On the Heegaard Floer homology of S^3_{-p/q}(K)

Heegaard Floer同调与具有非尖点奇点的平面曲线

Heegaard Floer homology and plane curves with non-cuspidal singularities

本文利用Floer理论给出链环knotification的knot Floer复形上H₁-作用的一般公式，并借此研究复射影平面中一般代数曲线非尖点奇点的可能构型。

Immersed concordances of links and Heegaard Floer homology

An immersed concordance between two links is a concordance with possible self-intersections. Given an immersed concordance we construct a smooth four-dimensional cobordism between surgeries on links. By applying $d$-invariant inequalities for this cobordism we obtain inequalities between the $H$-functions of links, which can be extracted from the link Floer homology package. As an application we show a Heegaard Floer theoretical criterion for bounding the splitting number of links. The criterion is especially effective for L-space links, and we present an infinite family of L-space links with …

实Heegaard Floer理论中的自然性

Naturality in real Heegaard Floer theory

本文证明了实Heegaard Floer同调、实纽结Floer同调及实缝合Floer同调的自然性，还定义了对合实Heegaard Floer同调，补充了相关理论性质。

On two types of Heegaard diagram used in knot Floer homology

In this brief note, we give an explicit sequence of Heegaard moves interpolating between local versions of the Kauffman-states Heegaard diagram and the planar Heegaard diagram used in knot Floer homology, and show how these local moves can be used to go between the global versions of the Heegaard diagrams.

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.

Heegaard-Floer Homology and a family of Brieskorn spheres

We compute the Heegaard-Floer homology for the family Sigma(2,3,6n+1) of Brieskorn spheres using the algorithm by P. Ozsvath and Z. Szabo

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.

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 $HF^+$.

Heegaard Floer homologies of pretzel knots

We compute the Ozsváth-Szabó Heegaard Floer homology of three stranded pretzel knots.

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…

A combinatorial approach to surgery formulas in Heegaard Floer homology

Using the combinatorial approach to Heegaard Floer homology we obtain a relatively easy formula for computation of hat Heegaard Floer homology for the three-manifold obtained by rational surgery on a knot K inside a homology sphere Y.

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.

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.

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.

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 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.

Seifert fibered homology spheres with trivial Heegaard Floer homology

We show that among Seifert fibered integer homology spheres, Poincare sphere (with either orientation) is the only non-trivial example which has trivial Heegaard Floer homology. Together with an earlier result, this shows that if an integer homology sphere has trivial Heegaard Floer homology, then it is a connected sum of a number of Poincare spheres and hyperbolic homology spheres.

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.

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。

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.

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 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。