$S^1$-equivariant symplectic homology and linearized contact homology

TL;DR

This paper introduces three equivalent definitions of $S^1$-equivariant symplectic homology, establishes an isomorphism with linearized contact homology under rational coefficients, and proposes a rigorous alternative to cylindrical contact homology.

Contribution

It provides new definitions, proves an isomorphism with linearized contact homology, and offers a rigorous substitute for cylindrical contact homology.

Findings

Positive $S^1$-equivariant symplectic homology is isomorphic to linearized contact homology with rational coefficients.

Several computations and applications demonstrate the utility of the new framework.

A rigorous alternative to cylindrical/linearized contact homology is introduced.

Abstract

We present three equivalent definitions of $S^1$-equivariant symplectic homology. We show that, using rational coefficients, the positive part of $S^1$-equivariant symplectic homology is isomorphic to linearized contact homology, when the latter is defined. We present several computations and applications, and introduce a rigorously defined substitute for cylindrical/linearized contact homology based on an $S^1$-equivariant construction.