Study on the structure of the group of homeomorphisms

同胚群的结构研究

• 批准号: 10640096
• 负责人: FUKUI Kazuhiro
• 金额: $ 2.05万
• 依托单位: Kyoto Sangyo University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640096/
• 关键词: foliated manifold; homeomorphism group; commutator; homology; Lipschitz homeomorphism group; G-manifold; compact. leaf; stable; リプシッツ同相写像; ホロモジー; 完全; 葉層構造; 軌道; リプシッツ写像

1. We considered the group of foliation preserving homeomorphisms of a foliated manifold and computed the first homologies of the groups for condimension one 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. Especially the group for the Reeb foliation on the 3-sphere is perfect. Furthermore we showed than if the foliation preserving Lipschitz homeomorphisms of a Lipschitz foliated manifold and computed the first homology of the group of foliation preserving Lipschitz homeomorphisms of a codimension one CィイD11ィエD1-foliated manifold. Then we have a phenomenon different from that in topological case.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 stability of compact leaves. Then we showed that all compact Hausdorff CィイD1rィエD1-foliations of 4-manifolds by hyperbolic surfaces are not stable (1 【less than or equal】 r【less than or equal】∞).

1.研究了叶状流形的叶状保同胚群，并计算了条件为1的叶状保同胚群的第一同调。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。尤其是3球上Reeb叶理的组是完美的。进一步证明了Lipschitz叶化流形的保叶化Lipschitz同胚，并计算了余维为1的C_（11）_（11）_（11）-叶化流形的保叶化Lipschitz同胚群的第一同调.这样我们就有了一个不同于拓扑情况下的现象.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.我们考虑了紧叶的稳定性。然后证明了4-流形上的所有紧Hausdorff C_（？）D_1 r_（？）D_1-叶解都是不稳定的（1 [小于或等于] r[小于或等于]∞）.

项目成果

K.Abe and K.Fukui: "On the structure of automorphisms of manifolds"Proceeding of International Conference on Geometry, Integrability, and Quantization (St. Constantine, Bulgaria). (to appear).

K.Abe 和 K.Fukui：“论流形自同构的结构”国际几何、可积性和量子化会议论文集（保加利亚圣康斯坦丁）。

山田 修司: "牧野書店"Mathematica で楽しむ数理科学. 224 (1999)

山田修二：“牧野书店”与 Mathematica 一起享受数学科学 224 (1999)。

福井和彦-今西英器: "On commutators of foliation preserving homeomorhisms" Jour.Math.Soc.Japan. 51-1. 227-236 (1999)

Kazuhiko Fukui-Hideki Imanishi：“论叶状保持同胚” Jour.Math.Soc.Japan 51-1（1999）。

阿部 孝順: "On the Structtcre of the group of equivaniant dideomoybinos of G-manifolds with codmeusion one orbit"Toplpgy.

Takajun Abe：“关于具有 codmeusion 一个轨道的 G 流形的等价双双核群的结构”Toplpgy。

K.Fukui and H.Imanishi: "On commutators of foliation preserving homecomorphisms"J.Math. Soc. Japan. 51-1. 227-236 (1999)

K.Fukui 和 H.Imanishi：“关于叶状保持同胚的交换子”J.Math。