Topics in Heegaard Floer homology
Sucharit Sarkar
这是作者提交给普林斯顿大学的博士论文。本文的结果已经出现在arXiv：math/0607777 v4、arXiv：math/0607691和arXiv：0901.2156中。
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
These are the lecture notes for a course on Heegaard Floer homology held at PCMI in Summer 2019. We describe Heegaard diagrams, Heegaard Floer homology, knot Floer homology, and the relationship between the knot and 3-manifold invariants.
Combinatorial Heegaard Floer homology and nice Heegaard diagrams
我们考虑3-流形Y的hat Heegaard弗洛尔同调的稳定化形式（即闭3-流形的Heegaard弗洛尔同调的U=0变体）。我们给出了一个组合算法来构造这个不变量，从Heegaard分解Y，并给出了其不变性的组合证明。
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。
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.
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
本文利用P. Ozsvath和Z. Szabo
A combinatorial description of the $U^2=0$ version of Heegaard Floer homology
我们表明，每个3-流形承认Heegaard图中的截断版本的Heegaard弗洛尔同调（当全息磁盘通过基点最多一次）可以计算组合。
Combinatorial Heegaard Floer homology and sign assignments
我们提供了一个完整的提升的组合定义的Heegaard弗洛尔同调漂亮的图，并表明，独立性的证明，使用方便的图适应这种设置。
Heegaard Floer Homologies: lecture notes
这些是2014年6月SMF暑期学校关于“三维空间中的几何和量子拓扑”的系列讲座的讲义。重点是从缝合弗洛尔同调的角度研究Heegaard的弗洛尔同调。
Heegaard Floer homology, L-spaces and smoothing order on links I
本文主要研究Heegaard Floer链复形的边界映射为零的L-空间。我们收集这样的流形系统地使用光滑阶链接。
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.
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
受L. P. Roberts关于Heegaard-弗洛尔同调中有理blow-downs的结果的启发，我们研究了沿着3-流形的这种操作，这些操作作为$S^{3}$沿着几个非交替切片结点的分支双覆盖而出现。
An algorithm to compute some Heegaard Floer homologies
前一篇文章试图构造一个算法来计算三维流形Y的Heegaard弗洛尔同调\hat{HF}（Y）。然而，有一个错误的证明，作者，截至目前，无法修复。
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。
We present some non-trivial calculations of Baldwin-Ozsváth-Szabó cohomology of links, and applications to Heegaard-Floer homology of branched double covers.
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。
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
大量重写。为清晰起见，表格和证明已重新格式化和/或重写。
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.
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
我们主要研究Heegaard Floer链复形的边界映射为零的L-空间。在以前的论文\cite{Usui}中，我们利用链上的光滑阶系统地收集了这样的流形。本文在适当的约束下对这类L-空间进行了分类。
Heegaard Floer Homology and Triple Cup Products
我们给出了所有三维流形和扭转Spin^c结构的模系数为2的Heegaard弗洛尔同调的无穷味的完整计算。计算结果与Ozsvath和Szabo的修正计算结果一致。因此，这建立了与马克杯模2同调的同构。
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…
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.
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
如果一个3-流形$Y$包含一个非分离球面，则$Y$的某个扭曲Heegaard弗洛尔同调为零.这个简单的事实使我们能够证明几个结果Dehn手术结在这样的流形。对于$L$ -空间中的纽结也证明了类似的结果。
Algebraic and Heegaard-Floer invariants of knots with slice Bing doubles
如果一个纽结K的Bing双曲是切片的，那么K是代数切片的。此外，Ozsvath-Szabo和Manolescu-Owens发展的Heegaard-Floer和谐不变量在K上消失了.
Rational blow-downs in Heegaard-Floer homology
本文描述了Heegaard-弗洛尔同调中的已知结果如何适用于所有已知的有理吹落的例子，并提供了几个新的四维块，当发现其中一个块嵌入光滑的四维流形中时，这些块可以交换，同时保留一些Ozsváth-Szabó四维流形不变量。
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.
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。
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
利用$H_1(Y)$在Heegaard弗洛尔同调同调中的作用，推广了具有标准$\operatorname{HF}^\infty$的$3$ -流形的Ozsváth-Szabó修正项.我们建立了这些不变量的基本性质：共轭不变性，方向反转下的行为，可加性，和自旋$^c$有理同调配边不变性。
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
利用陶布斯的周期性端点定理，奥克兰给出了环形和双曲不可约整数同调球面的例子，这些球面不是三球面中的结。我们对Heegaard弗洛尔同调中的一个纽结进行了同调球面的处理，给出了一个障碍.这是用来构建无穷多个小塞弗特纤维的例子。
A spanning tree model for the Heegaard Floer homology of a branched double-cover
给定S^3中的一个链K的图，我们为分支双覆盖Sigma（K）画出一个Heegaard图。关联的Heegaard Floer链复形的生成元对应于链图的考夫曼态。利用这个模型，我们对同调\hat{HF}（Sigma（K））作为分次群进行了一些计算.我们还推测存在类似于结Floer和Khovanov同调的delta-分级的\hat{HF}（Sigma（K））上的delta-分级。
A survey of Heegaard Floer homology
这项工作有两个目标。第一个是提供一个概念性的介绍Heegaard弗洛尔同调，第二个是调查目前的状态，该领域，而不是旨在完整性。在回顾了Heegaard弗洛尔同调的结构之后，我们列出了它的一些最重要的应用。其中许多是纯粹的拓扑结果，而不是指Heegaard弗洛尔同调本身。然后，我们简要概述了拉格朗日交弗洛尔同调的构造。我们构造了Heegaard Floer链复形作为上述的一个特例，并试图激发各种看似特别的特征，如可容许性，基点的选择和Spin^c-结构的作用。我们还讨论了不变性的证明下的同调同构的所有选择，以及如何定义Heegaard弗洛尔同调使用这在函子的方式（自然）。接下来，我们将解释为什么Heegaard弗洛尔同调是可计算的，以及它如何适用于各种组合描述。最后一章给出了缝合弗洛尔同调的定义和应用的概述，其中包括一些关键证明的草图。在整个过程中，我们一直试图收集该地区的一些重要的开放式文物。例如，对其中两个问题的肯定回答将给出庞加莱猜测的新证明。
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
我们证明了Ozsvath和Szabo的重整化欧拉示性数的一个外科手术公式。对于有理同调三球面，这个欧拉特征线和Seiberg-Witten不变量之间相等。
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
我们使用Ozsvath和Szabo的算法，找到封闭的公式的帽子版本的Heegaard弗洛尔同调组的非零Dehn手术结的3-球的行列。作为应用程序，我们提供了新的界限上的数量不同的行列的Heegaard Floer组Dehn手术可以有。这些又给出了有理同调3-球面的有理Dehn手术亏格的一个新的下界。我们还提供了新的障碍，一个结是一个潜在的反例布线猜想。
On the contact class in Heegaard Floer homology
我们提出了另一种描述的Ozsvath-Szabo接触类Heegaard弗洛尔同调。利用我们的切触类，我们证明了如果一个切触结构（M，M）有一个适应的开卷分解，其页S是一个一次穿孔环面，那么单值性是右转向的当且仅当该切触结构是紧的.
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.
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.
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
利用由$p$和$q$生成的半群计算了环面结$T_{p,q}$的（+1）手术$S^3_1(K)$的Heegaard Floer同调，并得到了相应的Ozsvath-Szabo d-不变量的紧致公式（包含Dedekind和）.我们的结果与已知的结不变量的$T_{p,q}$的属和Levine-Tristram签名。此外，我们强调了（+1）和（-1）环面结手术的Heegaard弗洛尔同调之间惊人的相似性。这种关系在$τ$函数级别上表现得最好。
An algorithm for computing some Heegaard Floer homologies
本文给出了一个计算闭定向三流形的Heegaard弗洛尔同调的帽子形式的算法。该方法还允许我们计算来自三流形中的零同调链接的过滤。
Rank inequalities for the Heegaard Floer homology of Seifert homology spheres
建立了Seifert纤维积分同调球的Heegaard弗洛尔同调的约化味的秩不等式.将这些不等式与Seifert纤维空间之间的非零次映射的已知分类相结合，我们证明了Seifert同调球面之间的从Y'到Y的映射f产生如下不等式：|华氏度|秩HFred（Y）至多是秩HFred（Y '）。这些不等式也适用于结合算法的Nemethi给出一种方法来解决植物学问题的Heegaard弗洛尔同调的这些流形。
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.
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.
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
这篇论文由于证明中的一个错误而被撤回。这篇论文隐含地假设每一个同伦平凡的纽结都通过负交叉与非纽结相关，这是错误的（正如Peter Ozsvath向作者指出的那样）。
Order in the concordance group and Heegaard Floer homology
利用Ozsváth-Szabó定义的Heegaard-弗洛尔同调修正项，构造了光滑协调群中纽结为有穷级的一个新的障碍.这个障碍承担着一个正式的相似卡森和戈登，但敏感的光滑与拓扑类别之间的差异。作为应用，我们得到了小交叉结点协调序的新的下界。
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。
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。
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.
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
我们使用加边Heegaard弗洛尔同调计算τ不变的一个家庭的卫星结通过扭曲感染沿两个组成部分的Borromean环，一个推广的Whitehead加倍。我们表明，所产生的结的τ只取决于两个扭曲参数和两个同伴结的τ值。我们还包括一些注边Heegaard弗洛尔同调，可能作为一个有用的介绍这个问题。
Heegaard Floer homology of (n,n)-torus links: computations and questions
本文研究了$(n, n)$ -torus链环的Heegaard Floer链环同调.在$\mathbb{R}^{n}$中，支持非平凡同调的Alexander多重分次构成了一串$n-1$单位超立方体，我们计算了几乎所有Alexander分次中同调的秩和分次.我们还猜想一个完整的描述链接同源性，并提供了一些支持这一猜想。本文摘自作者2007年的博士论文。论文，并包含几个开放的问题。
Unknotting information from Heegaard Floer homology
我们利用Heegaard弗洛尔同调得到了unknotting数的界.这是一个概括的Ozsvath和Szabo的阻碍解开第一。我们确定了9_10，9_13，9_35，9_38，10_53，10_101和10_120的解结数;这完成了交叉数为9或更少的素数结的解结数表。我们的障碍使用了Montesinos定理的改进版本，该定理给出了结的分支双重覆盖的Dehn手术描述。
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