Topological study on the structure of the group of homeomorphisms

同胚群结构的拓扑研究

基本信息

批准号：
12640094
负责人：
FUKUI Kazuhiko
金额：
$ 2.11万
依托单位：
Kyoto Sangyou University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2001
项目状态：
已结题

项目摘要

1. We considered the group of Lipschitz homeomorphisms of a Lipschitz manifold and showed that the group is locally contractible and perfect. As its application, we also showed that the group of equivariant Lipschitz homeomorphisms of a principal G-manifold is perfect when G is a compact Lie group. Furthermore we showed that the group of Lipschitz homeomorphisms of R^n leaving the origin fixed is perfect. As its application, we can show that the groups of Lipschitz homeomorphisms of a Lipschitz orbifold and of foliation preserving Lipschitz homeomorphisms of a compact Hausdorff C^1-foliated manifold are perfect.2. It is known that the equivariant diffeomorphism group of a principal G-manifold M is perfect. If M has at least two orbit types, then it is not true. We determined the first homology group of the equivariant diffeomorphism group of M when M is a G-manifold with codimension one orbit.3. We considered the group of foliation preserving Lipschitz homeomorphisms of a foliated manifold and computed the first homologies of the groups for codimension one C^2-foliations. We showed that if the foliation has no type D components and has only a finite number of type R components, then the group is perfect. Furthermore we showed that if the foliation has a type D component and the linearization map is a C^1-diffeomorphism, then the group is not perfect. But we showed that if the foliation has a type D component and the linearization map is not absolutely continuous, then the group is perfect. This phenomenon is different from that in topological case.

1。我们考虑了Lipschitz歧管的Lipschitz同构同构，并表明该组在本地易于签约且完美。作为其应用，我们还表明，当G是一个紧凑的谎言群体时，主要G-Manifold的一组同型同态非常完美。此外，我们证明了Lipschitz的同构同构，留下来源是完美的。作为其应用，我们可以证明Lipschitz Orbifold的Lipschitz同构和保存Lipchitz同构的叶叶的同构同态的同态形态非常完美。2。众所周知，主要G-Manifold M的差异群是完美的。如果M至少具有两种轨道类型，则不正确。当M是具有Codimension One Orbit的G-Manifold时，我们确定了M的模棱两可的第一个同源组。3。我们考虑了叶状歧管的Lipschitz同构的一组叶面，并计算了这些组的第一批同源性。我们表明，如果叶面没有D型组件，并且只有有限数量的R组件，则该组是完美的。此外，我们表明，如果叶面具有D型成分，并且线性化图是C^1-DiffEomormorlist，则该组并不完美。但是我们表明，如果叶面具有D型成分，并且线性化图不是绝对连续的，那么该组是完美的。在拓扑情况下，这种现象不同。