Spitting Semifree Finite Group Actions on Homotopy Spheres into Solid Tori

Let G be a finite group acting smoothly and semifreely on the homotopy sphere Σn+k with fixed point set Fn. Let Fn1⊂Fn be a smooth submanifold that has the Z(P)-homology of Sm(0<m<n), where P is the set of primes dividing the order of G. In this paper, the author investigates whether it is possible to decompose Σn+k into G-invariant solid tori Sn×Dk and Dn+1×Sk−1 such that the fixed set for the action of G on Sn×Dk is F1. 

Two necessary conditions for such a decomposition are that G have periodic cohomology of period r and that n≡m(modr). In addition, since F1 is a Z(P)-homology sphere, TorHi(F1) is ZG-cohomologically trivial and determines an element [TorHi(F1)]∈K~0(ZG). Another necessary condition for such a decomposition is shown to be that X~(F1)=∑(−1)i[Hi(F1)]=0 in K~0(ZG). 

The main result of this paper is then the following theorem: Let G, Σn+k, and F1 be as above. Suppose n+k≥6, k>n+2, and m≡n(modr). Then there exists a G-invariant Sn×Dk in Σn+k with fixed set F1 if and only if X~(F1)=0. Thus, when X~(F1)=0, Σn+k has an equivariant decomposition as Sn×Dk∪Dn+1×Sk−1 corresponding to the decomposition F=F1∪Cl(F−F1).