A characterization of heaviness in terms of relative symplectic cohomology

TL;DR

This paper characterizes heaviness of compact sets in symplectic manifolds using relative symplectic cohomology, linking geometric properties to algebraic invariants and exploring implications for unions and superheaviness.

Contribution

It establishes a new criterion for heaviness via relative symplectic cohomology and discusses related concepts like superheaviness, advancing understanding in symplectic topology.

Findings

Heavy sets correspond to non-zero relative symplectic cohomology.

Union of non-heavy, Poisson commuting sets is not heavy.

Partial results on superheaviness are provided.

Abstract

For a compact subset $K$ of a closed symplectic manifold $(M, \omega)$, we prove that $K$ is heavy if and only if its relative symplectic cohomology over the Novikov field is non-zero. As an application we show that if two compact sets are not heavy and Poisson commuting, then their union is also not heavy. A discussion on superheaviness together with some partial results are also included.