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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.

本文研究了以下三个问题：（1）（P（G），L（G））-控制的等变手术，协边与表示理论，以及控制流形上出现的各向同性子群的理论;（2）等变框架法映射的等变协边理论的Dress归纳;（3）有限群H <$G之间的注入映射Ind <$D ~ 3G（/）H <$D ~ 3;确定了球面上奥利弗群G的光滑G作用的G不动点流形.我们在研究中获得了以下结果。(1)证明了圆和球面上奥利弗群的不动点分支的一个去插入定理。在一项联合工作中，K。Pawalowski等人的工作，证明了有限G-CW复形上的（P（G），L（G））-向量丛的扩张定理。利用等变稠化理论和这种扩张理论，我们发展了一个控制圆盘上各向同性子群的理论。(2)我们证明了Bak-Morimoto的外科阻塞群是一个Mackey函子，一个绿色函子作用于该Mackey函子上，并且代数Dress的归纳法适用于该阻塞群.此外，我们还证明了手术阻塞的配边不变性，并证明了几何Dress归纳法的有效性。(3)与T. Sumi和M. Yanagihara等人研究了各种有限群H <$G的归纳映射Ind <$D ~ 3G（/）H <$<$D ~ 3，构造了许多G的（P（G），L（G））-匹配对和（P（G），L（G））-间隙模.综合以上结果，我们确定了球面上各种奥利弗群G的光滑G作用的G不动点流形。