Study of control theory of the fixed point sets on spheres

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 Ii D3G(/)H I D3among various finite groups H.(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 I D3G(/)H I D 3 for various finite groups HáG，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.

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.