Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Symplectic topology and ideal-valued measures

Adi Dickstein, Yaniv Ganor, Leonid Polterovich, Frol Zapolsky

Abstract

We adapt Gromov's notion of ideal-valued measures to symplectic topology, and use it for proving new results on symplectic rigidity and symplectic intersections. Furthermore, it enables us to discuss three "big fiber theorems", the Centerpoint Theorem in combinatorial geometry, the Maximal Fiber Inequality in topology, and the Non-displaceable Fiber Theorem in symplectic topology, from a unified viewpoint. Our main technical tool is an enhancement of the symplectic cohomology theory recently developed by Varolgunes.

Symplectic topology and ideal-valued measures

Abstract

We adapt Gromov's notion of ideal-valued measures to symplectic topology, and use it for proving new results on symplectic rigidity and symplectic intersections. Furthermore, it enables us to discuss three "big fiber theorems", the Centerpoint Theorem in combinatorial geometry, the Maximal Fiber Inequality in topology, and the Non-displaceable Fiber Theorem in symplectic topology, from a unified viewpoint. Our main technical tool is an enhancement of the symplectic cohomology theory recently developed by Varolgunes.

Paper Structure

This paper contains 45 sections, 49 theorems, 204 equations, 4 figures.

Table of Contents

  1. Introduction and main results
  2. Three big fiber theorems
  3. Gromov's ideal-valued measures: a review
  4. Symplectic ideal-valued quasi-measures
  5. Quantitative non-displaceable fiber theorem
  6. Rigid fibers of involutive maps and symplectic rigidity
  7. SH-heavy sets
  8. Quantum IVQM versus cohomology IVM
  9. Discussion
  10. Persistence modules and symplectic cohomology
  11. Rigidity of open covers and multi-intersections
  12. Lagrangian IVQMs
  13. Relative symplectic cohomology of pairs
  14. Relative symplectic cohomology
  15. Axioms for relative symplectic cohomology of pairs
  16. ...and 30 more sections

Key Result

Theorem 1.1

Let $Y$ be a metric space of covering dimension $d$, and let $p,n$ be positive integers such that $n \geq p(d+1)$. Then for any continuous map $f {:\ } {\mathbb{T}}^n \to Y$ there is $y_0 \in Y$ such that

Figures (4)

  • Figure 1: The Hamiltonian $H_n$ and the four distinguished sets of 1-periodic orbits
  • Figure 2: The Hamiltonians $L_n$ and $H_n$ with the distinguished sets of 1-periodic orbits
  • Figure 3: The Hamiltonians $L_n$ and $H_n$ with the distinguished sets of 1-periodic orbits
  • Figure 4: The action separation between the lower and upper orbits is demonstrated on the on the Hamiltonians $H_n$. The value $L$ is marked. The tangents whose $y$ intercepts are the actions are depicted in gray, as well as dots representing the actions of the Morse critical points.

Theorems & Definitions (143)

  • Theorem 1.1: Maximal fiber inequality, Gromov gromov2009singularities, gromov2010singularities
  • Theorem 1.2: Topological centerpoint theorem, Karasev, karasev2014covering
  • Definition 1.3
  • Theorem 1.4: Non-displaceable fiber theorem, Entov--Polterovich, entov2006quasi
  • Remark 1.5
  • Definition 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Example 1.10
  • ...and 133 more