Hamiltonian Floer theory on surfaces: linking, positively transverse foliations and spectral invariants

We develop connections between the qualitative dynamics of Hamiltonian isotopies on a surface Σ 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 × Σ which is positively transverse to the vector field ∂ t ⊕ X H and which is assembled in a straight-forward way from the relevant Floer moduli spaces. Additionally, we provide a purely topological characterization of those Floer chains which both represent the fundamental class in C F ∗ ( 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 Humilière-Le Roux-Seyfaddini in (Humilière et al. in Geom. Topol. 20(4):2253–2334, 2016 ).