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Normal subgroups of diffeomorphism and homeomorphism groups of R^n and other open manifolds

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 Added by Paul Schweitzer SJ
 Publication date 2009
  fields
and research's language is English




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We determine all the normal subgroups of the group of C^r diffeomorphisms of R^n, r = 1,2,...,infinity, except when r=n+1 or n=4, and also of the group of homeomorphisms of R^n (r=0). We also study the group A_0 of diffeomorphisms of an open manifold M that are isotopic to the identity. If M is the interior of a compact manifold with nonempty boundary, then the quotient of A_0 by the normal subgroup of diffeomorphisms that coincide with the identity near to a given end e of M is simple.



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For a non-compact n-manifold M let H(M) denote the group of homeomorphisms of M endowed with the Whitney topology and H_c(M) the subgroup of H(M) consisting of homeomorphisms with compact support. It is shown that the group H_c(M) is locally contractible and the identity component H_0(M) of H(M) is an open normal subgroup in H_c(M). This induces the topological factorization H_c(M) approx H_0(M) times M_c(M) for the mapping class group M_c(M) = H_c(M)/H_0(M) with the discrete topology. Furthermore, for any non-compact surface M, the pair (H(M), H_c(M)) is locally homeomorphic to (square^w l_2,cbox^w l_2) at the identity id_M of M. Thus the group H_c(M) is an (l_2 times R^infty)-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^infty-topology and the subgroup D_c(M) of D(M) consisting of all diffeomorphisms with compact support. It is shown that the pair (D(M),D_c(M)) is locally homeomorphic to (square^w l_2, cbox^w l_2) at the identity id_M of M. Hence the group D_c(M) is a topological (l_2 times R^infty)-manifold for any dimension n.
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