Note on the equivariant \(K\)-theory spectrum. (English) Zbl 0811.55010
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 H. Hauschild and S. Waner [Ill. J. Math. 27, 52-66 (1983; Zbl 0522.55017)].
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 H. Hauschild and S. Waner [Ill. J. Math. 27, 52-66 (1983; Zbl 0522.55017)].
Reviewer: R.Vogt (Osnabrück)
MSC:
| 55R91 | Equivariant fiber spaces and bundles in algebraic topology |
| 55P42 | Stable homotopy theory, spectra |
| 19L47 | Equivariant \(K\)-theory |
| 55R50 | Stable classes of vector space bundles in algebraic topology and relations to \(K\)-theory |
| 55Q50 | \(J\)-morphism |
Keywords:
equivariant periodic \(K\)-theory; equivariant spectra; equivariant spherical fibrations; finite group; equivariant real \(K\)-theory; connective \(G\)-spectra; equivariant \(J\)-homomorphisms; equivariant Adams conjectureCitations:
Zbl 0522.55017References:
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