Rabinowitz Floer homology and symplectic homology

Abstract

The Rabinowitz-Floer homology groups $RF{H}_{\ast }(M,W)$ are associated to an exact embedding of a contact manifold $(M,\xi )$ into a symplectic manifold $(W,\omega )$. They depend only on the bounded component $V$ of $W\setminus M$. 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 $RF{H}_{\ast }(M,W)$, which then maps to symplectic cohomology of $V$. We compute $RF{H}_{\ast }(S{T}^{\ast }L,T^{\ast }L)$, where $S{T}^{\ast }L$ is the unit cosphere bundle of a closed manifold $L$. As an application, we prove that the image of an exact contact embedding of $S{T}^{\ast }L$ (endowed with the standard contact structure) cannot be displaced away from itself by a Hamiltonian isotopy, provided $\dim L\ge 4$ and the embedding induces an injection on ${\pi }_1$. In particular, $S{T}^{\ast }L$ 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.

59 pages, 8 figures