Note on the equivariant \(K\)-theory spectrum (Q1311663)

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)].