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A characterization of heaviness in terms of relative symplectic cohomology

Cheuk Yu Mak, Yuhan Sun, Umut Varolgunes

Abstract

For a compact subset $K$ of a closed symplectic manifold $(M, ω)$, 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.

A characterization of heaviness in terms of relative symplectic cohomology

Abstract

For a compact subset of a closed symplectic manifold , we prove that 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.
Paper Structure (11 sections, 36 theorems, 67 equations)

This paper contains 11 sections, 36 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Spectral invariants
  4. Heaviness and superheaviness
  5. Relative symplectic cohomology
  6. Symplectic ideal-valued quasi-measures
  7. Chain level product structure
  8. Proof of results
  9. An algebraic lemma
  10. Heaviness
  11. Superheaviness

Key Result

Theorem 1.1

For a compact subset $K$ of $M$:

Theorems & Definitions (85)

  • Theorem 1.1: Theorem 1.4 EP09
  • Remark 1.2
  • Definition 1.3
  • Conjecture 1.4
  • Remark 1.5
  • Conjecture 1.6: Conjecture 1.52 DGPZ
  • Theorem 1.7
  • Corollary 1.8
  • proof
  • Corollary 1.9
  • ...and 75 more