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Hamiltonian Floer theory on surfaces

Dustin Connery-Grigg

Abstract

We develop connections between the qualitative dynamics of Hamiltonian isotopies on a surface $Σ$ and their chain-level Floer theory using ideas drawn from Hofer-Wysocki-Zehnder's theory of finite energy foliations. We associate to every collection of capped $1$-periodic orbits which is `maximally unlinked relative the Morse range' a singular foliation on $S^1 \times Σ$ which is positively transverse to the vector field $\partial_t \oplus X^H$ and which is assembled in a straight-forward way from the relevant Floer moduli spaces. This provides a Floer-theoretic method for producing foliations of the type which appear in Le Calvez's theory of positively transverse foliations for surface homeomorphisms. Additionally, we provide a purely topological characterization of those Floer chains which both represent the fundamental class in $CF_*(H,J)$, and which lie in the image of some chain-level PSS map. This leads to the definition of a novel family of spectral invariants which share many of the same formal properties as the Oh-Schwarz spectral invariants, and we compute the novel spectral invariant associated to the fundamental class in entirely dynamical terms. This significantly extends a project initiated by Humilière-Le Roux-Seyfaddini in arXiv:1502.03834.

Hamiltonian Floer theory on surfaces

Abstract

We develop connections between the qualitative dynamics of Hamiltonian isotopies on a surface and their chain-level Floer theory using ideas drawn from Hofer-Wysocki-Zehnder's theory of finite energy foliations. We associate to every collection of capped -periodic orbits which is `maximally unlinked relative the Morse range' a singular foliation on which is positively transverse to the vector field and which is assembled in a straight-forward way from the relevant Floer moduli spaces. This provides a Floer-theoretic method for producing foliations of the type which appear in Le Calvez's theory of positively transverse foliations for surface homeomorphisms. Additionally, we provide a purely topological characterization of those Floer chains which both represent the fundamental class in , and which lie in the image of some chain-level PSS map. This leads to the definition of a novel family of spectral invariants which share many of the same formal properties as the Oh-Schwarz spectral invariants, and we compute the novel spectral invariant associated to the fundamental class in entirely dynamical terms. This significantly extends a project initiated by Humilière-Le Roux-Seyfaddini in arXiv:1502.03834.

Paper Structure

This paper contains 41 sections, 92 theorems, 249 equations.

Table of Contents

  1. Introduction
  2. Structure of this paper
  3. Acknowledgments
  4. Capped braids and homological linking
  5. Capped Braids
  6. Linking of capped loops with close strands
  7. The homological linking number for capped braids
  8. Elements of Floer theory and linking
  9. Floer theory
  10. Continuation maps
  11. The PSS isomorphism
  12. The Gromov trick and positivity of intersections
  13. Asymptotic analysis for pseudoholomorphic cylinders
  14. Winding of eigenvectors of $A_{x,J}$
  15. Constructing chain-level continuation maps with prescribed behaviour
  16. ...and 26 more sections

Key Result

Theorem A

Let $(\Sigma,\omega)$ be a closed symplectic surface. $H \in C^\infty(S^1 \times \Sigma)$ be a non-degenerate Hamiltonian, and let $J \in C^\infty(S^1; \mathcal{J}_\omega(\Sigma))$ be such that $(H,J)$ is Floer regular. For any capped braid $\hat{X} \in murm(H)$, we may construct an oriented singula

Theorems & Definitions (210)

  • Theorem A
  • Theorem B
  • Theorem C
  • Corollary
  • Theorem D
  • Theorem E
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Remark 1
  • ...and 200 more