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 and Müller, and we construct an infinite-dimensional family of quasi-morphisms 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, via orbifold Floer homology, for links composed of parallel circles in the two-sphere. A particular feature of our work is that it avoids the orbifold setting and relies instead on ‘classical’ Floer homology. This not only substantially simplifies the technical background but seems essential for some aspects (such as the application to constructing quasi-morphisms).

Non-realizability of the pure braid group as area-preserving homeomorphisms

“…We prove Theorem 1 by using the quantitative Heegaard Floer homology that is introduced by D. Cristofaro-Gardiner, V. Humilière, C. Mak, S. Seyfaddini and I. Smith [3]. One may use Hutchings's holomorphic curve methods to prove the same result alternatively.…”

“…Remark 3. By P. Biran and O. Cornea's criterions (Proposition 6.1.4 of [2]) and the computations in [3], one should expect that HF (L, H) = 0. But we don't need this result because we only use the filtered version HF (a,b) (L, H).…”

“…We use this property in the energy estimate. Note that the function H satisfying the four conditions always exists (see Remark 6.8 of [3]).…”