Rabinowitz Floer homology and symplectic homology

Abstract

Abstract. The first two authors have recently defined RabinowitzFloer homology groups RF H * (M, W ) associated to an exact embedding of a contact manifold (M, ξ) into a symplectic manifold (W, ω). These depend only on the bounded component V of W \ M . We construct a long exact sequence in which symplectic cohomology of V maps to symplectic homology of V , which in turn maps to Rabinowitz-Floer homology RF H * (M, W ), which then maps to symplectic cohomology of V . We compute RF H * (ST * L, T * L), where ST * L is the unit cosphere bundle of a closed manifold L. As an application, we prove that the image of an exact contact embedding of ST * L (endowed with the standard contact structure) cannot be displaced away from itself by a Hamiltonian isotopy, provided dim L ≥ 4 and the embedding induces an injection on π1. In particular, ST * L does not admit an exact contact embedding into a subcritical Stein manifold if L is simply connected. We also prove that Weinstein's conjecture holds in symplectic manifolds which admit exact displaceable codimension 0 embeddings. IntroductionLet (W, λ) be a complete convex exact symplectic manifold, with symplectic form ω = dλ (see Section 3 for the precise definition). An embedding ι : M ֒→ W of a contact manifold (M, ξ) is called exact contact embedding if there exists a 1-form α on M such that such that ker α = ξ and α − λ| M is exact. We identify M with its image ι(M ). We assume that W \ M consists of two connected components and denote the bounded component of W \ M by V . One can classically [25] associate to such an exact contact embedding the symplectic (co)homology groups SH * (V ) and SH * (V ). We The first two authors have recently defined for such an exact contact embedding Floer homology groups RF H * (M, W ) for the Rabinowitz action functional [9]. We refer to Section 3 for a recap of the definition and of some useful properties. We will show in particular that these groups do not depend on W , but only on V (the same holds for SH * (V ) and SH * (V )). We shall use in this paper the notation RF H * (V ) and call them Rabinowitz Floer homology groups. Remark 1.1. All (co)homology groups are taken with field coefficients. Without any further hypotheses on the first Chern class c 1 (V ) of the tangent bundle, the symplectic (co)homology and Rabinowitz Floer homology groups are Z 2 -graded. If c 1 (V ) = 0 they are Z-graded, and if c 1 (V ) vanishes on π 2 (V ) the part constructed from contractible loops is Z-graded. This Zgrading on Rabinowitz Floer homology differs from the one in [9] (which takes values in 1 2 + Z) by a shift of 1/2 (see Remark 3.2).Our purpose is to relate these two constructions. The relevant object is a new version of symplectic homology, denotedŠH * (V ), associated to " -shaped" Hamiltonians like the one in Figure 1 on page 20 below. This version of symplectic homology is related to the usual ones via the long exact sequence in the next theorem. Theorem 1.2. There is a long exact sequence (1). . .The long exact sequence (1) can be seen ...

Estimates and Computations in Rabinowitz–floer Homology

“…When M is the sphere, HRF (D * M, λ) was computed in [CF09a], while in the case of an arbitrary closed manifold it has been determined by K. Cieliebak, U. Frauenfelder, and A. Oancea in [CFO09]. In the latter paper, the authors show that the Rabinowitz-Floer homology groups fit into an exact sequence of the form…”

“…Its homology does not depend on the choice of the auxiliary data and is called the Rabinowitz-Floer homology of (W, λ). Rabinowitz-Floer homology has the following important vanishing property: HRF (W, λ) = 0 whenever there is an embedding ϕ : W ֒→ W ′ into the interior part of another Liouville domain (W ′ , λ ′ ), such that ϕ * λ ′ − λ is exact and ϕ(W ) is displaceable within W ′ by a Hamiltonian isotopy (see Theorem 1.2 in [CF09a]; in order to prove this theorem and other invariance results, it is useful to define the Rabinowitz-Floer homology of (W, λ) by using more general ambient manifolds than the completionŴ ; the fact that the resulting homology does not depend on the choice of the ambient manifold is proved in [CFO09], Proposition 3.1). The fact that the Rabinowitz-Floer homology of (W, λ) vanishes implies the existence of closed Reeb orbits on ∂W , because otherwise HRF (W, λ) would be isomorphic to the singular homology of ∂W (see Corollary 3.3 in [Sch06] for a proof of this fact under weaker assumptions).…”

“…It provides obstructions for the existence of contact embeddings of unit cotangent sphere bundles (see [CF09a] and [CFO09]), it allows to prove existence and multiplicity results for leaf-wise intersections points (see [AF08a] and [AF08b]), and it plays a relevant role in the study of the dynamics and the symplectic topology of energy hypersurfaces of magnetic flows on cotangent bundles (see [CFP09], where Rabinowitz-Floer homology is extended to domains whose boundary need not be of contact type, but satisfies substantially weaker assumptions). An important ingredient in all these applications is to determine the Rabinowitz-Floer homology of (D * M, λ), the unit cotangent disk bundle of a closed Riemannian manifold (M, g), equipped with the restriction of the canonical Liouville one-form of the cotangent bundle T * M .…”

Erratum to: A Variational Approach to Givental’s Nonlinear Maslov Index

“…Thus, we can apply the techniques from [CFO10] to obtain L ∞ -bounds for the r-coordinate of solutions of the Rabinowitz-Floer equation. L ∞ -bounds for the Lagrange multiplier follow again by a standard scheme from the Fundamental Lemma 4.5.…”

“…Now we are in the position to construct Floer homology for A κ,R . We choose an almost complex structure J which on Σ × [1, ∞) is of SFT-type (see [CFO10]). We…”