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Poisson Bracket Invariants and Wrapped Floer Homology

Yaniv Ganor

TL;DR

The paper develops a bridge between interlinking phenomena in symplectic geometry and the algebraic structure of wrapped Floer homology. By encoding wrapped Floer data as a persistence module with a barcode, it derives concrete lower bounds for the Poisson bracket invariant pb^+ in terms of barcode features, thereby linking dynamical interlinking to Floer-theoretic invariants. The main results yield explicit bounds pb^+ ≥ 1/(μ(b−a)) (and pb^+ ≥ 1/(μb) in a limiting case) when the wrapped Floer barcode contains a bar of the form (μ, Cμ] or (μ, ∞), with applications to cotangent bundles showing interlinking of cotangent fibers and cosphere/zero-section pairs. This framework provides a robust method to certify interlinking via persistence theory and highlights deep connections between Morse-type invariants on path spaces and symplectic invariants.

Abstract

The Poisson bracket invariants, introduced by Buhovsky, Entov, and Polterovich and further studied by Entov and Polterovich, serve as invariants for quadruples of closed sets in symplectic manifolds. Their nonvanishing has significant implications for the existence of Hamiltonian chords between pairs of sets within the quadruple, with bounds on the time-length of these chords. In this work, we establish lower bounds on the Poisson bracket invariants for certain configurations arising in the completion of Liouville domains. These bounds are expressed in terms of the barcode of wrapped Floer homology. Our primary examples come from cotangent bundles of closed Riemannian manifolds, where the quadruple consists of two fibers over distinct points and two cosphere bundles of different radii, or a single cosphere bundle and the zero section.

Poisson Bracket Invariants and Wrapped Floer Homology

TL;DR

The paper develops a bridge between interlinking phenomena in symplectic geometry and the algebraic structure of wrapped Floer homology. By encoding wrapped Floer data as a persistence module with a barcode, it derives concrete lower bounds for the Poisson bracket invariant pb^+ in terms of barcode features, thereby linking dynamical interlinking to Floer-theoretic invariants. The main results yield explicit bounds pb^+ ≥ 1/(μ(b−a)) (and pb^+ ≥ 1/(μb) in a limiting case) when the wrapped Floer barcode contains a bar of the form (μ, Cμ] or (μ, ∞), with applications to cotangent bundles showing interlinking of cotangent fibers and cosphere/zero-section pairs. This framework provides a robust method to certify interlinking via persistence theory and highlights deep connections between Morse-type invariants on path spaces and symplectic invariants.

Abstract

The Poisson bracket invariants, introduced by Buhovsky, Entov, and Polterovich and further studied by Entov and Polterovich, serve as invariants for quadruples of closed sets in symplectic manifolds. Their nonvanishing has significant implications for the existence of Hamiltonian chords between pairs of sets within the quadruple, with bounds on the time-length of these chords. In this work, we establish lower bounds on the Poisson bracket invariants for certain configurations arising in the completion of Liouville domains. These bounds are expressed in terms of the barcode of wrapped Floer homology. Our primary examples come from cotangent bundles of closed Riemannian manifolds, where the quadruple consists of two fibers over distinct points and two cosphere bundles of different radii, or a single cosphere bundle and the zero section.
Paper Structure (21 sections, 21 theorems, 132 equations, 3 figures)

This paper contains 21 sections, 21 theorems, 132 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Structure of the Paper
  3. Acknowledgements
  4. Poisson Bracket Invariant
  5. Persistence Modules
  6. Wrapped Floer Homology
  7. Introduction and Definitions
  8. Action Filtration
  9. Viterbo's Transfer Map
  10. Continuation Maps Associated to an Exact Lagrangian Isotopy
  11. Energy Bounds and C0 bounds.
  12. C0 Bound for Floer Solutions with Moving Boundary Conditions
  13. Energy Bound
  14. Proof of Theorem \ref{['thm:mainTheorem']}
  15. pb+ and Deformations of the Symplectic Form.
  16. ...and 6 more sections

Key Result

Theorem A

Denote by $S_r^*N$ the cosphere bundle of radius $r$ inside $T^*N$. Pick $x,y\in N$, two points of distance $d$. Then, denoting by $T^*_q N$ the cotangent fiber over the corresponding point $q$, we have that for all $0<a<b$:

Figures (3)

  • Figure 1: The boundary decomposition of $\Gamma\cup\Delta$.
  • Figure 2: A pictorial representation of the diagram in the proof of Proposition \ref{['prop:iotaFactorization']}.
  • Figure 3: The map $\psi$ cannot map to the left of $\mu$ or we would have a factorizaiton of a non-zero morphism via the zero morphism.

Theorems & Definitions (50)

  • Definition 1.1
  • Theorem A
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Definition 1.5
  • Theorem B
  • Remark 1.6
  • Theorem 1.7: entov2017lagrangian
  • Remark 1.8
  • ...and 40 more