(Γ, Γ′)-Free bordisms, characteristic numbers and stationary point sets

Abstract

C. N. Lee and Wasserman [7] developed the notion of characteristic numbers for G-manifolds and proved their G-bordism invariance. In [2] we defined characteristic numbers for an unoriented singular G-bordism and proved their invariance with regard to singular G-bordism. The case of oriented singular G-bordism is considered in [3] and [4]. One of our primary aims in this paper is to develop these notions for (F, F')-free singular bordisms, F'cY being families of subgroups in a finite group G. (For definition see [6]). In [2] we tackled this problem for some special pairs of families (for so called "almost adjacent" pairs). In an effort to consider more general pairs of families, we get an analogue of Stong's result [6, Proposition 2] for finite abelian groups in w In this section we prove that if (M", 0) is a G-manifold with stationary point free induced action of the subgroup G~, then (M", 0) is a G-boundary, G a finite abelian group. Lastly in w this analogue has been used to show that if the fixed point set F of G2 in M" is nonempty and if F has an equivariant trivial normal bundle in M", then (M", 0) is a G-boundary. The author wishes to thank Dr. P. Jothilingam and Dr. R. Tandon for several helpful discussions and Dr. Kalyan Muldaerjea for this helpful comments. I am indebted to Prof. R. E. Stong for his invaluable suggestions.