Celebratio Mathematica

M. F. Atiyah and I. M. Singer: “The index of elliptic operators on compact manifolds,” Bull. Amer. Math. Soc. 69 : 3 (1963), pp. 422–433. MR 157392 Zbl 0118.31203 article

In [1960] Gel’fand posed the general problem of investigating the relationship between topological and analytical invariants of elliptic differential operators. In particular he suggested that it should be possible to express the index of an elliptic operator in topological terms. This problem has been taken up by Agranovič [1962a; 1962b], Dynin [1962b; 1961a; 1961b], Seeley [1961; 1963] and Vol’pert [1962] who have solved it in special cases. The purpose of this paper is to give a general formula for the index of an elliptic operator on any compact oriented differentiable manifold. As a special case of this formula we get the Hirzebruch–Riemann–Roch theorem for any compact complex manifold. This was previously known only for projective algebraic manifolds. Some other special cases, of interest in differential topology, are discussed.

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

M. F. Atiyah, V. K. Patodi, and I. M. Singer: “Spectral asymmetry and Riemannian geometry,” Bull. London Math. Soc. 5 (July 1973), pp. 229–234. MR 331443 Zbl 0268.58010 article

M. F. Atiyah	Related	Volume
Vijay Kumar Patodi	Related

M. F. Atiyah, V. K. Patodi, and I. M. Singer: “Spectral asymmetry and Riemannian geometry, I,” Math. Proc. Cambridge Philos. Soc. 77 (1975), pp. 43–69. MR 397797 Zbl 0297.58008 article

M. F. Atiyah	Related	Volume
Vijay Kumar Patodi	Related

M. F. Atiyah, V. K. Patodi, and I. M. Singer: “Spectral asymmetry and Riemannian geometry, II,” Math. Proc. Cambridge Philos. Soc. 78 : 3 (November 1975), pp. 405–432. MR 397798 Zbl 0314.58016 article

M. F. Atiyah	Related	Volume
Vijay Kumar Patodi	Related

M. F. Atiyah, V. K. Patodi, and I. M. Singer: “Spectral asymmetry and Riemannian geometry, III,” Math. Proc. Cambridge Philos. Soc. 79 : 1 (1976), pp. 71–99. MR 397799 Zbl 0325.58015 article

In Parts I and II of this paper [1975a; 1975b] we studied the ‘spectral asymmetry’ of certain elliptic self-adjoint operators arising in Riemannian geometry. More precisely, for any elliptic self-adjoint operator \( A \) on a compact manifold we defined \[ \eta_A(s) = \sum_{\lambda\neq 0}\operatorname{sign} \lambda \,|\lambda|^{-s}, \] where \( \lambda \) runs over the eigenvalues of \( A \). For the particular operators of interest in Riemannian geometry we showed that \( \eta_A(s) \) had an analytic continuation to the whole complex \( s \)-plane, with simple poles, and that \( s = 0 \) was not a pole. The real number \( \eta_A(0) \), which is a measure of ‘spectral asymmetry’, was studied in detail particularly in relation to representations of the fundamental group.

M. F. Atiyah	Related	Volume
Vijay Kumar Patodi	Related

M. F. Atiyah, N. J. Hitchin, and I. M. Singer: “Deformations of instantons,” Proc. Nat. Acad. Sci. U.S.A. 74 : 7 (July 1977), pp. 2662–2663. MR 458424 Zbl 0356.58011 article

Nigel J. Hitchin	Related
M. F. Atiyah	Related	Volume

M. F. Atiyah, N. J. Hitchin, and I. M. Singer: “Self-duality in four-dimensional Riemannian geometry,” Proc. Roy. Soc. London Ser. A 362 : 1711 (September 1978), pp. 425–461. MR 506229 Zbl 0389.53011 article

Nigel J. Hitchin	Related
M. F. Atiyah	Related	Volume

M. F. Atiyah, H. Donnelly, and I. M. Singer: “Eta invariants, signature defects of cusps, and values of \( L \)-functions,” Ann. Math. (2) 118 : 1 (July 1983), pp. 131–177. An addendum to this article was published in Ann. Math. (2) 119:3 (1984). MR 707164 Zbl 0531.58048 article

M. F. Atiyah	Related	Volume
Harold Gerard Donnelly	Related

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I. M. Singer: a brief biography

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