Differential Geometry
On the group of symplectic homeomorphisms
Comptes Rendus. Mathématique, Volume 346 (2008) no. 15-16, pp. 867-872.

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).

Soit (M,ω) une variété symplectique fermée. On définit à la Hofer une métrique d sur la composante connexe de l'identité dans le groupe Symp(M,ω) de tous les difféomorphismes symplectiques. Contrairement à la métrique de Hofer, la métrique d n'est pas bi-invariante. Nous montrons que la topologie métrique τ définie par d est naturelle (i.e. indépendante des choix faits pour la définir). Nous définissons la topologie symplectique comme une combinaison de la topologie τ et de la C0-topologie. Nous l'utilisons pour construire un sous-groupe SSympeo(M,ω) du groupe Sympeo(M,ω) des homéomorphismes symplectiques, qui contient le groupe Hameo(M,ω) des homéomorphismes hamiltoniens (introduits par Oh et Muller). Si M est simplement connexe, SSympeo(M,ω) coïncide avec Hameo(M,ω). De plus, son sous-groupe des commutateurs [SSympeo(M,ω),SSympeo(M,ω)] est contenu dans Hameo(M,ω).

Published online:
DOI: 10.1016/j.crma.2008.06.011
Banyaga, Augustin 1

1 Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA
     author = {Banyaga, Augustin},
     title = {On the group of symplectic homeomorphisms},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {867--872},
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     year = {2008},
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Banyaga, Augustin. On the group of symplectic homeomorphisms. Comptes Rendus. Mathématique, Volume 346 (2008) no. 15-16, pp. 867-872. doi : 10.1016/j.crma.2008.06.011. http://www.numdam.org/articles/10.1016/j.crma.2008.06.011/

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