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

对于CAT=DIFF，PL和TOP，紧2-流形的CAT-同胚群的同伦和拓扑类型已被许多作者研究过.本文将非紧2-流形的同胚群和嵌入空间的同胚群分类到TOP中的2-流形.设M是无边界2-流形，X是M的紧子多面体，设H（M）和ε（X，M）表示M的同胚群和X到M的嵌入空间。下标“0“表示单位元或包含的连通分支。证明了限制映射π：H（M）_0 →ε（X，M）_0是主丛，并得到了纤维连通的一个充分条件.（同伦型）在M是非紧连通的情形下，我们对H（M）_0的同伦型进行了分类，并证明了它们除了少数情形外都是可收缩的。在X是连通的情况下，对ε（X，M）_0的同伦类型进行了分类.（拓扑型）证明了当M是非紧连通的时，H（M）_0是l_2-流形，ε（X，M）也是l_2-流形。因此，这些空间的拓扑类型可以根据同伦类型进行分类. (PL Lipschitz拟共形情形）我们得到了PL Lipschitz拟共形同胚和嵌入的子群的相应结果。例如：（1）当M是非紧连通PL 2-流形时，PL-同胚的子群H^<PL>（M）_0是σ^∞-流形;（2）当M是Ricmann曲面时，拟共形同胚的子群H^<QC>（M）_0是σ ^∞-流形。在这些情形下，包含H^<PL>（M）_0 &lt;$H（M）_0和H^<QC>（M）_0 &lt;$H（M）_0是细同伦等价的.

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.