Transfers in the group of multiplicative units of the classical cohomology ring and Stiefel-Whitney classes
For a space X, let G(x) denote the group of series of the form \(1+x_ 1+x_ 2+...\), with \(x_ i\in H^ i(X;{\mathbb{Z}}/2)\), where group operation is given by multiplication of series. The group G(X) can be identified with the set of degree preserving ring homomorphisms \(Hom_{Ring}(H^*(BO;{\mathbb{Z}}/2),H^*(X;{\mathbb{Z}}/2))\) by assigning to a homomorphism f the element \(f(1+w_ 1+w_ 2+...)\in G(X)\) where \(w_ i\) denotes the ith Stiefel-Whitney class. This identification provides the natural transformation of functors w: KO(X)\(\to G(X)\), by \(w(\phi)=\phi^*(1+w_ 1+...)\), where \(\phi\) : \(X\to BO\). The author defines for a finite covering \(\pi\) : \(X\to Y\) a group homomorphism \(N^ w_{\pi}G(X)\to G(Y)\) which, a) satisfies all properties of transfer for finite coverings, b) is uniquely characterized by them and a normalization in terms of Evens transfer for double coverings, and c) satisfies \(w\circ \pi !=N^ w_{\pi}\circ w\), where \(\pi\) !: KO(X)\(\to KO(Y)\) denotes the transfer in KO.
scientific article; zbMATH DE number 4126305