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

2、3流形同胚群及其子群的同伦类型和拓扑类型研究

基本信息

批准号：
11640074
负责人：
YAGASAKI Tatsuhiko
金额：
$ 2.3万
依托单位：
Kyoto Institute of Technology
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2000
项目状态：
已结题

项目摘要

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.

紧2-流形的CAT同胚群的同伦和拓扑类型由不同的作者针对CAT=DIFF、PL和TOP进行了研究。在这项研究中，我们将非紧2-流形的同胚群和TOP中的2-流形的嵌入空间分类。假设M是一个无边界的2-流形，X是一个紧次多面体M·莱特·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”表示恒等或包含的连通分量。 1．（束）证明限制映射π：H(M)_0→ε(X，M)_0是主束，并得到了光纤连通的充分条件。 2． (同伦型) 在M非紧且连通的情况下，我们对H(M)_0的同伦型进行了分类，并表明除了少数情况外它们都是可收缩的。我们还对X连通情况下ε(X,M)_0的同伦类型进行了分类。3.(拓扑类型)证明了在M非紧连通情况下H(M)_0是l_2流形，并且ε(X,M)也是l_2流形。因此，这些空间的拓扑类型可以根据它们的同伦类型进行分类。 4．（PL Lipschitz 拟共形案例）我们获得了 PL Lipschitz 拟共形同态和嵌入子群的相应结果。例如，(1) 当 M 是非紧连通 PL 2-流形时，PL-同态子群 H^<PL>(M)_0 是 σ^∞-流形，并且 (2) 当 M 是 Ricmann 曲面时，拟共形同态子群 H^<QC>(M)_0 是 Σ-流形。在这些情况下，包含项 H^<PL>(M)_0⊂H(M)_0 和 H^<QC>(M)_0⊂H(M)_0 是精细同伦等价。