Vanishing of Rabinowitz Floer homology on negative line bundles

TL;DR

This paper constructs Rabinowitz Floer homology for negative line bundles over symplectic manifolds and demonstrates its vanishing, contrasting with the non-vanishing of symplectic homology in these cases.

Contribution

It extends Rabinowitz Floer homology to negative line bundles and establishes a vanishing result, highlighting differences from symplectic homology.

Findings

Rabinowitz Floer homology vanishes for negative line bundles

Symplectic homology does not vanish in these cases

The vanishing relates to the symplectically aspherical condition

Abstract

Following [Fra08, AF14] we construct Rabinowitz Floer homology for negative line bundles over symplectic manifolds and prove a vanishing result. In [Rit14] Ritter showed that symplectic homology of these spaces does not vanish, in general. Thus, the theorem $\mathrm{SH}=0\Leftrightarrow\mathrm{RFH}=0$, [Rit13], does not extend beyond the symplectically aspherical situation. We give a conjectural explanation in terms of the Cieliebak-Frauenfelder-Oancea long exact sequence [CFO10].