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.

V2: Improvements to exposition and minor edits addressing referees' remarks. Corrected an error, spotted by Ibrahim Trifa, in the statement and proof of Theorem 7.7(ii); none of the main results are affected

References in corpus (7)

• Quantum Structures for Lagrangian Submanifolds
• Hamiltonian handleslides for Heegaard Floer homology
• Construction of spectral invariants of Hamiltonian paths on closed symplectic manifolds
• Periodic Floer homology and the smooth closing lemma for area-preserving surface diffeomorphisms
• PFH spectral invariants and $C^{\infty }$ closing lemmas
• Lagrangian configurations and Hamiltonian maps
• Subleading asymptotics of link spectral invariants and homeomorphism groups of surfaces

Cited by in corpus (3)

• On the Hofer-Zehnder conjecture on weighted projective spaces
• Exact orbifold fillings of contact manifolds
• Closed-open morphisms on periodic Floer homology