The Evens-Kahn formula for the total Stiefel-Whitney class

Abstract

Let G(X) denote the (augmented) multiplicative group of classical cohomology ring of a space X, with coefficients in Z/2. The (augmented) total Stiefel-Whitney class is a natural homomorphism w: KO(X) -* G(X). We show that the functor G( ) possesses a 'transfer homomorphism' for double coverings such that w commutes with the transfer. This is related to a question of G. Segal. As a special case, we obtain a formula for the total Stiefel-Whitney class of a representation of a finite group induced from a (real) representation of a subgroup of index 2, which is analogous to the one obtained by Evens and Kahn for the total Chern class.