K-théorie des corps finis et homotopie stable du classifiant d'un groupe de Lie
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\(\in {\mathbb{N}})\).
coaction in the K-homology of the classifying space of a compact Lie group
localization with respect to the Adams self-map of the Moore spectrum
Hurewicz map from stable homotopy to K-homology
scientific article; zbMATH DE number 3899738
scientific article