Project/Area Number  09640110 
Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Geometry

Research Institution  OKAYAMA UNIVERSITY 
Principal Investigator 
MORIMOTO Masaharu Faculty of Env. Sci. Tech. Okayama University, Associate Professor, 環境理工学部, 助教授 (30166441)

CoInvestigator(Kenkyūbuntansha) 
NAKAJIMA Atsushi Faculty of Env. Sci. Tech. Okayama University, Professor, 環境理工学部, 教授 (30032824)
NODA Ryuzaburo Faculty of Env. Sci. Tech. Okayama University, Professor, 環境理工学部, 教授 (70029726)
SHIMOKAWA Kazuhisa Faculty of Science, Okayama University, Professor, 理学部, 教授 (70109081)
TANAKA Katsumi Faculty of Science, Okayama University, Associate Professor, 理学部, 助教授 (60207082)
IKEHATA Shuichi Faculty of Env. Sci. Tech. Okayama University, Professor, 環境理工学部, 教授 (20116429)
佐々木 徹 岡山大学, 環境理工学部, 講師 (20260664)

Project Period (FY) 
1997 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥3,500,000 (Direct Cost : ¥3,500,000)
Fiscal Year 1999 : ¥1,000,000 (Direct Cost : ¥1,000,000)
Fiscal Year 1998 : ¥1,100,000 (Direct Cost : ¥1,100,000)
Fiscal Year 1997 : ¥1,400,000 (Direct Cost : ¥1,400,000)

Keywords  fixed point / Gaction / sphere 
Research Abstract 
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 Gfixed point manifolds of smooth Gactions on spheres for Oliver groups G. We obtained the following results in the research. (1) We proved a deletinginserting 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 GCW complexes. Using the equivariant thickening theory with this extension theory, we developed a theory to control isotropy subgroups on disks. (2) We proved that BakMorimoto'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 Gfixed point manifolds of smooth Gactions on spheres for various Oliver groups G.
