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The index of elliptic operators on compact manifolds
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Introduction.In his paper [ló] 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 (see §1 for the definition) in topological terms.This problem has been taken up by Agranovic [2; 3], Dynin [3; 14; 15], Seeley [20;21 ] and Vol'pert [22 ] 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 (Theorem 1).As a special case of this formula we get the Hirzebruch-Riemann-Roch theorem for any compact complex manifold (Theorem 3).This was previously known only for projective algebraic manifolds.Some other special cases, of interest in differential topology, are discussed in §3.We are greatly indebted to A. P. Calderon, L. Nirenberg, and R. T. Seeley for their generous help. Elliptic operators.Let X be a compact oriented smooth mani-
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“A systematic theory of characteristic classes begins with the universal bundle theorem.It says that with a given compact manifold M as base space the fiber bundles with a structural group G (Lie group) are in one-one correspondence with the homotopy classes of mappings of M into a classifying space B G ; the latter depends only on G and on the dimension of M.”
From The geometry of G-structures (1966) · cites this paper as [12]
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Research works printed but not linked: 19
- Agmon, S., 1959, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math., 12, 623, 10.1002/cpa.3160120405 · DOI
- 2. M. S. Agranovic, On the index of elliptic operators, Soviet Math. Dokl. 3 (1962), 194.
- Agranovič, M. S., 1962, General boundary-value problems for elliptic systems in higher-dimensional regions, Dokl. Akad. Nauk SSSR, 146, 511
- Atiyah, M. F., 1962, Analytic cycles on complex manifolds, Topology, 1, 25, 10.1016/0040-9383(62)90094-0 · DOI
- Borel, A., 1958, Characteristic classes and homogeneous spaces. I, Amer. J. Math., 80, 458, 10.2307/2372795 · DOI
- Bott, Raoul, 1959, The stable homotopy of the classical groups, Ann. of Math. (2), 70, 313, 10.2307/1970106 · DOI
- Brauer, Richard, 1935, Spinors in 𝑛 Dimensions, Amer. J. Math., 57, 425, 10.2307/2371218 · DOI
- Calderón, A.-P., 1957, Singular integral operators and differential equations, Amer. J. Math., 79, 901, 10.2307/2372441 · DOI
- Conner, P. E., 1964, Differentiable periodic maps
- Douglis, Avron, 1955, Interior estimates for elliptic systems of partial differential equations, Comm. Pure Appl. Math., 8, 503, 10.1002/cpa.3160080406 · DOI
- Dynin, A. S., 1961, Singular operators of arbitrary order on a manifold, Dokl. Akad. Nauk SSSR, 141, 21
- 15. A. S. Dynin, Multidimensional elliptic boundary value problems with a single unknown function, Soviet Math. Dokl. 2 (1961), 1431-1433.
- Gel′fand, I. M., On elliptic equations, Uspehi Mat. Nauk, 15, 121, 10.1070/RM1960v015n03ABEH004094 · DOI
- Hirzebruch, F., 1956, Neue topologische Methoden in der algebraischen Geometrie
- Kodaira, Kunihiko, 1951, The theorem of Riemann-Roch on compact analytic surfaces, Amer. J. Math., 73, 813, 10.2307/2372120 · DOI
- Seeley, R. T., 1959, Singular integrals on compact manifolds, Amer. J. Math., 81, 658, 10.2307/2372921 · DOI
- Seeley, R. T., 1961, Regularization of singular integral operators on compact manifolds, Amer. J. Math., 83, 265, 10.2307/2372955 · DOI
- Seeley, R. T., 1963, The index of elliptic systems of singular integral operators, J. Math. Anal. Appl., 7, 289, 10.1016/0022-247X(63)90054-4 · DOI
- Vol′pert, A. I., 1962, The index of systems of two-dimensional singular integral equations, Dokl. Akad. Nauk SSSR, 142, 776
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The index of elliptic operators, II
Atiyah, M. F., & Segal, G. B. (1968) · The Annals of Mathematics · doi:10.2307/1970716 · Cited by 461
Self-citationChern, S. S. (1966) · Bulletin of the American Mathematical Society · doi:10.1090/s0002-9904-1966-11473-8 · Cited by 159
Metadata
- Authors
- Atiyah, M. F., & Singer, I. M.
- Venue
- Bulletin of the American Mathematical Society
- Publisher
- American Mathematical Society (AMS)
- Published
- 1963
- Volume
- 69
- Issue
- 3
- Pages
- 422-433
- ISSN
- 0002-9904 (ISSN-L)
- 1088-9485
- 0273-0979
- OpenAlex
- W2145744586
- Cited by
- 679
- References
- 0
- Citations by year
- 2026: 18 · 2025: 36 · 2024: 26 · 2023: 31 · 2022: 15 · 2021: 37 · 2020: 39 · 2019: 25 · 2018: 27 · 2017: 23 · 2016: 28 · 2015: 17 · 2014: 24 · 2013: 22 · 2012: 19
- FWCI
- 8.57
- Type
- journal article
- Language
- en
- Open access
- diamond
- doi.org
- Locations
- Bulletin of the American Mathematical Society (publishedVersion)
- Topics
- Physical Sciences › Mathematics › Mathematical Physics › Homotopy and Cohomology in Algebraic Topology
- Physical Sciences › Mathematics › Mathematical Physics › Advanced Operator Algebra Research
- Physical Sciences › Mathematics › Geometry and Topology › Geometric and Algebraic Topology
- Keywords
- index theorem, elliptic operators, compact manifolds, topological invariants, Hirzebruch-Riemann-Roch theorem, complex manifolds, differential topology
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