Celebratio Mathematica — Atiyah

Papers on index theory (1963–1984)

Works connected to Graeme Bryce Segal

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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.