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Quantitative Heegaard Floer cohomology and the Calabi invariant

Daniel Cristofaro-Gardiner, Vincent Humilière, Cheuk Yu Mak, Sobhan Seyfaddini, Ivan Smith

Abstract

We define a new family of spectral invariants associated to certain Lagrangian links in compact and connected surfaces of any genus. We show that our invariants recover the Calabi invariant of Hamiltonians in their limit. As applications, we resolve several open questions from topological surface dynamics and continuous symplectic topology: we show that the group of Hamiltonian homeomorphisms of any compact surface with (possibly empty) boundary is not simple; we extend the Calabi homomorphism to the group of Hameomorphisms constructed by Oh-Müller; and, we construct an infinite dimensional family of quasimorphisms on the group of area and orientation preserving homeomorphisms of the two-sphere. Our invariants are inspired by recent work of Polterovich and Shelukhin defining and applying spectral invariants for certain classes of links in the two-sphere.

Quantitative Heegaard Floer cohomology and the Calabi invariant

Abstract

We define a new family of spectral invariants associated to certain Lagrangian links in compact and connected surfaces of any genus. We show that our invariants recover the Calabi invariant of Hamiltonians in their limit. As applications, we resolve several open questions from topological surface dynamics and continuous symplectic topology: we show that the group of Hamiltonian homeomorphisms of any compact surface with (possibly empty) boundary is not simple; we extend the Calabi homomorphism to the group of Hameomorphisms constructed by Oh-Müller; and, we construct an infinite dimensional family of quasimorphisms on the group of area and orientation preserving homeomorphisms of the two-sphere. Our invariants are inspired by recent work of Polterovich and Shelukhin defining and applying spectral invariants for certain classes of links in the two-sphere.

Paper Structure

This paper contains 35 sections, 45 theorems, 197 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Recovering the Calabi invariant
  3. The algebraic structure of the group of area-preserving homeomorphisms
  4. Quasimorphisms on the sphere
  5. Quantitative Heegard Floer cohomology and link spectral invariants
  6. Preliminaries
  7. Recollections
  8. Hameomorphisms and finite energy homeomorphisms
  9. The mass-flow and flux homomorphisms
  10. Non-simplicity and the extension of Calabi
  11. The Calabi property
  12. Link spectral invariants for Hamiltonian diffeomorphisms and homeomorphisms
  13. Infinite twists on positive genus surfaces
  14. Calabi on Hameo
  15. Heegaard tori and Clifford tori
  16. ...and 20 more sections

Key Result

Theorem 1.1

Let $\ul{L}^m$ be a sequence of equidistributed Lagrangian links in a closed symplectic surface $(\Sigma, \omega)$. Then, for any $H\in C^\infty([0,1]\times \Sigma)$ we have

Figures (3)

  • Figure 1: In blue, an example of a Lagrangian link with $k=9$ components on a surface $\Sigma$ of genus $2$.
  • Figure 2: A typical example of a link $\ul {L}^m$ for $m$ large in an equidistributed sequence. Here, $\Sigma$ has genus 2, there are 4 non-contractible components in $\ul {L}^m$ (in blue). The disc components in $\Sigma\setminus \ul {L}^m$ are colored in grey.
  • Figure 3: After sliding $L_1$ across $L_2$ along the red arc, we get $L_1'$.

Theorems & Definitions (98)

  • Theorem 1.1: Calabi property
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.6
  • Example 1.7
  • Theorem 1.9
  • Definition 1.12
  • Theorem 1.13
  • Definition 2.1
  • Proposition 2.2
  • ...and 88 more