Rabinowitz Floer homology and symplectic homology

TL;DR

This paper explores Rabinowitz-Floer homology in symplectic geometry, establishing a long exact sequence linking it with symplectic homology, computing specific examples, and applying results to contact embeddings and Weinstein's conjecture.

Contribution

It constructs a new long exact sequence connecting Rabinowitz-Floer homology with symplectic (co)homology and applies it to contact embedding obstructions and Weinstein's conjecture.

Findings

Computed Rabinowitz-Floer homology for unit cosphere bundles.

Proved non-displaceability of certain contact embeddings.

Confirmed Weinstein's conjecture in specific symplectic manifolds.

Abstract

The Rabinowitz-Floer homology groups $RFH_*(M,W)$ 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 $RFH_*(M,W)$, which then maps to symplectic cohomology of $V$. We compute $RFH_*(ST^*L,T^*L)$, 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 $\dim L\ge 4$ and the embedding induces an injection on $\pi_1$. 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.