On the group of symplectic homeomorphisms

Abstract

Let ( M , ω ) be a closed symplectic manifold. We define a Hofer-like metric d on the identity component Sym ( M , ω ) 0 in the group Symp ( M , ω ) of all symplectic diffeomorphisms of ( M , ω ) . Unlike the Hofer metric on the group Ham ( M , ω ) of Hamiltonian diffeomorphisms, the metric d is not bi-invariant. We show that the metric topology τ defined by d is natural (i.e. independent of the choice involved in its definition). We define the symplectic topology as a blend of the Hofer-like topology τ and the C 0 -topology. We use it to construct a subgroup SSympeo ( M , ω ) of the group Sympeo ( M , ω ) of all symplectic homeomorphisms, containing the group Hameo ( M , ω ) of Hamiltonian homeomorphisms (introduced by Oh and Muller). If M is simply connected SSympeo ( M , ω ) coincides with Hameo ( M , ω ) . Moreover its commutator subgroup [ SSympeo ( M , ω ) , SSympeo ( M , ω ) ] is contained in Hameo ( M , ω ) . To cite this article: A. Banyaga, C. R. Acad. Sci. Paris, Ser. I 346 (2008).