On the Rabinowitz Floer homology of twisted cotangent bundles

Let (M, g) be a closed connected orientable Riemannian manifold of dimension n????2. Let ??:?=??? ₀?+??? ^* ?? denote a twisted symplectic form on T ^* M, where ${\sigma\in\Omega^{2}(M)}$ is a closed 2-form and ?? ₀ is the canonical symplectic structure ${dp\wedge dq}$ on T ^* M. Suppose that ?? is weakly exact and its pullback to the universal cover ${\widetilde{M}}$ admits a bounded primitive. Let ${H:T^{*}M\rightarrow\mathbb{R}}$ be a Hamiltonian of the form ${(q,p)\mapsto\frac{1}{2}\left|p\right|^{2}+U(q)}$ for ${U\in C^{\infty}(M,\mathbb{R})}$ . Let ?? _k :?=?H ^?1(k), and suppose that k?>?c(g, ??, U), where c(g, ??, U) denotes the Ma?é critical value. In this paper we compute the Rabinowitz Floer homology of such hypersurfaces. Under the stronger condition that k?>?c ₀(g, ??, U), where c ₀(g, ??, U) denotes the strict Ma?é critical value, Abbondandolo and Schwarz (J Topol Anal 1:307?C405, 2009) recently computed the Rabinowitz Floer homology of such hypersurfaces, by means of a short exact sequence of chain complexes involving the Rabinowitz Floer chain complex and the Morse (co)chain complex associated to the free time action functional. We extend their results to the weaker case k?>?c(g, ??, U), thus covering cases where ?? is not exact. As a consequence, we deduce that the hypersurface ?? _k is never (stably) displaceable for any k?>?c(g, ??, U). This removes the hypothesis of negative curvature in Cieliebak et?al. (Geom Topol 14:1765?C1870, 2010, Theorem 1.3) and thus answers a conjecture of Cieliebak, Frauenfelder and Paternain raised in Cieliebak et?al. (2010). Moreover, following Albers and Frauenfelder (2009; J Topol Anal 2:77?C98, 2010) we prove that for k?>?c(g, ??, U), any ${\psi\in\mbox{Ham}_{c}(T^{*}M,\omega)}$ has a leaf-wise intersection point in ?? _k , and that if in addition ${\dim\, H_{*}(\Lambda M;\mathbb{Z}_{2})=\infty}$ , dim M????2, and the metric g is chosen generically, then for a generic ${\psi\in\mbox{Ham}_{c}(T^{*}M,\omega)}$ there exist infinitely many such leaf-wise intersection points.     

Lagrangian Rabinowitz Floer homology and twisted cotangent bundles

We study the following rigidity problem in symplectic geometry: can one displace a Lagrangian submanifold from a hypersurface? We relate this to the Arnold Chord Conjecture, and introduce a refined question about the existence of relative leaf-wise intersection points, which are the Lagrangian-theoretic analogue of the notion of leaf-wise intersection points defined by Moser (Acta. Math. 141(1–2):17–34, 1978). Our tool is Lagrangian Rabinowitz Floer homology, which we define first for Liouville domains and exact Lagrangian submanifolds with Legendrian boundary. We then extend this to the ‘virtually contact’ setting. By means of an Abbondandolo–Schwarz short exact sequence we compute the Lagrangian Rabinowitz Floer homology of certain regular level sets of Tonelli Hamiltonians of sufficiently high energy in twisted cotangent bundles, where the Lagrangians are conormal bundles. We deduce that in this situation a generic Hamiltonian diffeomorphism has infinitely many relative leaf-wise intersection points.     

Index computations in Rabinowitz Floer homology

In this note we study two index questions. In the first we establish the relationship between the Morse indices of the free time action functional and the fixed time action functional. The second is related to Rabinowitz Floer homology. Our index computations are based on a correction term which is defined as follows: around a nondegenerate Hamiltonian orbit lying in a fixed energy level a well-known theorem says that one can find a whole cylinder of orbits parametrized by the energy. The correction term is determined by whether the periods of the orbits are increasing or decreasing as one moves up the orbit cylinder. We also provide an example to show that, even above the Ma?é critical value, the periods may be increasing thus producing a jump in the Morse index of the free time action functional in relation to the Morse index of the fixed time action functional.     

Translated points and Rabinowitz Floer homology

We prove that if a contact manifold admits an exact filling, then every local contactomorphism isotopic to the identity admits a translated point in the interior of its support, in the sense of Sandon [Internat. J. Math. 23 (2012), 1250042]. In addition, we prove that if the Rabinowitz Floer homology of the filling is nonzero, then every contactomorphism isotopic to the identity admits a translated point, and if the Rabinowitz Floer homology of the filling is infinite dimensional, then every contactomorphism isotopic to the identity has either infinitely many translated points, or a translated point on a closed leaf. Moreover, if the contact manifold has dimension greater than or equal to 3, the latter option generically does not happen. Finally, we prove that a generic compactly supported contactomorphism on ${\mathbb{R}^{2n+1}}$ has infinitely many geometrically distinct iterated translated points all of which lie in the interior of its support.     

On the Floer homology of cotangent bundles

This paper concerns Floer homology for periodic orbits and for a Lagrangian intersection problem on the cotangent bundle T* M of a compact orientable manifold M. The first result is a new L^∞ estimate for the solutions of the Floer equation, which allows us to deal with a larger—and more natural—class of Hamiltonians. The second and main result is a new construction of the isomorphism between the Floer homology and the singular homology of the free loop space of M in the periodic case, or of the based loop space of M in the Lagrangian intersection problem. The idea for the construction of such an isomorphism is to consider a Hamiltonian that is the Legendre transform of a Lagrangian on T M and to construct an isomorphism between the Floer complex and the Morse complex of the classical Lagrangian action functional on the space of W^1,2 free or based loops on M. © 2005 Wiley Periodicals, Inc.     

On the colored Jones polynomial, sutured Floer homology, and knot Floer homology

Let K⊂S³, and let denote the preimage of K inside its double branched cover, Σ(S³,K). We prove, for each integer n>1, the existence of a spectral sequence whose E² term is Khovanov's categorification of the reduced n-colored Jones polynomial of (mirror of K) and whose E^∞ term is the knot Floer homology of (when n odd) and of (S³,K#Kr) (when n even). A corollary of our result is that Khovanov's categorification of the reduced n-colored Jones polynomial detects the unknot whenever n>1.     

Vanishing homology

In this paper, we introduce a new homology theory devoted to the study of families such as semialgebraic or subanalytic families, and in general, to any family definable in an o-minimal structure (such as Denjoy–Carleman definable, or ln-exp definable sets). The idea is to study the cycles that are vanishing when we approach a special fiber. This also enables us to derive local metric invariants for germs of definable sets. We prove that the homology groups are finitely generated.     

K-theoretic invariants for Floer homology

This paper defines two K-theoretic invariants, Wh ₁ and Wh ₂, for individual and one-parameter families of Floer chain complexes. The chain complexes are generated by intersection points of two Lagrangian submanifolds of a symplectic manifold, and the boundary maps are determined by holomorphic curves connecting pairs of intersection points. The paper proves that Wh ₁ and Wh ₂ do not depend on the choice of almost complex structures and are invariant under Hamiltonian deformations. The proof of this invariance uses properties of holomorphic curves, parametric gluing theorems, and a stabilization process. Submitted: April 2001, Revised: December 2001, Final version: February 2002.     

Vanishing and nonvanishing theorems for numerically effective line bundles on complex spaces

Summary We prove some vanishing and nonvanishing theorems for numerically effective line bundles, generalizing to complex spaces and morphisms between complex spaces results of Kawamata, Viehweg and Shokurov.A Guido Zappa per il suo settantesimo compleannoSupported by SFB 170 «Geometrie und Analysis», Göttingen (FRG), and by Ministero Pubblica Istruzione.     

Knot Floer homology detects fibred knots

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 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. Dedicated to Professor Boju Jiang on the occasion of his 70th birthday Mathematics Subject Classification (2000) 57R58, 57M27, 57R30     

Instanton Floer homology for lens spaces

Floer constructed instanton homology for integral homology three-spheres. In this paper, we extend instanton Floer homology to lens spaces L(p, q). Moreover we show a gluing formula for a variant of Donaldson invariant along lens spaces. As an application, we prove that ${X = \mathbb {CP}^2 \# \mathbb {CP}^2}$ does not admit a decomposition ${X = X_1 \cup X_2}$ . Here X ₁ and X ₂ are oriented, simply connected, non-spin four-manifolds with b ⁺ = 1 and with boundary L(p, 2), and p is a prime number of the form 16N + 1.     

The growth rate of symplectic Floer homology

The main theme of this paper is to study for a symplectomorphism of a compact surface, the asymptotic invariant which is defined to be the growth rate of the sequence of the total dimensions of symplectic Floer homologies of the iterates of the symplectomorphism. We prove that the asymptotic invariant coincides with asymptotic Nielsen number and with asymptotic absolute Lefschetz number. We also show that the asymptotic invariant coincides with the largest dilatation of the pseudo-Anosov components of the symplectomorphism and its logarithm coincides with the topological entropy. This implies that symplectic zeta function has a positive radius of convergence. This also establishes a connection between Floer homology and geometry of 3-manifolds.     

Vanishing of local homology

 We study the vanishing properties of local homology of complexes of modules without assuming that its homology is artinian. Using vanishing results for local homology and cohomology we prove new vanishing results for Ext- and Tor-modules. Received: 1 August 2002; in final form: 23 September 2002 / Published online: 16 May 2003 Mathematics Subject Classification (1991): 13C12, 13D07, 13D45.     

Floer homology for almost Hamiltonian isotopies

Seidel introduced a homomorphism from the fundamental group ${\pi }_1(\mathrm{Ham}(M))$ of the group of Hamiltonian diffeomorphisms of certain compact symplectic manifolds $(M,\omega )$ to a quotient of the automorphism group $\mathrm{Aut}({\mathit{HF}}_1(M,\omega ))$ of the Floer homology ${\mathit{HF}}_1(M,\omega )$. We prove a rigidity property: if two Hamiltonian loops represent the same element in ${\pi }_1(\mathrm{Diff}(M))$, then the image under the Seidel homomorphism of their classes in ${\pi }_1(\mathrm{Ham}(M))$ coincide. The proof consists in showing that Floer homology can be defined by using ‘almost Hamiltonian’ isotopies, i.e. isotopies that are homotopic relatively to endpoints to Hamiltonian isotopies. To cite this article: A. Banyaga, C. Saunders, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Conley conjecture and local Floer homology

In this paper we connect algebraic properties of the pair-of-pants product in local Floer homology and Hamiltonian dynamics. We show that for an isolated periodic orbit, the product is non-uniformly nilpotent and use this fact to give a simple proof of the Conley conjecture for closed manifolds with aspherical symplectic form. More precisely, we prove that on a closed symplectic manifold, the mean action spectrum of a Hamiltonian diffeomorphism with isolated periodic orbits is infinite.     

Instanton Floer homology and contact structures

We define an invariant of contact 3-manifolds with convex boundary using Kronheimer and Mrowka’s sutured instanton Floer homology theory. This is the first invariant of contact manifolds—with or without boundary—defined in the instanton Floer setting. We prove that our invariant vanishes for overtwisted contact structures and is nonzero for contact manifolds with boundary which embed into Stein fillable contact manifolds. Moreover, we propose a strategy by which our contact invariant might be used to relate the fundamental group of a closed contact 3-manifold to properties of its Stein fillings. Our construction is inspired by a reformulation of a similar invariant in the monopole Floer setting defined by Baldwin and Sivek (arXiv:1403.1930, 2014).     

Mutative hyperbolic homology 3-spheres with the same Floer homology

We construct a large class of finitely many hyperbolic homology 3-spheres making the following invariants equal, simultaneously, the integral homology, the quantum SU(2) invariants, the hyperbolic volume, the hyperbolic isometry group, the -invariant, the Chern-Simons invariant, and the Floer homology.