Equivariant surgery theory: Deleting-inserting theorems of fixed point manifolds on spheres and disks

The paper gives a tool to delete and insert fixed point manifolds for smooth actions of finite Oliver groups on spheres and disks. A similar result was already given in a joint article with E. Laitinen and K. Pawalowski for those of finite nonsolvable groups on spheres. It is useful in classifying smooth actions on spheres from the view point of fixed point data. The methods employed in the present paper are equivariant surgery and equivariant connected sum associated with elements in the Burnside ring. The idea of killing surgery obstructions is as follows: Let G be a finite group not of prime power order, C a contractible, finite G-CW complex, and σ an element in a K-theoretic group arising as an obstruction class of geometric object f. It often holds that (1 - [C])^mσ becomes trivial for large integers m, where [C] is the element represented by C in the Burnside ring Ω(G). One expects that the algebraic object (1 - [C])^mσ is realizable as the obstruction class of G-connected sum of f's related to (1 - [C])^m. Since it is true for the case here, we can kill the obstruction σ by taking G-connected sum of f's.