Study of homotopy types and topological types of homeomorphism groups and their subgroups of 2 and 3-manifolds

点击翻译按钮获取中文摘要

The homotopy ad topological types of groups of CAT-homeomorphism of compact 2-manifolds were studied by various authors for CAT=DIFF, PL and TOP.In this research we classified those of the homeomorphism groups of the noncompact 2-manifolds and the embedding spaces into 2-manifolds in TOP.Suppose M is a 2-manifold without boundary and X is a compact subpolyhedron of M.Let H (M) and ε(X,M) denote the homeomorphism group of M and the space of embeddings of X into M.The subscript "0"denotes the connected component of the identity or the inclusion.1. (Bundle) We showed that the restriction map π : H (M) _0 →ε (X,M)_0 is a principal bundle, and obtained a sufficient condition for the fiber to be connected.2. (Homotopy Type) In the case where M is noncompact and connected, we classified the homotopy type of H (M)_0 and showed that they are contractible except for a few cases. We also classified the homotopy types of ε(X,M) _0 in the case where X is connected.3.(Topological Type) We showed that H(M)_0 is a l_2-manifold in the case where M is noncompact and connected, and thatε(X,M) is also a l_2-manifold. Therefore, the topological types of these spaces can be classified with based upon their homotopy types.4. (PL Lipschitz Quasiconformal case) We obtained the corresponding results for the subgroups of PL Lipschitz Quasiconformal homeomorphisms and embeddings. For example, (1) when M is a noncompact connected PL 2-manifold, the subgroup of PL-homeomorphisms H^<PL>(M)_0 is a σ^∞-manifold, and (2) when M is a Ricmann surface, the subgroup of quasiconformal homeomorphisms H^<QC>(M)_0 is a Σ-manifold. In these cases, the inclusions H^<PL>(M)_0⊂H(M)_0 and H^<QC>(M)_0⊂H(M)_0 are fine homotopy equivalences.