On Stable Homotopy Types of Some Stunted Spaces

In this note we shall study the stable homotopy types (S-types) of the stunted spaces N'{(G) = N»(G)/N-(Gl where JV»(G) = S" modG are quotients of S* by free orthogonal actions of a closed subgroup G of S. In §2, we show that N'l(G) are homeomorphic to the Thorn spaces N(G)^. If G is not finite, then G is S, S or the normalizer N(S*) of S in S. The case with G = S or S has been treated by Feder and Gitler [8], [9]. We consider the case with G = N(S) in §3. The case with G = Zm (cyclic group of order m) has been treated in [12], [15]. On and after §4, we consider the remaining cases, i.e. the cases with G the binary dihedral or binary polyhedral groups (see §2 for definitions). We examine the representation groups of the generalized quaternion groups D*(2) in §4 and evaluate the orders of some elements of KF(N (D*(2TMyj) in §5 or J(N«(D*(2))) in §§6-7 and study the S-types of NjJ(G) in the final section §8.