Celebratio Mathematica — Atiyah

Sir Michael Atityah OM
22 April 1929–11 January 2019

Works connected to Graeme Bryce Segal

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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.

Sir Michael Atityah OM22 April 1929–11 January 2019

Works connected to Graeme Bryce Segal

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Sir Michael Atityah OM
22 April 1929–11 January 2019