Bijan Sahamie
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.
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Heegaard Floer homologies of pretzel knots
我们计算了三个链节的Ozsváth-Szabó Heegaard弗洛尔同调。
A combinatorial approach to surgery formulas in Heegaard Floer homology
利用Heegaard弗洛尔同调的组合方法，我们得到了一个相对简单的计算hat Heegaard弗洛尔同调的公式，该公式是通过对同调球面Y中的纽结K进行有理外科手术而得到的。
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.
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
我们给出了一个精确的描述拼接公式从以前的文件中的结Floer复杂的结在同调球。
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。
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.
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。
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.
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
在这个简短的说明中，我们观察到Heegaard Floer接触不变量是组合的，通过应用Sarkar-Wang的算法来描述Honda-Kazez-Matic的接触不变量。我们包括这种组合计算的一个例子。
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。
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.
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…
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.
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.
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
我们引入了一个额外的过滤$\CFK(Y,K)$和使用它，以获得的弗洛尔同调的$(Y,K)$，这是从$(Y_i,K_i), i=1,2$通过粘合上的框架环面边界的结补公式。
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
我们用圆柱流形Sigma x [0，1] x R中的全纯曲线重新表述Heegaard弗洛尔同调，其中Sigma是Heegaard曲面，而不是Sym^g（Sigma）.然后，我们表明，整个不变性证明可以在我们的设置。在此过程中，我们推导出了Heegaard弗洛尔同调中dbar算子指数的新公式，并缩短了几个证明过程。在证明了不变性之后，我们证明了我们的构造等价于Ozsvath-Szabo的原始构造。最后，我们讨论的阐述Heegaard弗洛尔同调建议我们的建设，以及一个简短的讨论与C Taubes计划的关系。
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
在以前的论文中，Sarkar和第三作者给出了三流形的Heegaard弗洛尔同调的帽子版本的组合描述。给定两个连通三流形之间的配边关系，在它们的Heegaard弗洛尔同调之间存在一个诱导映射.假设每个边界分支的第一个同调群上接于配边（模挠）的第一个同调群上。在此假设下，我们提出了一个程序，以组合的方式找到诱导Heegaard Floer映射的秩，在帽子的版本。
This note corrects one serious mistake and several smaller mistakes from arXiv:math/0502404. The main results of that paper are unchanged.
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…
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
设$L\subset S^3$为链接。我们研究了$S^3$的分支双覆盖$Σ(L)$的Heegaard弗洛尔同调，沿着$L$分支。当$L$是交替连接时，其分支双覆盖的$\HFa$具有特别简单的形式，完全由连接的行列式确定。对于一般情况下，我们推导出一个谱序列的$E^2$项是一个合适的变体Khovanov的同源性的链接$L$，收敛到Heegaard弗洛尔同调的$Σ(L)$。
Slicing planar grid diagrams: a gentle introduction to bordered Heegaard Floer homology
我们描述了一些代数的分解平面网格图。这为把Heegaard弗洛尔同调推广到具有参数化边界的三维流形提供了一个有用的模型。本文旨在作为一个温和的介绍这个问题，本身并没有立即拓扑应用。
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
给定一个封闭的，有向的三流形$Y$上的切触结构，我们描述了一个不变量，它在三流形的弗洛尔同调$\HFa$中取值。这个不变量为零过扭曲的切触结构和非零的斯坦可填充的。该构造使用Giroux在开卷分解方面对切触结构的解释，以及在数学GT/0209056中引入的结弗洛尔同调。
Knots with unknotting number one and Heegaard Floer homology
我们使用Heegaard弗洛尔同调来给用一个单一的交叉变化解开一个结的障碍。这些限制在所讨论的结是交替的情况下特别有用。作为一个例子，我们使用它们来分类所有的交叉数小于或等于9和unknotting数等于1的结。我们还分类交替结与十个交叉和unknotting数等于一。
Hamiltonian handleslides for Heegaard Floer homology
亏格为$g$的黎曼曲面中的不相交线性无关圆的$g$元组在其$g$重对称积中确定"Heegaard环面“。通过手柄滑动改变圆会产生一个新的环面。证明了对于具有某些性质的辛形式，这两个环面是哈密顿-同位素拉格朗日子流形。这为Ozsvath-Szabo的Heegaard弗洛尔同调的handleslide不变性提供了另一条途径。
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
本文综述了闭三维流形以及三维球面上的纽结和链环的Heegaard弗洛尔同调的构造。我们还讨论了这个不变量的三个应用纽结理论：研究的Thurston范数的一个链接补充，切片属的一个结，和unknotting数的一个结。我们强调应用瑟斯顿规范，并说明了理论的情况下，康威链接。
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.
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…
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。
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
通过研究S^3中纽结K在其双分支覆盖内的原像的Heegaard弗洛尔同调，我们给出了经典光滑协调群中K具有有穷级的简单障碍.作为应用，我们证明了所有交叉数不超过12且光滑协调阶未知的2-桥纽结都具有无穷光滑协调阶。
Heegaard Floer Invariants and Tight Contact Three--Manifolds
设Y（r）是通过对三球面中的右手三叶结进行有理r-外科手术而得到的封闭定向三流形。利用接触外科学和Heegaard Floer接触不变量，我们在Y（r）上构造了正的紧切触结构，其中r不等于1。这特别意味着，正E6和E7铅垂的定向边界承载正的紧密切触结构，解决了公知的公开问题。
Covering spaces and Q-gradings on Heegaard Floer homology
Heegaard弗洛尔同调是由P. Ozsvath和Z. Szabo将一个3-流形Y与一族相对分次的阿贝尔群HF（Y，t）联系起来，这些群在Y上由Spin^c结构t索引。在Y是有理同调球的情形下，Ozsvath和Szabo将相对Z-分次提升为绝对Q-分次。这导致\bigoplus_{t\in Spin^c(Y)} HF（Y，t）的相对Q分级。本文通过研究覆盖空间的Heegaard弗洛尔同调，给出了这种相对Q-分次的另一种构造.
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.
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
我们探讨了某些限制结在三个领域承认非平凡的塞弗特纤维手术。这些限制源于塞弗特纤维空间的Heegaard弗洛尔同调，因此它们对这种纽结的Alexander多项式和纽结的弗洛尔同调都有影响。特别是，我们表明，某些多项式从来没有亚历山大多项式的结承认同源三球塞弗特纤维手术。另一方面，纽结弗洛尔同调限制也适用于亚历山大多项式没有给出信息的情况，例如木下寺坂纽结。
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.
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
我们计算了任意有理同调3-球面的Heegaard弗洛尔同调，其中有理同调3-球面具有形式为（T，n）的开卷分解，其中T是亏格为1的曲面，具有1个边界分支.此外，我们还计算了S^1上第一个贝蒂数等于1的T^2-丛的Heegaard弗洛尔同调，并将我们的结果与Lebow关于这类环面丛的嵌入接触同调的结果进行了比较.我们使用这些计算的限制斯坦填充的切触结构兼容，这样的开放的书籍，缩小了一些类的3-编织结有限的和谐秩序，并确定所有准交替链接与编织指数最多为3。
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
给定标准切触3-球面中的一个横链，我们研究了作为球面的一个分支双覆盖而产生的切触流形。我们给出了一个接触手术描述这样的流形，它允许确定Heegaard Floer接触不变量的一些。通过Birman-Menasco的结的例子，我们表明，这些接触流形可能无法区分非同位素的横结。我们还研究了分支双覆盖的Heegaard Floer接触不变量与Khovanov同调之间的关系，特别是我们在相关论文中引入的横链不变量。
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.
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.
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。
Real Heegaard Floer homology with integral coefficients
本文将Guth-Manolescu的实Heegaard Floer理论升级为整数系数的ℤ/2-分次不变量，给出相对分次与提升存在的充要障碍，其障碍理论对Floer理论有独立价值。
Bordered Heegaard Floer homology is a three-manifold invariant which associates to a surface F an algebra A(F) and to a three-manifold Y with boundary identified with F a module over A(F). In this paper, we establish naturality properties of this invariant. Changing the diffeomorphism between F and the boundary of Y tensors the bordered invariant with a suitable bimodule over A(F). These bimodules give an action of a suitably based mapping class group on the category of modules over A(F). The Hochschild homology of such a bimodule is identified with the knot Floer homology of the associated op…
Bordered Heegaard Floer homology is an invariant for 3-manifolds, which associates to a surface F an algebra A(Z), and to a 3-manifold Y with boundary, together with an orientation-preserving diffeomorphism ϕfrom F to \bdy Y, a module over A(Z). We study the Grothendieck group of modules over A(Z), and define an invariant lying in this group for every bordered 3-manifold. We prove that this invariant recovers the kernel of the inclusion of H_1(\bdy Y; Z) into H_1(Y; Z) if H_1(Y, \bdy Y; Z) is finite, and is 0 otherwise. We also study the properties of this invariant corresponding to gluing. As…
The hat and plus version of the Heegaard Floer contact invariant are not equivalent
研究揭示了3流形上接触结构的特性，展示存在紧致且零扭转的接触结构，其中Heegaard Floer不变量非零而加号不变量为零。
We define an invariant of contact structures in dimension three from Heegaard Floer homology. This invariant takes values in the set $\mathbb{Z}_{\geq0}\cup\{\infty\}$. It is zero for overtwisted contact structures, $\infty$ for Stein fillable contact structures, non-decreasing under Legendrian surgery, and computable from any supporting open book decomposition. As an application, we obstruct Stein fillability on contact 3-manifolds with non-vanishing Ozsváth-Szabó contact class.
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…
Heegaard diagrams and holomorphic disks
配备有Heegaard图的三流形可以用来建立弗洛尔同调理论，其微分计数Heegaard曲面的$g$ -重对称积中的伪全纯圆盘。这导致了三流形的一个拓扑不变量，Heegaard弗洛尔同调，它在配边下是函子的。在这篇综述文章中，我们勾勒出这种结构，并描述了它的一些拓扑应用。
Heegaard Floer homology and alternating knots
在前面的文章中，我们在定向三流形Y中引入了零同调纽结K的纽结不变量，它与Y的Heegaard弗洛尔同调密切相关。本文研究了三球面上纽结的这些纽结同调群的一些性质。给出了链复形的生成元及其分次的组合描述。在这种描述的帮助下，我们确定了交替结的结同调，表明在这种特殊情况下，它只取决于签名和结的亚历山大多项式（推广了Rasmussen的结果，两桥结）。应用程序包括新的限制亚历山大多项式交替结。
Contact Structures with Distinct Heegaard Floer Invariants
证明了由4-流形X上的非同伦Stein结构诱导的Y= dX上的切触结构具有不同的Heegaard Floer不变量.
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