectral sequence, between the reduced Khovanov homology of (the mirror of) a link L in S^3 and the Heegaard Floer homology of its double-branched cover. This relationship has since been recast by the authors as a specific instance of a broader connection between Khovanov- and Heegaard Floer-type homology theories, using a version of Heegaard Floer homology for sutured manifolds developed by Juhasz. In the present work we prove the naturality of the spectral sequence under certain elementary TQFT operations, using a…

Notes on bordered Floer homology

This is a survey of bordered Heegaard Floer homology, an extension of the Heegaard Floer invariant HF-hat to 3-manifolds with boundary. Emphasis is placed on how bordered Heegaard Floer homology can be used for computations.

卫星纽结与浸入式Heegaard Floer同调

Satellite knots and immersed Heegaard Floer homology

本文提出计算卫星纽结$UV=0$纽结Floer复形的新方法，通过叠加伴纽结浸入曲线与模式Heegaard图得到浸入式图简化计算，推广了已有结论并扩展了相关构造。

A tour of bordered Floer theory

Heegaard Floer theory is a kind of topological quantum field theory, assigning graded groups to closed, connected, oriented 3-manifolds and group homomorphisms to smooth, oriented 4-dimensional cobordisms. Bordered Heegaard Floer homology is an extension of Heegaard Floer homology to 3-manifolds with boundary, with extended-TQFT-type gluing properties. In this survey, we explain the formal structure and construction of bordered Floer homology and sketch how it can be used to compute some aspects of Heegaard Floer theory.