Sur les transformations de contact au-dessus des surfaces

Emmanuel Giroux

Published 2001-02-01Version 1

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 $Z^2 \times \{0\}$).