Lagrangian Floer homology of the Clifford torus and real projective space in odd dimensions

Combinatorial Floer Homology

We define combinatorial Floer homology of a transverse pair of noncontractibe nonisotopic embedded loops in an oriented 2-manifold without boundary, prove that it is invariant under isotopy, and prove that it is isomorphic to the original Lagrangian Floer homology.

Rabinowitz Floer homology for tentacular Hamiltonians

This paper extends the definition of Rabinowitz Floer homology to non-compact hypersurfaces. We present a general framework for the construction of Rabinowitz Floer homology in the non-compact setting under suitable compactness assumptions on the periodic orbits and the moduli spaces of Floer trajectories. We introduce a class of hypersurfaces being the level sets of specific Hamiltonians: strongly tentacular Hamiltonians, for which the compactness conditions are satisfied, thus enabling us to define the Rabinowitz Floer homology for this class. Rabinowitz Floer homology in turn serves as a to…

Floer Homology for Symplectomorphism

Let (M,ω) be a compact symplectic manifold, and ϕbe a symplectic diffeomorphism on M, we define a Floer-type homology FH_*(ϕ) which is a gen- eralization of Floer homology for symplectic fixed points defined by Dostoglou and Salamon for monotone symplectic manifolds. These homology groups are modules over a suitable Novikov ring and depend only on ϕup to a Hamiltonian isotopy.

刘维尔域自同构的Floer同调

Floer homology of automorphisms of Liouville domains

本研究结合刘维尔域的不动点Floer同调与辛同调，并用该方法检测出刘维尔域辛映射类群中的非平凡元素。

On Heegaard Floer homology of plumbed three-manifolds with b_1=1

In this paper we extend the results of \cite{Plumbing} to calculate Ozsváth-Szabó Floer homology group HF+ for a class of negative-semidefinite plumbings with b_1 = 1.

Hypercontact structures and Floer homology

We introduce a new Floer theory associated to a pair consisting of a Cartan hypercontact 3-manifold M and a hyperkaehler manifold X.

The homology of path spaces and Floer homology with conormal boundary conditions

We define the Floer complex for Hamiltonian orbits on the cotangent bundle of a compact manifold satisfying non-local conormal boundary conditions. We prove that the homology of this chain complex is isomorphic to the singular homology of the natural path space associated to the boundary conditions.

A note on local Floer homology

In general, Lagrangian Floer homology - if well-defined - is not isomorphic to singular homology. For arbitrary closed Lagrangian submanifolds a local version of Floer homology is defined in [Flo89, Oh96] which is isomorphic to singular homology. This construction assumes that the involved Hamiltonian function $H$ is sufficiently $C^2$-small and the almost complex structure is sufficiently standard. In this note we develop a new construction of local Floer homology which works for any (compatible) almost complex structure and all Hamiltonian function with Hofer norm less than the minimal (symp…

Corrigendum: "On the Floer homology of cotangent bundles"

We fix an orientation issue which appears in our previous paper about the isomorphism between Floer homology of cotangent bundles and loop space homology. When the second Stiefel-Whitney class of the underlying manifold does not vanish on 2-tori, this isomorphism requires the use of a twisted version of the Floer complex.

A Lagrangian Piunikhin-Salamon-Schwarz morphism and two comparison homomorphisms in Floer homology

In this article we address two issues. First, we explore to what extent the techniques of Piunikhin, Salamon and Schwarz in [PSS96] can be carried over to Lagrangian Floer homology. In [PSS96] an isomorphism between Hamiltonian Floer homology and the singular homology is established. In contrast, Lagrangian Floer homology is not isomorphic to the singular homology of the Lagrangian submanifold, in general. Depending on the minimal Maslov number, we construct for certain degrees two homomorphisms between Lagrangian Floer homology and singular homology. In degrees where both maps are defined we …

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.

The Arnold-Givental conjecture and moment Floer homology

We prove the Arnold-Givental conjecture for a class of Lagrangian submanifolds in Marsden-Weinstein quotients which are fixpoint sets of some antisymplectic involution. For these Lagrangians the Floer homology cannot in general be defined by standard means due to the bubbling phenomenon. To overcome this difficulty we consider moment Floer homology whose boundary operator is defined by counting solutions of the symplectic vortex equations on the strip which have better compactness properties than the original Floer equations.

Gauged Floer homology and spectral invariants

We define a version of spectral invariant in the vortex Floer theory for a $G$-Hamiltonian manifold $M$. This defines potentially new (partial) symplectic quasi-morphism and quasi-states when $M//G$ is not semi-positive. We also establish a relation between vortex Hamiltonian Floer homology and Woodward's quasimap Floer homology by constructing a closed-open string map between them. This yields applications to study non-displaceability problems of subsets in $M//G$

Note on coisotropic Floer homology and leafwise fixed points

For an adiscal or monotone regular coisotropic submanifold $N$ of a symplectic manifold I define its Floer homology to be the Floer homology of a certain Lagrangian embedding of $N$. Given a Hamiltonian isotopy $ϕ=(ϕ^t)$ and a suitable almost complex structure, the corresponding Floer chain complex is generated by the $(N,ϕ)$-contractible leafwise fixed points. I also outline the construction of a local Floer homology for an arbitrary closed coisotropic submanifold. Results by Floer and Albers about Lagrangian Floer homology imply lower bounds on the number of leafwise fixed points. This repro…

周期Floer同调上的闭-开态射

Closed-open morphisms on periodic Floer homology

本文研究从周期Floer同调（PFH）到定量Heegaard Floer同调的闭-开态射，基于Lagrangian链假设构造该态射并证明其非零性，进而推导出PFH-谱不变量与HF-谱不变量之间的关系。

Floer homology of Lagrangian submanifolds

This is a translation of an article appeared in Japanese in Suugaku 63 (2011), no. 1, 43-66 (MR2790665) and is a survey of Lagrangian Floer homology which the author studies jointly with Y.-G.Oh, H. Ohta, and K. Ono. It also contains some explanation on its relation to (homological) mirror symmetry.

Holomorphic disc, spin structures and Floer cohomology of the Clifford torus

We compute the Bott-Morse Floer cohomology of the Clifford torus in $\CP^n$ with all possible spin-structures. Each spin structure is known to determine an orientation of the moduli space of holomorphic discs, and we analyze the change of orientation according to the change of spin structure of the Clifford torus. Also, we classify all holomorphic discs with boundary lying on the Clifford torus by establishing a Maslov index formula for such discs. As a result, we show that in odd dimensions there exist two spin structures which give non-vanishing Floer cohomology of the Clifford torus, and in…

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.

Knot Floer homology of fibred knots and Floer homology of surface diffeomorphisms

We prove that the Knot Floer homology group of a fibred knot of genus g in the Alexander grading 1-g is isomorphic to a version of the fixed point Floer homology of an area-preserving representative of the monodromy.

Periodic Floer homology and Seiberg-Witten Floer cohomology

Various Seiberg-Witten Floer cohomologies are defined for a closed, oriented 3-manifold; and if it is the mapping torus of an area-preserving surface automorphism, it has an associated periodic Floer homology as defined by Michael Hutchings. We construct an isomorphism between a certain version of Seiberg-Witten Floer cohomology and the corresponding periodic Floer homology, and describe some immediate consequences.

Algebras with matchings and knot Floer homology

Knot Floer homology is a knot invariant defined using holomorphic curves. In more recent work, taking cues from bordered Floer homology,the authors described another knot invariant, called "bordered knot Floer homology", which has an explicit algebraic and combinatorial construction. In the present paper, we extend the holomorphic theory to bordered Heegaard diagrams for partial knot projections, and establish a pairing result for gluing such diagrams, in the spirit of the pairing theorem of bordered Floer homology. After making some model calculations, we obtain an identification of a variant…

HF=HM V: Seiberg-Witten-Floer homology and handle addition

This is the last 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.

A cube of resolutions for knot Floer homology

We develop a skein exact sequence for knot Floer homology, involving singular knots. This leads to an explicit, algebraic description of knot Floer homology in terms of a braid projection of the knot.

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.

HF=HM IV: The Seiberg-Witten Floer homology and ech correspondence

This is the fourth 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. The second isomorphism relates the relevant version of the embedded contact homology on the auxilliary manifold with a version of the Seiberg-Witten Floer homology on this same manifold. The third isomorphism relat…

Introduction to sutured Floer homology

This article is a standalone introduction to sutured Floer homology for graduate students in geometry and topology. It is divided into three parts. The first part is an introductory level exposition of Lagrangian Floer homology. The second part is a construction of Heegaard Floer homology as a special, and slightly modified, case of Lagrangian Floer homology. The third part covers the background on sutured manifolds, the definition of sutured Floer homology, as well as a discussion of its most basic properties and implications (it detects the product, behaves nicely under surface decomposition…

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.

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.

通过乘性流树构造等变拉格朗日Floer同调

Equivariant Lagrangian Floer homology via multiplicative flow trees

本文通过在Floer复形上构造并利用$A_\infty$-模结构，提供了等变拉格朗日Floer同调群的构造方法。

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.

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.

An overview of knot Floer homology

Knot Floer homology is an invariant for knots discovered by the authors and, independently, Jacob Rasmussen. The discovery of this invariant grew naturally out of studying how a certain three-manifold invariant, Heegaard Floer homology, changes as the three-manifold undergoes Dehn surgery along a knot. Since its original definition, thanks to the contributions of many researchers, knot Floer homology has emerged as a useful tool for studying knots in its own right. We give here a few selected highlights of this theory, and then move on to some new algebraic developments in the computation of k…

Splicing knot complements and bordered Floer homology

We show that the integer homology sphere obtained by splicing two nontrivial knot complements in integer homology sphere L-spaces has Heegaard Floer homology rank strictly greater than one. In particular, splicing the complements of nontrivial knots in the 3-sphere never produces an L-space. The proof uses bordered Floer homology.

Floer homology of algebraically finite mapping classes

We compute the Floer homology of mapping classes which do not have any pseudo-Anosov components in the sense of Thurston's theory of surface diffeomorphisms. The formula for the Floer homology is obtained from a topological separation of fixed points and a separation mechanism for Floer connecting orbits. As examples, we consider the geometric monodromy of isolated plane curve singularities.

Legendrian knots, transverse knots and combinatorial Floer homology

Using the combinatorial approach to knot Floer homology, we define an invariant for Legendrian knots in the three-sphere, which takes values in link Floer homology. This invariant can be used to also construct an invariant of transverse knots.

Knot Floer homology and integer surgeries

Let $K$ be a null-homologous knot in a three-manifold $Y$. We give a description of the Heegaard Floer homology of integer surgeries on $Y$ along $K$ in terms of the filtered homotopy type of the knot invariant for $K$. As an illustration, we calculate the Heegaard Floer homology groups of non-trivial circle bundles over Riemann surfaces.

Sutured Floer homology and invariants of Legendrian and transverse knots

Using contact-geometric techniques and sutured Floer homology, we present an alternate formulation of the minus and plus version of knot Floer homology. We further show how natural constructions in the realm of contact geometry give rise to much of the formal structure relating the various versions of Heegaard Floer homology. In addition, to a Legendrian or transverse knot K in a contact manifold (Y,ξ), we associate distinguished classes EHL(K) in the minus-version of knot floer homology and EHIL(K) in the plus version, which are each invariant under Legendrian or transverse isotopies of K. Th…

Algebras with matchings and link Floer homology

We explain how to use bordered algebras to compute a version of link Floer homology. As a corollary, we also give a fast computation of the Thuston polytope for links in the three-sphere.

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.

Combinatorial tangle Floer homology

In this paper we extend the idea of bordered Floer homology to knots and links in $S^3$: Using a specific Heegaard diagram, we construct gluable combinatorial invariants of tangles in $S^3$, $D^3$ and $I\times S^2$. The special case of $S^3$ gives back a stabilized version of knot Floer homology.

Bordered Floer homology and existence of incompressible tori in homology spheres

Let $K$ denote a knot inside the homology sphere $Y$. The zero-framed longitude of $K$ gives the complement of $K$ in $Y$ the structure of a bordered three-manifold, which may be denoted by $Y(K)$. We compute the quasi-isomorphism type of the bordered Floer complex of $Y(K)$ in terms of the knot Floer complex associated with $K$. As a corollary, we show that if a homology sphere has the same Heegaard Floer homology as $S^3$ it does not contain any incompressible tori. Consequently, if $Y$ is an irreducible homology sphere $L$-space then $Y$ is either $S^3$, or the Poicaré sphere $Σ(2,3,5)$, or…

On combinatorial link Floer homology

Link Floer homology is an invariant for links defined using a suitable version of Lagrangian Floer homology. In an earlier paper, this invariant was given a combinatorial description with mod 2 coefficients. In the present paper, we give a self-contained presentation of the basic properties of link Floer homology, including an elementary proof of its invariance. We also fix signs for the differentials, so that the theory is defined with integer coefficients.

Indefinite Stein fillings and Pin(2)-monopole Floer homology

Given a spin$^c$ rational homology sphere $(Y,\mathfrak{s})$ with $\mathfrak{s}$ self-conjugate and for which the reduced monopole Floer homology $\mathit{HM}_{\bullet}(Y,\mathfrak{s})$ has rank one, we provide obstructions to the intersection forms of its Stein fillings which are not negative definite. The proof of this result (and of its natural generalizations we discuss) uses $\mathrm{Pin}(2)$-monopole Floer homology.

A note on knot Floer homology of links

Ozsvath and Szabo proved that knot Floer homology determines the genera of knots in S^3. We will generalize this deep result to links in homology 3-spheres, by adapting their method. Our proof relies on a result of Gabai and some constructions related to foliations. We also interpret a theorem of Kauffman in the world of knot Floer homology, hence we can compute the top filtration term of the knot Floer homology for alternative links.

Künneth Formula in Rabinowitz Floer homology

Rabinowitz Floer homology has been investigated on a submanifold of contact type. The contact condition, however, is quite restrictive. For example, a product of contact hypersurfaces is rarely of contact type. In this article, we study Rabinowitz Floer homology for a class of non-contact submanifolds. We show for this example that there are infinitely many leafwise intersection points by proving a Künneth formula for Rabinowitz Floer homology.

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.

Transverse braids and combinatorial knot Floer homology

We describe a new method for combinatorially computing the transverse invariant in knot Floer homology. Previous work of the authors and Stone used braid diagrams to combinatorially compute knot Floer homology of braid closures. However, that approach was unable to explicitly identify the invariant of transverse links that naturally appears in braid diagrams. In this paper, we improve the previous approach in order to compute the transverse invariant. We define a new combinatorial complex that computes knot Floer homology and identify the BRAID invariant of transverse knots and links in the ho…

Nielsen theory, Floer homology and a generalisation of the Poincare-Birkhoff theorem

The purpose of this mostly expository paper is to discuss a connection between Nielsen fixed point theory and symplectic Floer homology theory for symplectomorphisms of surface and a calculation of Seidel's symplectic Floer homology for different mapping classes. We also describe symplectic zeta functions and asympltotic symplectic invariant. A generalisation of the Poincare- Birkhoff fixed point theorem and Arnold conjecture is proposed.

An involution acting nontrivially on Heegard-Floer homology

We show that a certain involution on the well known homology sphere (the boundary of the Mazur manifold) induces a nontrivial homomorphism on its Heegard-Floer homology groups (recently defined by Ozsvath and Szabo). We discuss a possible application of this to constructing exotic smooth structures on 4-manifolds.

Symmetries and adjunction inequalities for knot Floer homology

We derive symmetries and adjunction inequalities of the knot Floer homology groups which appear to be especially interesting for homologically essential knots. Furthermore, we obtain an adjunction inequality for cobordism maps in knot Floer homologies. We demonstrate the adjunction inequalities and symmetries in explicit calculations which recover some of the main results from [1] on longitude Floer homology and also give rise to vanishing results on knot Floer homologies. Furthermore, using symmetries we prove that the knot Floer homology of a fiber distinguishes $\stwo\times\sone$ from other…

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.

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.

Knot Floer homology and rational surgeries

Let $K$ be a rationally null-homologous knot in a three-manifold $Y$. We construct a version of knot Floer homology in this context, including a description of the Floer homology of a three-manifold obtained as Morse surgery on the knot $K$. As an application, we express the Heegaard Floer homology of rational surgeries on $Y$ along a null-homologous knot $K$ in terms of the filtered homotopy type of the knot invariant for $K$. This has applications to Dehn surgery problems for knots in $S^3$. In a different direction, we use the techniques developed here to calculate the Heegaard Floer homolo…

The periodic Floer homology of a Dehn twist

The periodic Floer homology of a surface symplectomorphism, defined by the first author and M. Thaddeus, is the homology of a chain complex which is generated by certain unions of periodic orbits, and whose differential counts certain embedded pseudoholomorphic curves in R cross the mapping torus. It is conjectured to recover the Seiberg-Witten Floer homology of the mapping torus for most spin-c structures, and is related to a variant of contact homology. In this paper we compute the periodic Floer homology of some Dehn twists.