Paolo Ghiggini, Gilberto Spano
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.
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We correct a mistake on the citation of JSJ theory in \cite{Ni}. Some arguments in \cite{Ni} are also slightly modified accordingly.
Ozsváth and Szabó conjectured that knot Floer homology detects fibred knots in $S^3$. We will prove this conjecture for null-homologous knots in arbitrary closed 3--manifolds. Namely, if $K$ is a knot in a closed 3--manifold $Y$, $Y-K$ is irreducible, and $\hat{HFK}(Y,K)$ is monic, then $K$ is fibred. The proof relies on previous works due to Gabai, Ozsváth--Szabó, Ghiggini and the author. A corollary is that if a knot in $S^3$ admits a lens space surgery, then the knot is fibred.
Given a closed 3--manifold $Y$, we show that the Heegaard Floer homology determines whether $Y$ fibres over the circle with a fibre of negative Euler characteristic. This is an analogue of an earlier result about knots proved by Ghiggini and the author.
Ozsvath and Szabo conjectured that knot Floer homology detects fibred knots. We propose a strategy to approach this conjecture based on Gabai's theory of sutured manifold decomposition and contact topology. We implement this strategy for genus-one knots, obtaining as a corollary that, if rational surgery on a knot $K$ gives the Poincare homology sphere $Σ(2,3,5)$, then $K$ is the left-handed trefoil knot.
Let K \subset Y be a knot in a three manifold which admits a longitude-framed surgery such that the surgered manifold has first Betti number greater than that of Y. We find a formula which computes the twisted Floer homology of the surgered manifold, in terms of twisted knot Floer homology. Using this, we compute the twisted Heegaard Floer homology \underline{HF}^+ of the mapping torus of a diffeomorphism of a closed Riemann surface whose mapping class is periodic, giving an almost complete description of the structure of these groups. When the mapping class is nontrivial, we find in particula…
Ozsváth and Szabó conjectured that knot Floer homology detects fibred links. We will verify this conjecture for closed 3-braids, by classifying fibred closed 3-braids. In particular, given a nontrivial closed 3-braid, either it is fibred, or it differs from a fibred link by a half twist. The proof uses Gabai's method of disk decomposition.
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.
The symplectic Floer homology HF_*(f) of a symplectomorphism f:S->S encodes data about the fixed points of f using counts of holomorphic cylinders in R x M_f, where M_f is the mapping torus of f. We give an algorithm to compute HF_*(f) for f a surface symplectomorphism in a pseudo-Anosov or reducible mapping class, completing the computation of Seidel's HF_*(h) for h any orientation-preserving mapping class.
Heegaard Floer homology and maximal twisting numbers
本文通过调整Ozsváth-Szabó的完整路径算法，建立了Seifert纤维空间上负扭转紧致接触结构与Heegaard Floer同调群之间的对应关系，揭示了这些结构的分类及其接触不变量c^+的区分。
Bordered Floer homology, handlebody detection, and compressing diffeomorphisms
本研究证明带边Floer同调可在整数同调L-空间连通和的等价下检测柄体及映射类在给定柄体/压缩体的延拓性，还结合Casson-Long思路与轨线理论给出检测映射类是否可延拓到任意压缩体的算法。
We consider symplectic Floer homology in the lowest nontrivial dimension, that is to say, for area-preserving diffeomorphisms of surfaces. Particular attention is paid to the quantum cap product; we show that it distinguishes the trivial element of the mapping class group from any nontrivial one.
We prove the existence of a spectral sequence for Lagrangian Floer homology which converges to the Floer homology of the image of a Lagrangian submanifold under multiple fibred Dehn twists. The $E_1$ term of the sequence is given by the hypercube of "resolutions" of the Dehn twists involved. The proof relies on the exact triangle for fibered Dehn twists due to Wehrheim and Woodward. As applications we obtain a spectral sequences from Khovanov homology to symplectic Khovanov homology. Also when a 3-manifold $M$ is given by gluing two handlebodies by a surface diffeomorphism $ϕ$, we obtain a spe…
We calculate the Heegaard Floer homologies$HF^+(M,s) for mapping tori M associated to certain surface diffeomorphisms, where s is any Spin^c structure on M whose first Chern class is non-torsion. Let gamma and delta be a pair of geometrically dual nonseparating curves on a genus g Riemann surface Sigma_g, and let sigma be a curve separating Sigma_g into components of genus 1 and g-1. Write t-gamma, t_delta, and t_sigma for the right-handed Dehn twists about each of these curves. The examples we consider are the mapping tori of the diffeomorphisms t_gamma^m circ t_delta^n for m,n in Z and that …
We investigate the line between tight and overtwisted for surgeries on fibred transverse knots in contact 3-manifolds. When the contact structure $ξ_K$ is supported by the fibred knot $K \subset M$, we obtain a characterisation of when negative surgeries result in a contact structure with non-vanishing Heegaard Floer contact class. To do this, we leverage information about the contact structure $ξ_{\overline{K}}$ supported by the mirror knot $\overline{K} \subset -M$. We derive several corollaries about the existence of tight contact structures, L-space knots outside $S^3$, non-planar contact …
We show that the reduced quantum hyperbolic invariants of pseudo-Anosov diffeomorphisms of punctured surfaces are intertwiners of local representations of the quantum Teichmüller spaces. We characterize them as the only intertwiners that satisfy certain natural cut-and-paste operations of topological quantum field theories and such that their traces define invariants of mapping tori.
The main purpose of this paper is to provide a description of the fundamental group of a symplectic manifold in terms of Floer theoretic objects. As an application, we show that when counted with a suitable notion of multiplicity, non degenerate Hamiltonian diffeomorphisms have enough fixed points to generate the fundamental group.
If $K$ is a fibered knot in a closed, oriented $3$--manifold $Y$ with fiber $F$, and $\widehat{HFK}(Y,K,[F], g(F)-1;\mathbb Z/2\mathbb Z)$ has rank $r$, then the monodromy of $K$ is freely isotopic to a diffeomorphism with at most $r-1$ fixed points. This generalizes earlier work of Baldwin--Hu--Sivek and Ni. We also clarify a misleading formula in Cotton-Clay's computation of the symplectic Floer homology of mapping classes of surfaces.
Floer homology and right-veering monodromy
本文证明纽结Floer复形可检测纤维化纽结的单值是否右转向，从而给出紧接触结构的纯Floer理论刻画，并应用于Dehn手术与紧叶状结构。
We show that the properties of admitting a co-oriented taut foliation and having a left-orderable fundamental group are equivalent for rational homology $3$-sphere graph manifolds and relate them to the property of not being a Heegaard-Floer L-space. This is accomplished in several steps. First we show how to detect families of slopes on the boundary of a Seifert fibred manifold in four different fashions - using representations, using left-orders, using foliations, and using Heegaard-Floer homology. Then we show that each method of detection determines the same family of detected slopes. Next…
Lipshitz, Ozsváth and Thurston defined a bordered Heegaard Floer invariant CFDA for 3-manifolds with two boundary components, including mapping cylinders for surface diffeomorphisms. We define a related invariant for certain 4-dimensional cobordisms with corners, by associating a morphism F from CFDA(f) to CFDA(g) to each such cobordism between two mapping cylinders f and g. Like the Ozsváth-Szabó invariants of cobordisms between closed 3-manifolds, this morphism arises from counting holomorphic triangles on Heegaard triples. We demonstrate that the homotopy class of the morphism F only depend…
Area-preserving diffeomorphisms of a 2-disc can be regarded as time-1 maps of (non-autonomous) Hamiltonian flows on solid tori, periodic flow-lines of which define braid (conjugacy) classes, up to full twists. We examine the dynamics relative to such braid classes and define a braid Floer homology. This refinement of the Floer homology originally used for the Arnol'd Conjecture yields a Morse-type forcing theory for periodic points of area-preserving diffeomorphisms of the 2-disc based on braiding. Contributions of this paper include (1) a monotonicity lemma for the behavior of the nonlinear C…
We study the classification of slice disks of knots up to isotopy and diffeomorphism using an invariant in knot Floer homology. We compute the invariant of a slice disk obtained by deform-spinning, and show that it can be effectively used to distinguish non-isotopic slice disks with diffeomorphic complements. Given a slice disk of a composite knot, we define a numerical stable diffeomorphism invariant called the rank. This can be used to show that a slice disk is not a boundary connected sum, and to give lower bounds on the complexity of certain hyperplane sections of the slice disk.
We write down an explicit formula for the $+$ version of the Heegaard Floer homology (as an absolutely graded vector space over an arbitrary field) of the results of Dehn surgery on a knot $K$ in $S^3$ in terms of homological data derived from $CFK^{\infty}(K)$. This allows us to prove some results about Dehn surgery on knots in $S^3$. In particular, we show that for a fixed manifold there are only finitely many alternating knots that can produce it by surgery. This is an improvement on a recent result by Lackenby and Purcell. We also derive a lower bound on the genus of knots depending on the…
In this paper, we generalize the work of the second author and prove a grading shifting property, in sutured monopole and instanton Floer theories, for general balanced sutured manifolds. This result has a few consequences. First, we offer an algorithm that computes the Floer homologies of a family of sutured handle-bodies. Second, we obtain a canonical decomposition of sutured monopole and instanton Floer homologies and build polytopes for these two theories, which was initially achieved by Juhász for sutured (Heegaard) Floer homology. Third, we establish a Thurston-norm detection result for …
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…
In this paper, we extend the theory of sutured Floer homology developed by the author. We first prove an adjunction inequality, and then define a polytope P(M,g) in H^2(M,\partial M; R) that is spanned by the Spin^c-structures which support non-zero Floer homology groups. If (M,g) --> (M',g') is a taut surface decomposition, then a natural map projects P(M',g') onto a face of P(M,g); moreover, if H_2(M) = 0, then every face of P(M,g) can be obtained in this way for some surface decomposition. We show that if (M,g) is reduced, horizontally prime, and H_2(M) = 0, then P(M,g) is maximal dimension…
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…
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.
Sutured Floer homology, denoted by SFH, is an invariant of balanced sutured manifolds previously defined by the author. In this paper we give a formula that shows how this invariant changes under surface decompositions. In particular, if (M, γ)--> (M', γ') is a sutured manifold decomposition then SFH(M',γ') is a direct summand of SFH(M, γ). To prove the decomposition formula we give an algorithm that computes SFH(M,γ) from a balanced diagram defining (M,γ) that generalizes the algorithm of Sarkar and Wang. As a corollary we obtain that if (M, γ) is taut then SFH(M,γ) is non-zero. Other applica…
Braid Stability and a Floer theory of Braid Isotopies
本文在闭辛曲面哈密顿流周期轨道辫子空间上建立Floer理论，证明定量辫子稳定性，并应用于二维环面，仅用可缩轨道证明拓扑熵约束集的Hofer距离趋于无穷。
We show that a decorated knot concordance $C$ from $K$ to $K'$ induces a homomorphism $F_C$ on knot Floer homology that preserves the Alexander and Maslov gradings. Furthermore, it induces a morphism of the spectral sequences to $\widehat{HF}(S^3) \cong \mathbb{Z}_2$ that agrees with $F_C$ on the $E^1$ page and is the identity on the $E^\infty$ page. It follows that $F_C$ is non-vanishing on $\widehat{HFK}_0(K, τ(K))$. We also obtain an invariant of slice disks in homology 4-balls bounding $S^3$. If $C$ is invertible, then $F_C$ is injective, hence $\dim \widehat{HFK}_j(K,i) \le \dim \widehat{…
We construct Bott-type and equivariant Seiberg-Witten Floer homology and cohomology for 3-manifolds, in particular rational homology spheres, and prove their diffeomorphism invariance. This paper is a revised version of math.DG/9701010. Some typos are removed.
We analyze the effect of adding, removing, and moving basepoints on link Floer homology. We prove that adding or removing basepoints via a procedure called quasi-stabilization is a natural operation on a certain version of link Floer homology, which we call $CFL_{UV}^\infty$. We consider the effect on the full link Floer complex of moving basepoints, and develop a simple calculus for moving basepoints on the link Floer complexes. We apply it to compute the effect of several diffeomorphisms corresponding to moving basepoints. Using these techniques we prove a conjecture of Sarkar about the map …
A knot \widetilde{K} \subset S^3 is q-periodic if there is a \mathbb Z_q-action preserving \widetilde{K} whose fixed set is an unknot U. The quotient of \widetilde{K} under the action is a second knot K. We construct equivariant Heegaard diagrams for q-periodic knots, and show that Murasugi's classical condition on the Alexander polynomials of periodic knots is a quick consequence of these diagrams. For \widetilde{K} a two-periodic knot, we show there is a spectral sequence whose E^1 page is \hat{\mathit{HFL}}(S^3,\widetilde{K}\cup U)\otimes V^{\otimes (2n-1)})\otimes \mathbb Z_2((θ)) and whos…
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 construct equivariant and Bott-type Seiberg-Witten Floer homology and cohomology for 3-manifolds, in particular rational homology spheres, and prove their diffeomorphism invariance. We present several versions of the equivariant theory: the singular version, the de Rham version and the Cartan version, with the first playing the most important role. These versions are shown to be equivalent to each other. A few typos are removed.
Using Heegaard Floer homology, we construct a numerical invariant for any smooth, oriented $4$-manifold $X$ with the homology of $S^1 \times S^3$. Specifically, we show that for any smoothly embedded $3$-manifold $Y$ representing a generator of $H_3(X)$, a suitable version of the Heegaard Floer $d$ invariant of $Y$, defined using twisted coefficients, is a diffeomorphism invariant of $X$. We show how this invariant can be used to obstruct embeddings of certain types of $3$-manifolds, including those obtained as a connected sum of a rational homology $3$-sphere and any number of copies of $S^1 …
Bordered Heegaard Floer homology is an invariant for 3-manifolds, which associates to a surface F an algebra A(F), and to a 3-manifold Y with boundary, together with an orientation-preserving diffeomorphism from F to the boundary of Y, a module over A(F). In a previous paper, we defined relative Z/2 differential gradings on the algebra A(F) and the modules over it. In this paper, we turn the relative grading into an absolute one, and show that the resulting Z/2-graded module is an invariant of the bordered 3-manifold.
Gompf's cork and Heegaard Floer homology
本文利用 Heegaard Floer 同调推导出双扭转结 K 上的一般条件，使 swallow-follow 操作下的 1/n-手术成为软木塞边界，从而构造出更多软木塞例子并推广到其他微分同胚。
Kronheimer and Mrowka defined invariants of balanced sutured manifolds using monopole and instanton Floer homology. Their invariants assign isomorphism classes of modules to balanced sutured manifolds. In this paper, we introduce refinements of these invariants which assign much richer algebraic objects called projectively transitive systems of modules to balanced sutured manifolds and isomorphisms of such systems to diffeomorphisms of balanced sutured manifolds. Our work provides the foundation for extending these sutured Floer theories to other interesting functorial frameworks as well, and …
Projective Naturality in Heegaard Floer Homology
该研究在Heegaard Floer同调领域，证明了Heegaard Floer不变量可给出到$\text{Mod}_{\text{Mod}}$射影化范畴中可迁系统范畴的函子，揭示了其底层射影同伦范畴结构，并给出对合同调应用及推广方向。
Given a self-diffeomorphism h of a closed, orientable surface S and an embedding f of S into a three-manifold M, we construct a mutant manifold N by cutting M along f(S) and regluing by h. We will consider whether there are any gluings such that for any embedding, the manifold and its mutant have isomorphic Heegaard Floer homology. In particular, we will demonstrate that if the gluing is not isotopic to the identity, then there exists an embedding of S into a three-manifold M such that the rank of the non-torsion summands of the Heegaard Floer homology of M differs from that of its mutant. We …
Suppose S is a compact surface with boundary, and let g be a diffeomorphism of S which fixes the boundary pointwise. We denote by (M_{S,g},ξ_{S,g})$ the contact 3-manifold compatible with the open book (S,g). In this article, we construct a Stein cobordism from the contact connected sum (M_{S,h},ξ_{S,h}) # (M_{S,g},ξ_{S,g}) to (M_{S,hg},ξ_{S,hg}), for any two boundary-fixing diffeomorphisms h and g. This cobordism accounts for the comultiplication map on Heegaard Floer homology discovered in an earlier paper by the author, and it illuminates several geometrically interesting monoids in the map…
An Excision Theorem in Heegaard Floer Theory
本文证明亏格一情形下以泛Novikov环系数扭曲的Heegaard Floer同调切除定理，并用于区分流形及计算双分支链环0手术的同调群。
Let M be a weakly monotone symplectic manifold, and H be a time-dependent Hamiltonian; we assume that the periodic orbits of the corresponding time-dependent Hamiltonian vector field are non-degenerate. We construct a refined version of the Floer chain complex associated to these data and any regular covering of M, and derive from it new lower bounds for the number of periodic orbits. We prove in particular that if the fundamental group of M is finite and solvable or simple, then the number of periodic orbits is not less than the minimal number of generators of the fundamental group. For a gen…
Corks, involutions, and Heegaard Floer homology
基于已有代数框架，本文提出一组探测软木塞的Floer理论不变量，通过局部等价形式体系阻碍对合在同调球上的扩展，并定义同调配边群修改以证明其包含强不可扩展软木塞子群。
Small Dehn surgery and SU(2)
本文证明三维球面中非平凡纽结的3-手术基本群总有不可约SU(2)-表示，回答Kronheimer-Mrowka问题，并推广至区间[3,5)的无穷多斜率。
This paper's theme is the relation between several classical and well-known objects: triangle Fuchsian groups, quasi-homogeneous singularities of plane curves, torus knot complements in the 3-sphere. Torus knots are the only nontrivial knots whose complements admit transitive Lie group actions. In fact S^3\K_{p,q} is diffeomorphic to a coset space of the universal covering group of PSL_2(R) with respect to a discrete subgroup G contained in the preimage of a (p,q,\infty)-triangle Fuchsian group. The existence of such a diffeomorphism between is known from a general topological classification o…
On four-dimensional Dehn twists and Milnor fibrations
该研究借助磁单极Floer同调进展计算族Seiberg--Witten不变量，证明极小椭圆奇点链不定辛填充的边界Dehn扭转无限阶非平凡，揭示了四维空间中三类全新的拓扑与辛几何现象。
We give constraints on smooth families of 4-manifolds with boundary using Manolescu's Seiberg-Witten Floer stable homotopy type, provided that the fiberwise restrictions of the families to the boundaries are trivial families of 3-manifolds. As an application, we show that, for a simply-connected oriented compact smooth 4-manifold $X$ with boundary with an assumption on the Frøyshov invariant or the Manolescu invariants $α, β, γ$ of $\partial X$, the inclusion map $\mathrm{Diff}(X,\partial) \hookrightarrow \mathrm{Homeo}(X,\partial)$ between the groups of diffeomorphisms and homeomorphisms whic…
We study quasi-morphisms on the groups Pn of pure braids on n strings and on the group D of compactly supported area-preserving diffeomorphisms of an open two-dimensional disc. We show that it is possible to build quasi-morphisms on Pn by using knot invariants which satisfy some special properties. In particular, we study quasi-morphisms which come from knot Floer homology and Khovanov-type homology. We then discuss possible variations of the Gambaudo-Ghys construction, using the above quasi-morphisms on Pn to build quasi-morphisms on the group D of diffeomorphisms of a 2-disc.
From a handlebody-theoretic perspective, the simplest compact, contractible 4-manifolds, other than the 4-ball, are Mazur manifolds. We produce the first pairs of Mazur manifolds that are homeomorphic but not diffeomorphic. Our diffeomorphism obstruction comes from our proof that the knot Floer homology concordance invariant $ν$ is an invariant of the smooth 4-manifold associated to a knot in the 3-sphere by attaching an n-framed 2-handle to the 4-ball along the knot. In contrast, we also show (modulo forthcoming work of Ozsváth and Szabó) that the concordance invariants $τ$ and $ε$ are not in…
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