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，我们对非紧2-流形的同胚群和嵌入空间进行了分类分成 TOP 中的 2-流形。假设 M 是无边界的 2-流形，X 是的紧次多面体M.设H(M)和ε(X,M)表示M的同胚群和X到M中的嵌入空间。下标“0”表示恒等或包含的连通分量。1． ) 我们证明了限制映射 π : H (M) _0 → ε (X,M)_0 是一个主丛，并得到了光纤连通的充分条件。 2． M 是非紧连通的，我们对 H (M)_0 的同伦类型进行了分类，并表明除了少数情况外它们是可收缩的。我们还对 X 连通的情况下的 ε(X,M) _0 的同伦类型进行了分类。 .3.（拓扑类型）我们证明，在M非紧且连通的情况下，H(M)_0是一个l_2流形，并且ε(X,M)也是一个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拟共形情况）我们获得了PL Lipschitz拟共形同胚和嵌入的相应结果。 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，并且 (2) 当 M 是黎曼曲面时，拟共形同胚的子群 H^<QC>(M)_0 是 Σ-流形。情况下，包含 H^<PL>(M)_0⊂H(M)_0 和 H^<QC>(M)_0⊂H(M)_0是精细同伦等价。