On the group of symplectic homeomorphisms

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 C⁰-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).