C0-transport of flux geometry

Abstract

The goal of this paper is to study, in a large scale point of view, the flux geometry of a closed symplectic manifold (M,ω): namely, the topological counterpart of the flux homomorphism. Using metrics arising from the decomposition of closed 1- forms with respect to an arbitrary linear section S, we generalize the construction of the group of strong symplectic homeomorphisms. The flux homomorphism for symplectomorphisms is extended to a surjective group homomorphism Sω0 on the group of S-homeomorphisms. We prove that the kernel of Sω0 is path connected, coincides with the subgroup Hameo(M, ω) of all Hamiltonian homeomorphisms and investigate the discreteness of the corresponding flux group SΓω. Later on, without appealing to any lifting map, we give an alternative proof of a result from the classical flux geometry saying that any smooth symplectic isotopy in Ham(M,ω) is a Hamiltonian isotopy. Furthermore under some hypothesis, we prove that any S-topological isotopy in Hameo(M, ω) is a continuous Hamiltonian isotopy. We also proved that any S-topological isotopy with trivial flux is homotopic to a continuous Hamiltonian isotopy, relatively to fixed endpoints.