K-ring of the orbit spaces of spheres by finite free actions (Q756194)

If a finite group G operates freely on a sphere \(S^ n\) and the operation is piecewise linear, then its quotient space \(S^ n/G\) is a PL-manifold and K-ring \(K(S^ n/G)\) is defined. The purpose of this paper is to give a complete description of \(K(S^ n/G)\) by terms of the representation ring of G, R(G). This is obtained by showing that \textit{M. F. Atiyah}'s conjecture in Publ. Math. Inst. Hautes Étud. Sci. 9, 247- 288 (1961; Zbl 0107.023) is true for the Artin-Tate groups (with periodic cohomology). We may consider the Atiyah's conjecture in two steps. (A) For p-group G, universal cycles in the Atiyah's spectral sequence \(H^*(G)\to R(G)\) are generated by Chern classes of representations of G. (B) If p-group \(G_ p\) satisfies (A), then for every group G which contains \(G_ p\) as a p-Sylow subgroup, p-primary component in universal cycles in the spectral sequence \(H^*(G)\to R(G)\) is generated by Chern classes of representations of G. I believe that (A) is still an unsolved problem. We prove that (B) is true for \(G_ p\) which is isomorphic to a cyclic group or a generalized quaternion group but (B) is not true in general.