Mayer-Vietoris property for relative symplectic cohomology

TL;DR

This paper develops a relative symplectic cohomology invariant for compact subsets of symplectic manifolds and proves a Mayer-Vietoris property under specific geometric conditions involving barriers.

Contribution

It introduces a geometric framework for the Mayer-Vietoris property in relative symplectic cohomology, utilizing barriers and deformation arguments.

Findings

Established the existence of restriction maps for the invariant.

Proved the Mayer-Vietoris property under integrability assumptions.

Utilized deformation techniques focusing on energy-zero Floer solutions.

Abstract

In this paper, we construct a Hamiltonian Floer theory based invariant called relative symplectic cohomology, which assigns a module over the Novikov ring to compact subsets of closed symplectic manifolds. We show the existence of restriction maps, and prove some basic properties. Our main contribution is to identify a natural geometric situation in which relative symplectic cohomology of two subsets satisfy the Mayer-Vietoris property. This is tailored to work under certain integrability assumptions, the weakest of which introduces a new geometric object called a barrier - roughly, a one parameter family of rank 2 coisotropic submanifolds. The proof uses a deformation argument in which the topological energy zero (i.e. constant) Floer solutions are the main actors.