K-théorie des corps finis et homotopie stable du classifiant d'un groupe de Lie
type: Item
instance of: scholarly article
title: K-théorie des corps finis et homotopie stable du classifiant d'un groupe de Lie
zbMATH Open document ID: 0564.55003
DOI: 10.1016/0022-4049(84)90042-2
author: Lionel Schwartz
published in: Journal of Pure and Applied Algebra
publication date: 1984-01-01T00:00:00Z
review text: Let G be a compact connected Lie group with classifying space BG and \(h: \pi\) \({}_*^ s(X)\to K_*(X)\) the Hurewicz map from stable homotopy to K-homology. The author studies the relation between \(\pi_*^ s(BG)/tor\) and the subgroup \(PK_ m(BG)\) of primitive elements with respect to the coaction in \(K_*(BG)\). Over the rationals both groups are isomorphic but beyond this not much is known on im(h). The paper contains a very nice and explicit description of \(K_*(BG)\) and \(PK_*(BG)\) in terms of numerical polynomials. These are elements in \({\mathbb{Q}}[x_ 1,...,x_ k]\), \((k=rank(G))\), which are invariant under the action of the Weyl group and satisfy certain integrability conditions. The main result of this paper is then as follows: The author defines an element \(U\in PK_*(BG)\) and shows that for every \(b\in PK_*(BG)_{(p)}\) the product \(U^ n\cdot b\) (polynomial multiplication) is in \(im(h)_{(p)}\) provided n is large enough and p is an odd prime. For \(G=T^ k\), a torus, the proof relies on a theorem of \textit{H. Miller} [J. Pure Appl. Algebra 20, 287-312 (1981; Zbl 0459.55012)] which computes the localization of \(\pi_*^ s(- ;{\mathbb{Z}}/p)\) with respect to the Adams self-map of the Moore spectrum. To obtain the proof in general, the author generalizes the result of Miller from \({\mathbb{Z}}/p\) to the \({\mathbb{Z}}/p^ m\)-case (m\(\in {\mathbb{N}})\).
Mathematics Subject Classification ID: 55N15
Mathematics Subject Classification ID: 55R40
Mathematics Subject Classification ID: 55Q10
Mathematics Subject Classification ID: 55R35
Mathematics Subject Classification ID: 55P42
Mathematics Subject Classification ID: 55R50
Mathematics Subject Classification ID: 55P60
zbMATH DE Number: 3899738
zbMATH Keywords: coaction in the K-homology of the classifying space of a compact Lie group
zbMATH Keywords: localization with respect to the Adams self-map of the Moore spectrum
zbMATH Keywords: Hurewicz map from stable homotopy to K-homology
zbMATH Keywords: action of the Weyl group
Wikidata QID: Q115364463
MaRDI profile type: Publication
full work available at URL: 0022-4049(84)90042-2
OpenAlex ID: W1997823835
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description: scientific article; zbMATH DE number 3899738