Hamiltonian Floer theory on surfaces

Abstract

We develop connections between the qualitative dynamics of Hamiltonian isotopies on a surface $\Sigma$ 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 $S^1 \times \Sigma$ 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 Humili\`{e}re-Le Roux-Seyfaddini in arXiv:1502.03834.