2001/02/01 by Emmanuel Giroux, Giroux, Emmanuel
Mathematics · #53D10 (secondary) #57M50 #57R17 (primary) 53D35 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Symplectic Geometry (math.SG) #math.DG #math.GT #math.SG #msc:53D10 #msc:53D35 #msc:57M50 #msc:57R17
paper · pdf · doi:10.48550/arxiv.math/0102009
15 pages, LaTeX
arxiv created 2001/02/01 · openalex publication_date 2001/02/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 H1(V) ≃ Z3 has connected fibers and the image is (known to be) the stabilizer of Z2 × \0\).