The group of Hamiltonian homeomorphisms and C(0)-symplectic topology

Abstract

The main purpose of this paper is to carry out some of the foundational study of C-0-Hamiltonian geometry and C-0-symplectic topology. We introduce the notion of Hamiltonian topology on the space of Hamiltonian paths and on the group of Hamiltonian diffeomorphisms. We then de. ne the group, denoted by Hameo(M, omega), consisting of Hamiltonian homeomorphisms such that Ham(M, omega) not subset of Hameo(M, omega) subset of Sympeo(M, omega), where Sympeo(M, omega) is the group of symplectic homeomorphisms. We prove Hameo(M, omega) is a normal subgroup of Sympeo( M, omega) and contains all the time-one maps of Hamiltonian vector fields of C-1,C-1-functions, and Hameo(M, omega) is path-connected and so contained in the identity component Sympeo(0)(M, omega) of Sympeo(M, omega). We also prove that the mass flow of any Hamiltonian homeomorphism vanishes. In the case of a closed orientable surface, this implies that Hameo(M, omega) is strictly smaller than the identity component of the group of area-preserving homeomorphisms when M not equal S-2. For M = S-2, we conjecture that Hameo(S-2, omega) is still a proper subgroup of Sympeo(0)(S-2, omega).