Note on the equivariant \(K\)-theory spectrum
type: Item
instance of: scholarly article
title: Note on the equivariant \(K\)-theory spectrum
zbMATH Open document ID: 0811.55010
author: Kazuhisa Shimakawa
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1995-05-02T00:00:00Z
review text: Let \(G\) be a finite group, \(KO_ G(X)\) the equivariant real \(K\)-theory of a finite \(G\)-CW complex \(X\) and \(Sph_ G(X)\) the stable equivalence classes of spherical \(G\)-fibrations over \(X\). There are connective \(G\)- spectra \(kO_ G\) and \(kF_ G\) representing the theories \(KO_ G(-)\) and \(Sph_ G(-)\) respectively and a map of \(G\)-spectra \(kO_ G\to kF_ G\) inducing the equivariant \(J\)-homomorphisms \(J_ G: KO_ G(X)\to Sph_ G(X)\). The author proves that there is a \(G\)-spectrum \(KO_ G\) representing the periodic \(KO_ G\)-theory and a map of \(G\)-spectra \(KO_ G\to KO_ G\) inducing an equivalence between \(kO_ G\) and the \((-1)\)-connected cover of \(KO_ G\).The result is then used to give an alternative proof of the equivariant Adams conjecture by deducing it from the one- and two- dimensional cases proved by \textit{H. Hauschild} and \textit{S. Waner} [Ill. J. Math. 27, 52-66 (1983; Zbl 0522.55017)].
Mathematics Subject Classification ID: 55R91
Mathematics Subject Classification ID: 55P42
Mathematics Subject Classification ID: 19L47
Mathematics Subject Classification ID: 55R50
Mathematics Subject Classification ID: 55Q50
zbMATH DE Number: 487015
zbMATH Keywords: equivariant periodic \(K\)-theory
zbMATH Keywords: equivariant spectra
zbMATH Keywords: equivariant spherical fibrations
zbMATH Keywords: finite group
zbMATH Keywords: equivariant real \(K\)-theory
zbMATH Keywords: connective \(G\)-spectra
zbMATH Keywords: equivariant \(J\)-homomorphisms
zbMATH Keywords: equivariant Adams conjecture
reviewed by: Rainer Vogt
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195167052
OpenAlex ID: W1989981934
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DOI: 10.2977/PRIMS/1195167052
description: scientific article; zbMATH DE number 487015