Study of control theory of the fixed point sets on spheres

球面上不动点集控制理论研究

基本信息

批准号：
09640110
负责人：
MORIMOTO Masaharu
金额：
$ 2.24万
依托单位：
OKAYAMA UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1999
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640110/
关键词：
fixed point G-action sphere

项目摘要

The purpose of this research was to study the following three : (1) (P(G), L(G))-controlled equivariant surgery, cobordism and representation theories and a theory to control isotropy subgroups appearing on manifolds ; (2) Dress'induction of the equivariant cobordism theory of equivariant framed normal maps ; (3) the injection maps IndィイD3G(/)HィエD3 among various finite groups H ⊂ G ; and determine the G-fixed point manifolds of smooth G-actions on spheres for Oliver groups G. We obtained the following results in the research. (1) We proved a deleting-inserting theorem of fixed point components on disks and spheres for Oliver Groups. In a joint work K. Pawalowski, we proved an extension theory of (P(G), L(G))-vector bundles on finite G-CW complexes. Using the equivariant thickening theory with this extension theory, we developed a theory to control isotropy subgroups on disks. (2) We proved that Bak-Morimoto's surgery obstruction group is a Mackey functor on which a Green functor acts, and algebraic Dress'induction works for the obstruction group. In addition, we proved that the cobordism invariance of the surgery obstruction and show that geometric Dress'induction works. (3) In joint works with T. Sumi and M. Yanagihara, we studied the induction maps IndィイD3G(/)HィエD3 for various finite groups H ⊂ G, we constructed (P(G), L(G))-matched pairs and (P(G), L(G))-gap modules for many G. Putting all this together, we determined the G-fixed point manifolds of smooth G-actions on spheres for various Oliver groups G.

这项研究的目的是研究以下三个：（1）（p（g），l（g）） - 受控的等效手术，恢复性和表示理论以及控制流形上出现的各向同性亚组的理论；（2）诱导等效框架正常地图的等效恢复理论；（3）在各个有限组中，注射图D3G（/）hie d3；并确定了Oliver组G的平滑G acTions的G固定点歧管。我们在研究中获得了以下结果。（1）我们为Oliver群体提供了固定点成分的固定点成分的删除插入理论。在K. Pawalowski的联合工作中，我们提供了（P（g），L（g）） - 有限G-CW复合物的矢量束的扩展理论。使用等效的增厚理论与这种扩展理论，我们开发了一种理论来控制磁盘上的各向同性亚组。（2）我们规定，Bak-Morimoto的手术目标小组是Mackey函数，绿色函数在其上表现出来，代数着装的诱导性适用于目标组。此外，我们还提供了手术目标的恢复不变性，并表明了几何着装诱导作品。 (3) In joint works with T. Sumi and M. Yanagihara, we studied the introduction maps Indie D3G(/)Hie D3 for various finite groups H ⊂ G, we constructed (P(G), L(G))-matched pairs and (P(G), L(G))-gap modules for many G. Putting all this together, we determined the G-fixed point manifolds of smooth G-actions on spheres for various奥利弗组G.