On the group of strong symplectic homeomorphisms

TL;DR

This paper introduces a new group of strong symplectic homeomorphisms, extending the concept of Hamiltonian homeomorphisms, and explores its topological and algebraic properties within symplectic geometry.

Contribution

It generalizes the Hamiltonian topology to an intrinsic symplectic topology and defines the group SSympeo, which extends and relates to known groups of symplectic homeomorphisms.

Findings

SSympeo(M,ω) is arcwise connected

It contains Hameo(M,ω) as a normal subgroup

Its commutator subgroup is contained in Hameo(M,ω)

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 $Hameo(M,\omega)$.