Homeomorphism and Diffeomorphism Groups of Non-compact Manifolds with the Whitney Topology

For a topological manifold M let H(M) be 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. The identity component H0(M) of H(M) is an open normal subgroup in Hc(M) and it induces the factorization Hc(M) ≈ H0(M)×Mc(M) for the mapping class group Mc(M) = Hc(M)/H0(M). For each connected non-compact surface M , it is shown that H0(M) ≈ R∞× l2 and that (a) Mc(M) is trivial ifM ≈ X\K, where X is the disk, the Möbius band, or the annulus, and K is a non-empty closed subset of a boundary circle of X and (b) Mc(M) is an infinite discrete space in all other cases. When M is a smooth n-manifold, let D(M) denote the group of diffeomorphisms of M endowed with the Whitney C∞-topology, D0(M) the identity component of D(M) and Dc(M) the subgroup of D(M) consisting of diffeomorphisms with compact support. It is shown that the group Dc(M) is an (R∞ × l2)-manifold and for n = 2 we obtain a similar result as in the C-case. Furthermore, it is shown that if M is a connected non-compact orientable irreducible 3-manifold without boundary, then D0(M) ≈ R∞× l2 and that if M is a connected compact n-manifold with boundary, then D0(IntM) ≈ D0(M,∂M) × R∞ × l2, where D0(M,∂M) is the identity component of the subgroup D(M,∂M) = {h ∈ D(M) : h|∂M = id∂M}. To prove these results, we develop a general method for recognizing topological groups (locally) homeomorphic to (small) box powers of l2