On the group of strong symplectic homeomorphisms

Abstract

We generalize the "hamiltonian topology" on hamiltonian isotopies to an intrinsic "symplectic topology" on the space of symplectic isotopies. We use it to define the group $SSympeo(M,\omega)$ of strong symplectic homeomorphisms, which generalizes the group $Hameo(M,\omega)$ of hamiltonian homeomorphisms introduced by Oh and Muller. The group $SSympeo(M,\omega)$ is arcwise connected, is contained in the identity component of $Sympeo(M,\omega)$; it contains $Hameo(M,\omega)$ as a normal subgroup and coincides with it when $M$ is simply connected. Finally its commutator subgroup $[SSympeo(M,\omega),SSympeo(M,\omega)]$ is contained in $Hameo(M,\omega)$.