Homeomorphism and diffeomorphism groups of non-compact manifolds with the Whitney topology
Abstract
For a non-compact n-manifold M let H(M) denote the group of homeomorphisms of M endowed with the Whitney topology and Hc(M) the subgroup of H(M) consisting of homeomorphisms with compact support. It is shown that the group Hc(M) is locally contractible and the identity component H0(M) of H(M) is an open normal subgroup in Hc(M). This induces the topological factorization Hc(M) ≈ H0(M) × c(M) for the mapping class group c(M) = Hc(M)/H0(M) with the discrete topology. Furthermore, for any non-compact surface M, the pair (H(M), Hc(M)) is locally homeomorphic to (w l2,w l2) at the identity idM of M. Thus the group Hc(M) is an (l2 × R∞)-manifold. We also study topological properties of the group D(M) of diffeomorphisms of a non-compact smooth n-manifold M endowed with the Whitney C∞-topology and the subgroup Dc(M) of D(M) consisting of all diffeomorphisms with compact support. It is shown that the pair (D(M),Dc(M)) is locally homeomorphic to (w l2, w l2) at the identity idM of M. Hence the group Dc(M) is a topological (l2 × R∞)-manifold for any dimension n.