Celebratio Mathematica

M. F. Atiyah and I. M. Singer: “The index of elliptic operators, I,” Ann. Math. (2) 87 : 3 (May 1968), pp. 484–530. A Russian translation was published in Uspehi Mat. Nauk 23:5(143). MR 236950 Zbl 0164.24001 article

This is the first of a series of papers which will be devoted to a study of the index of elliptic operators on compact manifolds. The main result was announced in [Atiyah and Singer 1963] and, for manifolds with boundary, in [Atiyah 1964]. The long delay between these announcements and the present paper is due to several factors. On the one hand, a fairly detailed exposition has already appeared in [Palais 1965]. On the other hand, our original proof, reproduced with minor modifications in [Palais 1965], had a number of drawbacks. In the first place the use of cobordism, and the computational checking associated with this, were not very enlightening. More seriously, however, the method of proof did not lend itself to certain natural generalizations of the problem where appropriate cobordism groups were not known. The reader who is familiar with the Riemann–Roch theorem will realize that our original proof of the index theorem was modelled closely on Hirzebruch’s proof of the Riemann–Roch theorem. Naturally enough we were led to look for a proof modelled more on that of Grothendieck. While we have not completely succeeded in this aim, we have at least found a proof which is much more natural, does not use cobordism, and lends itself therefore to generalization. In spirit, at least, it has much in common with Grothendieck’s approach.

M. F. Atiyah and I. M. Singer: “The index of elliptic operators, III,” Ann. Math. (2) 87 : 3 (May 1968), pp. 546–604. A Russian translation was published in Uspehi Mat. Nauk 24:1(145). MR 236952 Zbl 0164.24301 article

In [1968a], paper I of this series, the index of an elliptic operator was computed in terms of \( K \)-theory. In this paper, we carry out what is essentially a routine exercise by passing from \( K \)-theory to cohomology. In this way, we end up with the explicit cohomological formula for the index announced in [1963].

In [1968a] we also considered elliptic operators (or complexes) compatible with a compact group \( G \) of transformations. The index in this case is a character of \( G \), and the main theorem of [1968a] gave a construction for this in \( KG \)-theory. In [1968b], paper II of this series, the value of this index-character at an element \( g\in G \) was expressed as the index of a new “virtual operator” on the fixed point set of \( g \). This was referred to as a Lefschetz fixed-point formula. By combining this formula with the cohomological formula for the index, we obtain finally an explicit cohomological formula for the index-character. We shall describe this formula in detail for a number of important operators. In particular we draw attention to the “integrality theorems” obtained in this way for actions of finite groups on manifolds. Most of these do not depend on the analysis in [1968a], but are a consequence of combining the purely topological results of [1968b] and the present paper.

M. F. Atiyah and I. M. Singer: “The index of elliptic operators, IV,” Ann. Math. (2) 93 : 1 (January 1971), pp. 119–138. A Russian translation was published in Uspehi Mat. Nauk 27:4(166) (1972). MR 279833 Zbl 0212.28603 article

M. F. Atiyah and I. M. Singer: “The index of elliptic operators, V,” Ann. Math. (2) 93 : 1 (January 1971), pp. 139–149. A Russian translation was published in Uspehi Mat. Nauk 27:4(166) (1972). MR 279834 article

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Michael F. Atiyah

Sir Michael Atityah OM
22 April 1929–11 January 2019

Works connected to Isadore M. Singer

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Sir Michael Atityah OM22 April 1929–11 January 2019

Works connected to Isadore M. Singer

Filter the Bibliography List

Sir Michael Atityah OM
22 April 1929–11 January 2019