Celebratio Mathematica — Atiyah

Papers on K-theory (1959–1970)

Works connected to Graeme Bryce Segal

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M. F. Atiyah and G. B. Segal: “Equivariant \( K \)-theory and completion,” J. Differential Geometry 3 (1969), pp. 1–18. MR 0259946 Zbl 0215.24403 article

It was shown in [Atiyah 1961] that, for any finite group \( G \), the completed character ring \( R(G)^{\wedge} \) was isomorphic to \( K^*(B_G) \) where \( B_G \) denotes a classifying space for \( G \). The corresponding result for compact connected Lie groups was established in [Atiyah and Hirzebruch 1961], and a combination of the methods of [Atiyah and Hirzebruch 1961] and [Atiyah 1961] (together with certain basic properties of \( R(G) \) given in [Segal 1968b]) can be used to derive the theorem for general compact Lie groups. Such a proof however would be extremely lengthy, the worst part being in fact the treatment for finite groups where one climbs up via cyclic and Sylow subgroups.

The purpose of this paper is to give a new and much simpler proof of the theorem about \( K^*(B_G) \) which applies directly to all compact Lie groups \( G \). The main feature of our new proof is that we generalize the whole problem in a rather natural way by working with the equivariant \( K \)-theory developed in [Segal 1968a]. We shall formulate and prove a general theorem about the completion \( K_G^*(X)^{\wedge} \) for any compact \( G \)-space \( X \). The theorem about \( R(G) \) then follows by taking \( X \) to be a point.

M. F. Atiyah and G. B. Segal: “Exponential isomorphisms for \( \lambda \)-rings,” Quart. J. Math. Oxford Ser. (2) 22 : 3 (1971), pp. 371–378. MR 0291250 Zbl 0226.13008 article