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).

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

1 Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA
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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/

[1] Banyaga, A. Sur la structure du groupe des difféomorphismes qui préservent une forme symplectique, Comment. Math. Helv., Volume 53 (1978), pp. 174-227

[2] A. Banyaga, A Hofer-like metric on the group of symplectic diffeomorphisms, 2008, preprint

[3] A. Banyaga, On the group of strong symplectic homeomorphisms, 2008, preprint

[4] Hofer, H. On the topological properties of symplectic maps, Proc. Royal Soc. Edinburgh A, Volume 115 (1990), pp. 25-38

[5] Hofer, H.; Zehnder, E. Symplectic Invariants and Hamiltonian Dynamics, Birkhäuser Advanced Texts, Birkhäuser Verlag, 1994

[6] Lalonde, F.; McDuff, D. The geometry of the symplectic energy, Ann. of Math., Volume 141 (1995), pp. 349-371

[7] Oh, Y.-G.; Muller, S. The group of Hamiltonian homeomorphisms and C0-symplectic topology, J. Symp. Geom., Volume 5 (2007), pp. 167-220

[8] Ono, K. Floer–Novikov cohomology and the flux conjecture, Geom. Funct. Anal., Volume 16 (2006) no. 5, pp. 981-1020

[9] Warner, F. Foundations of Differentiable Manifolds and Lie Groups, Scott, Foresman and Company, 1971

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