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).
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