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Comptes Rendus Mathématique
Volume 346, n° 15-16
pages 867-872 (août 2008)
Doi : 10.1016/j.crma.2008.06.011
Received : 20 Mars 2008 ;  accepted : 28 June 2008
On the group of symplectic homeomorphisms
Sur le groupe des homéomorphismes symplectiques

Augustin Banyaga
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA 


Let   be a closed symplectic manifold. We define a Hofer-like metric d on the identity component   in the group   of all symplectic diffeomorphisms of  . Unlike the Hofer metric on the group   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  -topology. We use it to construct a subgroup   of the group   of all symplectic homeomorphisms, containing the group   of Hamiltonian homeomorphisms (introduced by Oh and Muller). If M is simply connected   coincides with  . Moreover its commutator subgroup   is contained in  . To cite this article: A. Banyaga, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

The full text of this article is available in PDF format.

Soit   une variété symplectique fermée. On définit à la Hofer une métrique d sur la composante connexe de lʼidentité dans le groupe   de tous les difféomorphismes symplectiques. Contrairement à la métrique de Hofer, la métrique d nʼest 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  -topologie. Nous lʼutilisons pour construire un sous-groupe   du groupe   des homéomorphismes symplectiques, qui contient le groupe   des homéomorphismes hamiltoniens (introduits par Oh et Muller). Si M est simplement connexe,   coïncide avec  . De plus, son sous-groupe des commutateurs   est contenu dans  . Pour citer cet article : A. Banyaga, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

The full text of this article is available in PDF format.

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