Transfers in the group of multiplicative units of the classical cohomology ring and Stiefel-Whitney classes
type: Item
instance of: scholarly article
title: Transfers in the group of multiplicative units of the classical cohomology ring and Stiefel-Whitney classes
zbMATH Open document ID: 0687.55005
author: Andrzej Kozłowski
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1989-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 55R12
Mathematics Subject Classification ID: 57R20
Mathematics Subject Classification ID: 55N15
zbMATH DE Number: 4126305
zbMATH Keywords: finite covering
zbMATH Keywords: Evens transfer
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195173762
OpenAlex ID: W1986211113
cites work: Infinite Loop Spaces (AM-90)
cites work: Characters and cohomology of finite groups
cites work: A Generalization of the Transfer Map in the Cohomology of Groups
cites work: On the Chern Classes of Representations of Finite Groups
cites work: Chern Classes of Certain Representations of Symmetric Groups
cites work: Characteristic classes of direct image bundles for covering maps
cites work: Classes de Stiefel-Whitney de formes quadratiques et de représentations galoisiennes réelles
cites work: Q3207739
cites work: The Evens-Kahn Formula for the Total Stiefel-Whitney Class
cites work: The transfer in Segal's cohomology
cites work: The geometry of iterated loop spaces
cites work: Homology of the infinite symmetric group
cites work: THE SQUARE-PRESERVING ALGEBRA ENDOMORPHISMS OF H*(BO, Z2)
cites work: THE MULTIPLICATIVE GROUP OF CLASSICAL COHOMOLOGY
cites work: Multiplicative transfers in ordinary cohomology
DOI: 10.2977/PRIMS/1195173762
description: scientific article; zbMATH DE number 4126305