Celebratio Mathematica

M. F. Atiyah and G. B. Segal: “The index of elliptic operators, II,” Uspehi Mat. Nauk 23 : 6 (144) (1968), pp. 135–149. Russian translation of article in Ann. Math. 87:3 (1968). MR 0236953 article

M. F. Atiyah and G. B. Segal: “The index of elliptic operators, II,” Ann. Math. (2) 87 : 3 (1968), pp. 531–545. Russian translation published in Uspehi Mat. Nauk 23:6(144) (1968). MR 0236951 Zbl 0164.24201 article

The purpose of this paper is to show how the index theorem of [Atiyah and Singer 1963] can be reformulated as a general “Lefschetz fixed-point theorem” on the lines of [Atiyah and Bott 1967]. In this way we shall obtain the main theorem of [Atiyah and Bott 1967], generalized to deal with arbitrary fixed-point sets, but only for transformations belonging to a compact group.

The content of this paper is essentially topological, and it should be viewed as a paper on the equivariant \( K \)-theory of manifolds. The analysis has all been done in [Atiyah and Singer 1968], and what we do here is simply to express the topological index in terms of fixed-point sets. This is quite independent of the main theorem of [Atiyah and Singer 1968] asserting the equality of the topological and analytical indices.

As in [Atiyah and Singer 1968], we avoid cohomology and use only \( K \)-theory. In paper III of this series, we shall pass over to cohomology obtaining explicit formulas in terms of characteristic classes.

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

M. Atiyah and G. Segal: “On equivariant Euler characteristics,” J. Geom. Phys. 6 : 4 (1989), pp. 671–677. MR 1076708 Zbl 0708.19004 article

M. Atiyah and G. Segal: “Twisted \( K \)-theory,” Ukr. Mat. Visn. 1 : 3 (2004), pp. 287–330. MR 2172633 Zbl 1151.55301 article

M. Atiyah and G. Segal: “Twisted \( K \)-theory and cohomology,” pp. 5–43 in Inspired by S. S. Chern. Edited by P. Griffiths. Nankai Tracts in Mathematics 11. World Scientific (Hackensack, NJ), 2006. MR 2307274 Zbl 1138.19003 ArXiv math/0510674 incollection

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