The Gysin exact sequence for $S^1$-equivariant symplectic homology

TL;DR

This paper introduces an $S^1$-equivariant symplectic homology for symplectically aspherical manifolds with contact boundary, establishing a Gysin exact sequence relating it to standard symplectic homology.

Contribution

It defines a new $S^1$-equivariant symplectic homology and proves its relation to classical symplectic homology via a Gysin exact sequence, using a parametrized Floer-theoretic approach.

Findings

Established the $S^1$-equivariant symplectic homology for certain manifolds.

Proved the Gysin exact sequence relating equivariant and non-equivariant symplectic homology.

Introduced a parametrized version of symplectic homology for families of Hamiltonians.

Abstract

We define $S^1$-equivariant symplectic homology for symplectically aspherical manifolds with contact boundary, using a Floer-type construction first proposed by Viterbo. We show that it is related to the usual symplectic homology by a Gysin exact sequence. As an important ingredient of the proof, we define a parametrized version of symplectic homology, corresponding to families of Hamiltonian functions indexed by a finite dimensional smooth parameter space.