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Chapitre D'ouvrage Année : 2001

Sur les transformations de contact au-dessus des surfaces

Emmanuel Giroux

Résumé

Let S be a compact surface - or the interior of a compact surface - and let V be the manifold of cooriented contact elements of S equiped with its canonical contact structure. A diffeomorphism of V that preserves the contact structure and its coorientation is called a contact transformation over S. We prove the following results. 1) If S is neither a sphere nor a torus then the inclusion of the diffeomorphism group of S into the contact transformation group is 0-connected. 2) If S is a sphere then the contact transformation group is connected. 3) if S is a torus then the homomorphism from the contact transformation group of S to the automorphism group of $H_1(V) \\simeq Z^3$ has connected fibers and the image is (known to be) the stabilizer of $Z^2 \\times \\{0\\}$).

Dates et versions

hal-00009387 , version 1 (02-10-2005)

Identifiants

Citer

Emmanuel Giroux. Sur les transformations de contact au-dessus des surfaces. Ghys, Etienne et al. Essays on geometry and related topics Mémoires dédiés à André Haefliger, Genève: L'Enseignement Mathématique, pp.329-350, 2001, Monogr. Enseign. Math. 38, 2-940264-05-8. ⟨hal-00009387⟩

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