Stiefel-Whitney classes, united \(K\)-theory and real embeddings of number rings
type: Item
instance of: scholarly article
title: Stiefel-Whitney classes, united \(K\)-theory and real embeddings of number rings
zbMATH Open document ID: 1119.19004
author: Stephen A. Mitchell
published in: \(K\)-Theory
publication date: 2007-02-13T00:00:00Z
review text: The author studies the relations among the various Stiefel-Whitney classes associated to the real embeddings of a number ring \(A\), using a computation of the united \(K\)-theory of the algebraic \(K\)-theory spectrum \(KA\) of \(A\). More precisely, let \(F\) be a number field, let \(S\subset {\mathcal O}_F\) be a set of primes in its ring of integers \({\mathcal O}_F\), and let \(A= S^{-1}{\mathcal O}_F\) be its localization. In addition assume that \(1/2\in A\). Given a real embedding \(\alpha: F \subset {\mathbb R}\) there is an induced map \(BGL(A) \to BGL({\mathbb R}) \simeq BO\), which yields a sequence of Stiefel-Whitney classes \(w_*(\alpha)\). For a fixed real embedding \(\alpha\) the classes \(w_*(\alpha)\) are algebraically independent, but in general they do depend on the choice of \(\alpha\). The central matter of the paper is determining the relations between the various \(w_*(\alpha)\) for different \(\alpha\). This is equivalent to investigating the image of the map \(\varrho: H_*(BGL(A)) \to \prod_\alpha H_*(BO)\) in homology with coefficients in \(\mathbb{F}_2\), the field with two elements, which is induced by the set of the distinct real embeddings \(\alpha\). The motivation for computing \(\text{im}(\varrho)\) comes from the fact that it yields the relevant input data for a Serre spectral sequence that converges to \(H^*(BGL(A))\). To get a hand on the size of \(\text{im}(\varrho) \subset \prod_\alpha H_*BO\) the author makes use of the factorization \(BO \to BSC \to BU\) of the map \(BO \to BU\) induced by complexification, with \(BSC\) the classifying space of self-conjugate \(K\)-theory. More specifically, the author uses the fact that the image of a map \(H_*BGL(A) \to H_*BO\) of bicommutative Hopf algebras over \(\mathbb{F}_2\) whose composition to \(H_*BSC\) becomes trivial must lie in a suitable subgroup of \(H_*BO\). In order to get control on this vanishing property the author computes the united \(K\)-theory of \(KA\), which among ot…
reviewed by: Michael Joachim
Mathematics Subject Classification ID: 19F99
Mathematics Subject Classification ID: 55R40
Mathematics Subject Classification ID: 19L99
Mathematics Subject Classification ID: 55N15
zbMATH DE Number: 5125365
zbMATH Keywords: algebraic \(K\)-theory
zbMATH Keywords: narrow Picard group
zbMATH Keywords: number ring
zbMATH Keywords: Stiefel-Whitney classes
zbMATH Keywords: united \(K\)-theory
MaRDI profile type: Publication
full work available at URL: s10977-006-7105-z
OpenAlex ID: W1965256929
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DOI: 10.1007/S10977-006-7105-Z
description: scientific article; zbMATH DE number 5125365