Topological study on the structure of the group of homeomorphisms

同胚群结构的拓扑研究

• 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
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640094/
• 关键词: Lipschitz homeomorphism group; 1-dimensional homology; commutator; perfect; foliated manifold; G-manifold; orbifold; 1次元ホモロジー; 群作用; 軌道体; 葉層構造; 同相群; ホモロジー

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流形上的Lipschitz同胚群，并证明了这个群是局部可收缩的和完备的。作为应用，我们还证明了当G是紧李群时，主G-流形的等变Lipschitz同胚群是完备的。此外，我们还证明了R^n的原点不动的Lipschitz同胚群是完备的。作为应用，我们证明了Lipschitz轨道流形的Lipschitz同胚群和紧致Hausdorff C^1-叶流形的保叶Lipschitz同胚群是完备的.已知主G-流形M的等变同同态群是完备的。如果M至少有两种轨道类型，则它不为真。当M是余维为1轨道的G-流形时，我们确定了M的等变同同态群的第一同调群.我们考虑了叶化流形的保Lipschitz同胚的叶化群，并计算了余维为1的C^2-叶化群的第一同调。我们证明了如果叶理没有D型分支，只有有限个R型分支，则群是完全群。进一步证明了如果叶理有D型分支且线性化映射是C^1-单同态，则群不是完美群。但我们证明了如果叶理有D型分量且线性化映射不是绝对连续的，则群是完美的。这一现象与拓扑情况下的现象不同。

项目成果

阿部孝順, 福井和彦: "On the structure of the group of Lipschitz homeomorphisms and its subgroups"J. Math. Soc. Japan. 53-3. 501-511 (2001)

Takajun Abe，Kazuhiko Fukui：“关于 Lipschitz 同胚群及其子群的结构”J. Soc. 501-511。

大山淑之,谷山公規,山田修司: "Realization of Vassilier invariants by unknotting number one knots"Tokyo J.of Math.. (近刊).

Yoshiyuki Oyama、Kiminori Taniyama、Shuji Yamada：“通过解开第一结来实现 Vassilier 不变量”Tokyo J.of Math..（即将出版）。

河野進, 牛瀧文宏: "Geometry of finite spaces and equivariant finite spaces"Current Trends in Transformation Groups(K-theory Monograph Series). (近刊).

Susumu Kono，Fumihiro Ushitaki：“有限空间和等变有限空间的几何”变换群的当前趋势（K理论专着系列）（即将出版）。

阿部孝順, 福井和彦: "On the structure of the group of equivariant diffeomorphisms of G-manifolds with codimension one orbit"Topology. 40. 1325-1337 (2001)

Takajun Abe、Kazuhiko Fukui：“关于余维一轨道的 G 流形的等变微分同胚群的结构”拓扑学 40。1325-1337 (2001)

K. Fukui and E. Hirai: "A note on the first homology of the group of polynomial automorphisms of the coordinate space"Sci. Math. Japonicae. (to appear).

K. Fukui 和 E. Hirai：“关于坐标空间多项式自同构群的第一同调性的注释”Sci。