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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 

Abstract

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).

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Résumé

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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