Quantitative Heegaard Floer cohomology and the Calabi invariant

TL;DR

This paper introduces new spectral invariants for Lagrangian links in surfaces, connecting them to the Calabi invariant, and applies these to solve open problems in surface dynamics and symplectic topology.

Contribution

It defines a novel family of spectral invariants that recover the Calabi invariant and extends the Calabi homomorphism to broader groups, addressing key open questions.

Findings

The group of Hamiltonian homeomorphisms of any compact surface with boundary is not simple.

The Calabi homomorphism extends to the group of Hameomorphisms.

Constructs an infinite family of quasimorphisms on the sphere's homeomorphism group.

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-M\"uller; 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.