$S^1$-equivariant relative symplectic cohomology and relative symplectic capacities

TL;DR

This paper develops an $S^1$-equivariant framework for relative symplectic cohomology, introduces new capacities to measure subset properties, and compares these capacities under convexity conditions.

Contribution

It constructs an $S^1$-equivariant version of relative symplectic cohomology and introduces related capacities to analyze subset properties in symplectic manifolds.

Findings

Relative symplectic capacities can detect displaceability and heaviness.

The first relative Gutt-Hutchings capacity equals the relative symplectic (co)homology capacity under convexity.

The paper establishes an $S^1$-equivariant structure for the developed cohomology theory.

Abstract

In this paper, we construct an $S^1$-equivariant version of the relative symplectic cohomology developed by Varolgunes. As an application, we construct a relative version of Gutt-Hutchings capacities and a relative version of symplectic (co)homology capacity. We will see that these relative symplectic capacities can detect the diplaceability and the heaviness of a compact subset of a symplectic manifold. We compare the first relative Gutt-Hutchings capacity and the relative symplectic (co)homology capacity and prove that they are equal to each other under a convexity assumption.