Heegaard-Floer Homology and a family of Brieskorn spheres

Homology cobordism, knot concordance, and Heegaard Floer homology

We review some recent results in knot concordance and homology cobordism. The proofs rely on various forms of Heegaard Floer homology. We also discuss related open problems.

A combinatorial description of the $U^2=0$ version of Heegaard Floer homology

We show that every 3--manifold admits a Heegaard diagram in which a truncated version of Heegaard Floer homology (when the holomorpic disks pass through the basepoints at most once) can be computed combinatorially.

Almost simple linear graphs, homology cobordism and connected Heegaard Floer homology

Continuing our previous work, we effectively compute connected Heegaard Floer homologies of two families of Brieskorn spheres realized as the boundaries of almost simple linear graphs. Using Floer theoretic invariants of Dai, Hom, Stoffregen, and Truong, we show that these Brieskorn spheres also generate infinite rank summands in the homology cobordism group. Our computations also have applications to the concordance of classical knots and $0$-concordance of $2$-knots.

Combinatorial Heegaard Floer homology and sign assignments

We provide an intergral lift of the combinatorial definition of Heegaard Floer homology for nice diagrams, and show that the proof of independence using convenient diagrams adapts to this setting.

Heegaard Floer Homologies: lecture notes

These are lecture notes from a series of lectures at the SMF summer school on "Geometric and Quantum Topology in Dimension 3", June 2014. The focus is on Heegaard Floer homology from the perspective of sutured Floer homology.

Heegaard Floer homology, L-spaces and smoothing order on links I

In this paper, we focus on L-spaces for which the boundary maps of the Heegaard Floer chain complexes vanish. We collect such manifolds systematically by using the smoothing order on links.

A graph TQFT for hat Heegaard Floer homology

We construct maps on hat Heegaard Floer homology for cobordisms decorated with graphs. The graph TQFT allows for cobordisms with disconnected ends. Our construction uses Juhász's sutured Floer TQFT. We compute the maps for several elementary graph cobordisms. As an application, we compute the action of the fundamental group on hat Heegaard Floer homology.

Duality and mapping tori in Heegaard Floer homology

We show that the graph TQFT for Heegaard Floer homology satisfies a strong version of Atiyah's duality axiom for a TQFT. As an application, we compute some Heegaard Floer mixed invariants of 4-dimensional mapping tori in terms of Lefschetz numbers on $HF^+$.

On rational blow-downs in Heegaard-Floer homology

Motivated by a result of L.P. Roberts on rational blow-downs in Heegaard-Floer homology, we study such operations along 3-manifolds that arise as branched double covers of $S^{3}$ along several non-alternating, slice knots.

An algorithm to compute some Heegaard Floer homologies

The earlier article tried to construct an algorithm to compute the Heegaard Floer homology \hat{HF}(Y) for a 3-manifold Y. However there is an error in a proof which the author, as of now, is unable to fix.

论Heegaard Floer同调理论中的Whitney圆盘

On the Whitney disks in Heegaard Floer homology theory

本文利用Nielsen理论为Heegaard Floer同调中连接两个给定交点的Whitney圆盘存在性提供了一个简单判据。

Some non-trivial examples of the Baldwin-Ozsváth-Szabó twisted spectral sequence and Heegaard-Floer homology of branched double covers

We present some non-trivial calculations of Baldwin-Ozsváth-Szabó cohomology of links, and applications to Heegaard-Floer homology of branched double covers.

对合赫戈弗洛尔同调中的自然性与函子性

Naturality and functoriality in involutive Heegaard Floer homology

本文证明对合赫戈弗洛尔同调的一阶自然性，构造配边诱导的良定义映射，并将自然性与函子性结果推广到纽结和链环的对合弗洛尔理论。

Combinatorial Proofs in Bordered Heegaard Floer homology

Using bordered Floer theory, we give a combinatorial construction and proof of invariance for the hat version of Heegaard Floer homology. As a part of the proof, we also establish combinatorially the invariance of the linear-categorical representation of the strongly-based mapping class groupoid given by the same theory.

Finite knot surgeries and Heegaard Floer homology

Extensive rewrite. Tables and proofs have been reformatted and/or rewritten for clarity.

Triple linking numbers and Heegaard Floer homology

We establish some new relationships between Milnor invariants and Heegaard Floer homology. This includes a formula for the Milnor triple linking number from the link Floer complex, detection results for the Whitehead link and Borromean rings, and a structural property of the $d$-invariants of surgeries on certain algebraically split links.

检测 Heegaard Floer 同调实环

Detecting Heegaard Floer homology solid tori

本文证明有理同调实环为 Heegaard Floer 同调实环的充要条件是存在带非分离2-球的 Dehn 填充，并据此刻画了 Seifert 纤维化的 Heegaard Floer 实环。

Heegaard Floer homology, L-spaces and smoothing order on links II

We focus on L-spaces for which the boundary maps of the Heegaard Floer chain complexes vanish. In previous paper \cite{Usui}, we collect such manifolds systematically by using the smoothing order on links. In this paper, we classify such L-spaces under appropreate constraint.

Heegaard Floer Homology and Triple Cup Products

We give a complete calculation of the infinity flavor of Heegaard Floer homology with mod 2 coefficients for all three-manifolds and torsion Spin^c structures. The computation agrees with the conjectured calculation of Ozsvath and Szabo. This therefore establishes an isomorphism with Mark's cup homology mod 2.

Naturality and mapping class groups in Heegaard Floer homology

We show that all versions of Heegaard Floer homology, link Floer homology, and sutured Floer homology are natural. That is, they assign concrete groups to each based 3-manifold, based link, and balanced sutured manifold, respectively. Furthermore, we functorially assign isomorphisms to (based) diffeomorphisms, and show that this assignment is isotopy invariant. The proof relies on finding a simple generating set for the fundamental group of the "space of Heegaard diagrams," and then showing that Heegaard Floer homology has no monodromy around these generators. In fact, this allows us to give s…

Applications of involutive Heegaard Floer homology

We use Heegaard Floer homology to define an invariant of homology cobordism. This invariant is isomorphic to a summand of the reduced Heegaard Floer homology of a rational homology sphere equipped with a spin structure and is analogous to Stoffregen's connected Seiberg-Witten Floer homology. We use this invariant to study the structure of the homology cobordism group and, along the way, compute the involutive correction terms for certain families of three-manifolds.

Tight Planar Contact Manifolds with Vanishing Heegaard Floer Contact Invariants

In this note, we exhibit infinite families of tight non-fillable contact manifolds supported by planar open books with vanishing Heegaard Floer contact invariants. Moreover, we also exhibit an infinite such family where the supported manifold is hyperbolic.

Non-separating spheres and twisted Heegaard Floer homology

If a 3--manifold $Y$ contains a non-separating sphere, then some twisted Heegaard Floer homology of $Y$ is zero. This simple fact allows us to prove several results about Dehn surgery on knots in such manifolds. Similar results have been proved for knots in $L$--spaces.

Algebraic and Heegaard-Floer invariants of knots with slice Bing doubles

If the Bing double of a knot K is slice, then K is algebraically slice. In addition, Heegaard--Floer concordance invariants developed by Ozsvath-Szabo and by Manolescu-Owens vanish on K.

Rational blow-downs in Heegaard-Floer homology

The paper describes how known results in Heegaard-Floer homology apply to all known examples of rational blow-downs, and provides several new four dimensional pieces which could be exchanged while preserving some of the Ozsváth-Szabó four manifold invariants, when one of these pieces is found embedded in a smooth four manifold.

A survey on Heegaard Floer homology and concordance

In this survey article, we discuss several different knot concordance invariants coming from the Heegaard Floer homology package of Ozsvath and Szabo. Along the way, we prove that if two knots are concordant, then their knot Floer complexes satisfy a certain type of stable equivalence.

关于Heegaard Floer同调的讲义

Lecture notes on Heegaard Floer homology

本文是关于Heegaard Floer同调的系列讲座讲义，通过grid homology介绍代数结构，在Lagrangian Floer同调背景下定义Heegaard Floer同调并阐述其几何本质与关键性质。

Topics in Heegaard Floer homology

This is the author's PhD thesis, as submitted to the Princeton University. The results of this paper have already appeared in arXiv:math/0607777v4, arXiv:math/0607691 and arXiv:0901.2156.

A remark on the geography problem in Heegaard Floer homology

We give new obstructions to the module structures arising in Heegaard Floer homology. As a corollary, we characterize the possible modules arising as the Heegaard Floer homology of an integer homology sphere with one-dimensional reduced Floer homology. Up to absolute grading shifts, there are only two. We use this corollary to show that the chain complex depicted by Ozsváth, Stipsicz, and Szabó to argue that there is no algebraic obstruction to the existence of knots with trivial $ε$ invariant and non-trivial $Υ$ invariant cannot be realized as the knot Floer complex of a knot.

Generalized Heegaard Floer correction terms

We make use of the action of $H_1(Y)$ in Heegaard Floer homology to generalize the Ozsváth-Szabó correction terms for $3$-manifolds with standard $\operatorname{HF}^\infty$. We establish the basic properties of these invariants: conjugation invariance, behavior under orientation reversal, additivity, and spin$^c$ rational homology cobordism invariance.

Truncated Heegaard Floer Homology and Knot Concordance Invariants

In this paper we construct a sequence of integer-valued concordance invariants $ν_n(K)$ that generalize the Ozsváth-Szabó $ν$-invariant and the Hom-Wu $ν^+$-invariant.

Surgery obstructions and Heegaard Floer homology

Using Taubes' periodic ends theorem, Auckly gave examples of toroidal and hyperbolic irreducible integer homology spheres which are not surgery on a knot in the three-sphere. We give an obstruction to a homology sphere being surgery on a knot coming from Heegaard Floer homology. This is used to construct infinitely many small Seifert fibered examples.

A spanning tree model for the Heegaard Floer homology of a branched double-cover

Given a diagram of a link K in S^3, we write down a Heegaard diagram for the branched-double cover Sigma(K). The generators of the associated Heegaard Floer chain complex correspond to Kauffman states of the link diagram. Using this model we make some computations of the homology \hat{HF}(Sigma(K)) as a graded group. We also conjecture the existence of a delta-grading on \hat{HF}(Sigma(K)) analogous to the delta-grading on knot Floer and Khovanov homology.

A survey of Heegaard Floer homology

This work has two goals. The first is to provide a conceptual introduction to Heegaard Floer homology, the second is to survey the current state of the field, without aiming for completeness. After reviewing the structure of Heegaard Floer homology, we list some of its most important applications. Many of these are purely topological results, not referring to Heegaard Floer homology itself. Then, we briefly outline the construction of Lagrangian intersection Floer homology. We construct the Heegaard Floer chain complex as a special case of the above, and try to motivate the role of the various…

Heegaard Floer同调中的1度映射与秩不等式

Degree-1 maps and rank inequalities in Heegaard Floer homology

本文利用bordered Floer同调的浸入曲线解释，证明了特定1度映射在Heegaard Floer同调中产生的秩不等式，为Heegaard Floer同调与instanton Floer同调关系的猜想提供了进一步证据。

Surgery formula for the renormalized Euler characteristic of Heegaard Floer homology

We prove a surgery formula for the renormalized Euler characteristic of Ozsvath and Szabo. Equality between this Euler cahracteristic and the Seiberg-Witten invariant follows for rational homology three-spheres.

Heegaard Floer homology for manifolds with torus boundary: properties and examples

This is a companion paper to earlier work of the authors, which interprets the Heegaard Floer homology for a manifold with torus boundary in terms of immersed curves in a punctured torus. We prove a variety of properties of this invariant, paying particular attention to its relation to knot Floer homology, the Thurston norm, and the Turaev torsion. We also give a geometric description of the gradings package from bordered Heegaard Floer homology and establish a symmetry under spin$^c$ conjugation; this symmetry gives rise to genus one mutation invariance in Heegaard Floer homology for closed t…

Heegaard Floer groups of Dehn surgeries

We use an algorithm by Ozsvath and Szabo to find closed formulae for the ranks of the hat version of the Heegaard Floer homology groups for non-zero Dehn surgeries on knots in the 3-sphere. As applications we provide new bounds on the number of distinct ranks of the Heegaard Floer groups a Dehn surgery can have. These in turn give a new lower bound on the rational Dehn surgery genus of a rational homology 3-sphere. We also provide novel obstructions for a knot to be a potential counterexample to the Cabling Conjecture.

On the contact class in Heegaard Floer homology

We present an alternate description of the Ozsvath-Szabo contact class in Heegaard Floer homology. Using our contact class, we prove that if a contact structure (M,ξ) has an adapted open book decomposition whose page S is a once-punctured torus, then the monodromy is right-veering if and only if the contact structure is tight.

Heegaard Floer homology and splicing homology spheres

We prove a basic inequality for the d-invariants of a splice of knots in homology spheres. As a result, we are able to prove a new relation on the rank of reduced Floer homology under maps between Seifert fibered homology spheres, improving results of the first and second authors. As a corollary, a degree one map between two aspherical Seifert homology spheres is homotopic to a homeomorphism if and only if the Heegaard Floer homologies are isomorphic.

G-corks & Heegaard Floer Homology

Recently Auckly-Kim-Melvin-Ruberman showed that for any finite subgroup G of SO(4) there exists a contractible 4-manifold with an effective G-action on its boundary so that the twists associated to the non-trivial elements of G do not extend to diffeomorphisms of the entire manifold. We use a Heegaard Floer theoretic argument originating in work of Akbulut-Karakurt to give a different proof of this phenomenon.

Heegaard Floer 同调中的复合映射

Composition Maps in Heegaard Floer Homology

本文证明 Heegaard Floer 同调态射空间表述中的反向复合映射与 pair-of-pants 配边诱导映射同伦一致，并据此提出计算 hat 理论中任意配边映射的算法。

Heegaard Floer homologies for (+1) surgeries on torus knots

We compute the Heegaard Floer homology of $S^3_1(K)$ (the (+1) surgery on the torus knot $T_{p,q}$) in terms of the semigroup generated by $p$ and $q$, and we find a compact formula (involving Dedekind sums) for the corresponding Ozsvath--Szabo d-invariant. We relate the result to known knot invariants of $T_{p,q}$ as the genus and the Levine--Tristram signatures. Furthermore, we emphasize the striking resemblance between Heegaard Floer homologies of (+1) and (-1) surgeries on torus knots. This relation is best seen at the level of $τ$ functions.

An algorithm for computing some Heegaard Floer homologies

In this paper, we give an algorithm to compute the hat version of the Heegaard Floer homology of a closed oriented three-manifold. This method also allows us to compute the filtrations coming from a null-homologous link in a three-manifold.

Rank inequalities for the Heegaard Floer homology of Seifert homology spheres

We establish three rank inequalities for the reduced flavor of Heegaard Floer homology of Seifert fibered integral homology spheres. Combining these inequalities with the known classifications of non-zero degree maps between Seifert fibered spaces, we prove that a map f from Y' to Y between Seifert homology spheres yields the inequality that |deg f| rank HFred(Y) is at most rank HFred(Y'). These inequalities are also applied in conjunction with an algorithm of Nemethi to give a method to solve the botany problem for the Heegaard Floer homology of these manifolds.

New Heegaard Floer slice genus and clasp number bounds

We define several concordance invariants using knot Floer homology which give improvements over known slice genus and clasp number bounds from Heegaard Floer homology. We also prove that the involutive correction terms of Hendricks and Manolescu give both a slice genus and a clasp number bound.

Involutive Heegaard Floer homology

Using the conjugation symmetry on Heegaard Floer complexes, we define a three-manifold invariant called involutive Heegaard Floer homology, which is meant to correspond to $\mathbb{Z}_4$-equivariant Seiberg-Witten Floer homology. Further, we obtain two new invariants of homology cobordism, $\underline{d}$ and $\bar{d}$, and two invariants of smooth knot concordance, $\underline{V}_0$ and $\overline{V}_0$. We also develop a formula for the involutive Heegaard Floer homology of large integral surgeries on knots. We give explicit calculations in the case of L-space knots and thin knots. In partic…

Heegaard Floer theory and property P

This paper has been withdrawn due to an error in the proof. The paper implicitly assumed that every homotopically trivial knot is related to the unknot by negative crossings, which is false (as was pointed out to the author by Peter Ozsvath).

Order in the concordance group and Heegaard Floer homology

We use the Heegaard-Floer homology correction terms defined by Ozsváth--Szabó to formulate a new obstruction for a knot to be of finite order in the smooth concordance group. This obstruction bears a formal resemblance to that of Casson and Gordon but is sensitive to the difference between the smooth versus topological category. As an application we obtain new lower bounds for the concordance order of small crossing knots.

Khovanov同调与分支双覆盖的 involutive Heegaard Floer 同调

Khovanov homology and the Involutive Heegaard Floer homology of branched double covers

本文利用 involutive Heegaard Floer 同调，借助 bordered Floer 同调重构与手术正合三角，推广了 Ozsváth-Szabó 分支双覆盖谱序列，建立 Khovanov 同调与 Heegaard Floer 同调间的联系。

实缝合希盖尔弗勒同调

Real sutured Heegaard Floer homology

该研究发展实缝合流形理论及实希盖尔弗勒理论，提出实优美图概念，证明不变量可组合计算，其理论与已有理论有相同结构性质，且不变量有新性质，与缝合弗勒同调不同。

Connected Heegaard Floer homology of sums of Seifert fibrations

We compute the connected Heegaard Floer homology (defined by Hendricks, Hom, and Lidman) for a large class of 3-manifolds, including all linear combinations of Seifert fibered homology spheres. We show that for such manifolds, the connected Floer homology completely determines the local equivalence class of the associated $ι$-complex. Some identities relating the rank of the connected Floer homology to the Rokhlin invariant and the Neumann-Siebenmann invariant are also derived. Our computations are based on combinatorial models inspired by the work of Némethi on lattice homology.

关于 sutured 流形中的 Heegaard Floer 极小纽结

On Heegaard Floer minimal knots in sutured manifolds

本文在 Heegaard Floer 同调框架下分类了平衡 sutured 流形中的极小纽结，与 instanton Floer 分类一致，支持两种同调同构的猜想，并研究了 $S^1\ imes S^2$ 中的链接 Floer 同调植物学问题。

Knot doubling operators and bordered Heegaard Floer homology

We use bordered Heegaard Floer homology to compute the tau invariant of a family of satellite knots obtained via twisted infection along two components of the Borromean rings, a generalization of Whitehead doubling. We show that tau of the resulting knot depends only on the two twisting parameters and the values of tau for the two companion knots. We also include some notes on bordered Heegaard Floer homology that may serve as a useful introduction to the subject.