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En geometría diferencial, el teorema del índice de Atiyah-Singer', demostrado por Michael Atiyah e Isadore Singer (1963),^[1] afirma que para un operador diferencial elíptico en una variedad compacta, el índice analítico (relacionado con la dimensión del espacio de soluciones) es igual al índice topológico (definido en términos de algunos datos topológicos). Incluye muchos otros teoremas, como el teorema de Gauss-Bonnet generalizado y el teorema de Riemann-Roch, como casos especiales, y tiene aplicaciones a la física teórica.^[2]

Historia

El problema del índice para operadores diferenciales elípticos fue planteado por Izrail Guelfand.^[3] Se dio cuenta de la invariabilidad homotópica del índice, y pidió una fórmula para él mediante invariantes topológicos. Algunos de los ejemplos motivadores fueron el teorema de Riemann-Roch y su generalización el teorema de Hirzebruch-Riemann-Roch, y el teorema de la firma de Hirzebruch. Friedrich Hirzebruch y Armand Borel habían demostrado la integrabilidad del Â género de una variedad de espín, y Atiyah sugirió que esta integralidad podría explicarse si fuera el índice del operador de Dirac (que fue redescubierto por Atiyah y Singer en 1961).

El teorema de Atiyah-Singer fue anunciado en 1963.^[1] La demostración esbozada en este anuncio nunca fue publicada por ellos, aunque aparece en el libro de Palais.^[4] También aparece en el "Séminaire Cartan-Schwartz 1963/64"^[5] que se celebró en París simultáneamente con el seminario dirigido por Richard Palais en la Universidad de Princeton. La última charla en París fue la de Atiyah sobre las variedades con límite. Su primera prueba publicada^[6] sustituyó la teoría del cobordismo de la primera prueba por la K-teoría, y la utilizó para dar pruebas de varias generalizaciones en otra secuencia de trabajos. ^[7]

1965: Sergey P. Novikov publicó sus resultados sobre la invariancia topológica de las clases de Pontryagines racionales en las variedades lisas.^[8]
Los resultados de Robion Kirby y Laurent C. Siebenmann, ^[9] combinados con el trabajo de René Thom demostraron la existencia de clases racionales de Pontryagin en variedades topológicas. Las clases racionales de Pontryagin son ingredientes esenciales del teorema del índice en las variedades lisas y topológicas.
1969: Michael Atiyah definió los operadores elípticos abstractos en espacios métricos arbitrarios. Los operadores elípticos abstractos se convirtieron en protagonistas de la teoría de Kasparov y de la geometría diferencial no conmutativa de Connes.^[10]
1971: Isadore Singer propuso un programa completo para futuras extensiones de la teoría de índices.^[11]
1972: Gennadi G. Kasparov publicó su trabajo sobre la realización de la K-homología mediante operadores elípticos abstractos.
1973: Atiyah, Raoul Bott, y Vijay Patodi dieron una nueva prueba del teorema del índice^[12] utilizando la ecuación del calor, descrita en un trabajo de Melrose.^[13]
1977: Dennis Sullivan estableció su teorema sobre la existencia y unicidad de estructuras de Lipschitz y cuasiconformes en variedades topológicas de dimensión diferente a 4.^[14]
1983: Ezra Getzler^[15] motivado por las ideas de Edward Witten^[16] y Luis Álvarez Gaumé, dio una breve demostración del teorema del índice local para operadores que son localmente operadores de Dirac; esto cubre muchos de los casos útiles.
1983: Nicolae Teleman demostró que los índices analíticos de los operadores de firma con valores en haces vectoriales son invariantes topológicos.
1984: Teleman estableció el teorema del índice en las variedades topológicas.
1986: Alain Connes publicó su artículo fundamental sobre geometría no conmutativa.^[17]
1989: Simon K. Donaldson y Sullivan estudiaron la teoría de Yang-Mills en variedades cuasiconformes de dimensión 4. Introdujeron el operador de firma S definido en formas diferenciales de grado dos.^[18]
1990: Connes y Henri Moscovici demostraron la fórmula del índice local en el contexto de la geometría no conmutativa.
1994: Connes, Sullivan y Teleman demostraron el teorema del índice para operadores de firma en variedades cuasiconformes.

Notation

X es un compacto y colector liso. (sin límites).
E y F son un fibrado vectorial suave sobre X
D es un operador diferencial elíptico de E a F. Así que en coordenadas locales actúa como un operador diferencial, llevando secciones suaves de E a secciones suaves de F.

Referencias

Bibliografía

Los trabajos de Atiyah se reimprimen en los volúmenes 3 y 4 de sus obras recopiladas, ^[1]

Atiyah, M. F. (1970), «Global Theory of Elliptic Operators», Proc. Int. Conf. on Functional Analysis and Related Topics (Tokyo, 1969), University of Tokio, Zbl 0193.43601 .
Atiyah, M. F. (1976), «Elliptic operators, discrete groups and von Neumann algebras», Colloque "Analyse et Topologie" en l'Honneur de Henri Cartan (Orsay, 1974), Asterisque, 32–33, Soc. Math. France, Paris, pp. 43-72, MR 0420729 .
Atiyah, M. F.; Segal, G. B. (1968), «The Index of Elliptic Operators: II», Annals of Mathematics, Second Series 87 (3): 531-545, JSTOR 1970716, doi:10.2307/1970716 . This reformulates the result as a sort of Lefschetz fixed-point theorem, using equivariant K-theory.
Atiyah, Michael F.; Singer, Isadore M. (1963), «The Index of Elliptic Operators on Compact Manifolds», Bull. Amer. Math. Soc. 69 (3): 422-433, doi:10.1090/S0002-9904-1963-10957-X Parámetro desconocido |doi-access= ignorado (ayuda). An announcement of the index theorem.
Atiyah, Michael F.; Singer, Isadore M. (1968a), «The Index of Elliptic Operators I», Annals of Mathematics 87 (3): 484-530, JSTOR 1970715, doi:10.2307/1970715 . This gives a proof using K-theory instead of cohomology.
Atiyah, Michael F.; Singer, Isadore M. (1968b), «The Index of Elliptic Operators III», Annals of Mathematics, Second Series 87 (3): 546-604, JSTOR 1970717, doi:10.2307/1970717 . This paper shows how to convert from the K-theory version to a version using cohomology.
Atiyah, Michael F.; Singer, Isadore M. (1971a), «The Index of Elliptic Operators IV», Annals of Mathematics, Second Series 93 (1): 119-138, JSTOR 1970756, doi:10.2307/1970756 . This paper studies families of elliptic operators, where the index is now an element of the K-theory of the space parametrizing the family.
Atiyah, Michael F.; Singer, Isadore M. (1971b), «The Index of Elliptic Operators V», Annals of Mathematics, Second Series 93 (1): 139-149, JSTOR 1970757, doi:10.2307/1970757 .. This studies families of real (rather than complex) elliptic operators, when one can sometimes squeeze out a little extra information.
Atiyah, M. F.; Bott, R. (1966), «A Lefschetz Fixed Point Formula for Elliptic Differential Operators», Bull. Am. Math. Soc. 72 (2): 245-50, doi:10.1090/S0002-9904-1966-11483-0 .. This states a theorem calculating the Lefschetz number of an endomorphism of an elliptic complex.
Atiyah, M. F.; Bott, R. (1967), «A Lefschetz Fixed Point Formula for Elliptic Complexes: I», Annals of Mathematics, Second series 86 (2): 374-407, JSTOR 1970694, doi:10.2307/1970694 . and Atiyah, M. F.; Bott, R. (1968), «A Lefschetz Fixed Point Formula for Elliptic Complexes: II. Applications», Annals of Mathematics, Second Series 88 (3): 451-491, JSTOR 1970721, doi:10.2307/1970721 . These give the proofs and some applications of the results announced in the previous paper.
Atiyah, M.; Bott, R.; Patodi, V. K. (1973), «On the heat equation and the index theorem», Invent. Math. 19 (4): 279-330, Bibcode:1973InMat..19..279A, MR 0650828, S2CID 115700319, doi:10.1007/BF01425417 .. Atiyah, M.; Bott, R.; Patodi, V. K. (1975), «Errata», Invent. Math. 28 (3): 277-280, Bibcode:1975InMat..28..277A, MR 0650829, doi:10.1007/BF01425562 .
Atiyah, Michael; Schmid, Wilfried (1977), «A geometric construction of the discrete series for semisimple Lie groups», Invent. Math. 42: 1-62, Bibcode:1977InMat..42....1A, MR 0463358, S2CID 189831012, doi:10.1007/BF01389783 ., Atiyah, Michael; Schmid, Wilfried (1979), «Erratum», Invent. Math. 54 (2): 189-192, Bibcode:1979InMat..54..189A, MR 0550183, doi:10.1007/BF01408936 .
Atiyah, Michael (1988a), Collected works. Vol. 3. Index theory: 1, Oxford Science Publications, New York: The Clarendon Press, Oxford University Press, ISBN 978-0-19-853277-4, MR 0951894 .
Atiyah, Michael (1988b), Collected works. Vol. 4. Index theory: 2, Oxford Science Publications, New York: The Clarendon Press, Oxford University Press, ISBN 978-0-19-853278-1, MR 0951895 .
Baum, P.; Fulton, W.; Macpherson, R. (1979), «Riemann-Roch for singular varieties», Acta Mathematica 143: 155-191, S2CID 83458307, Zbl 0332.14003, doi:10.1007/BF02684299 .
Berline, Nicole; Getzler, Ezra; Vergne, Michèle (1992), Heat Kernels and Dirac Operators, Berlin: Springer, ISBN 978-3-540-53340-5 . This gives an elementary proof of the index theorem for the Dirac operator, using the heat equation and supersymmetry.
Bismut, Jean-Michel (1984), «The Atiyah–Singer Theorems: A Probabilistic Approach. I. The index theorem», J. Funct. Analysis 57: 56-99, doi:10.1016/0022-1236(84)90101-0 . Bismut proves the theorem for elliptic complexes using probabilistic methods, rather than heat equation methods.
Cartan-Schwartz (1965), Séminaire Henri Cartan. Théoreme d'Atiyah-Singer sur l'indice d'un opérateur différentiel elliptique. 16 annee: 1963/64 dirigee par Henri Cartan et Laurent Schwartz. Fasc. 1; Fasc. 2. (French), École Normale Supérieure, Secrétariat mathématique, Paris, Zbl 0149.41102 .
Connes, A. (1986), «Non-commutative differential geometry», Publications Mathématiques 62: 257-360, S2CID 122740195, Zbl 0592.46056, doi:10.1007/BF02698807 .
Connes, A. (1994), Noncommutative Geometry, San Diego: Academic Press, ISBN 978-0-12-185860-5, Zbl 0818.46076 .
Connes, A.; Moscovici, H. (1990), «Cyclic cohomology, the Novikov conjecture and hyperbolic groups», Topology 29 (3): 345-388, Zbl 0759.58047, doi:10.1016/0040-9383(90)90003-3 .
Connes, A.; Sullivan, D.; Teleman, N. (1994), «Quasiconformal mappings, operators on Hilbert space and local formulae for characteristic classes», Topology 33 (4): 663-681, Zbl 0840.57013, doi:10.1016/0040-9383(94)90003-5 .
Donaldson, S.K.; Sullivan, D. (1989), «Quasiconformal 4-manifolds», Acta Mathematica 163: 181-252, Zbl 0704.57008, doi:10.1007/BF02392736 .
Gel'fand, I. M. (1960), «On elliptic equations», Russ. Math. Surv. 15 (3): 113-123, Bibcode:1960RuMaS..15..113G, doi:10.1070/rm1960v015n03ABEH004094 . reprinted in volume 1 of his collected works, p. 65–75, ISBN 0-387-13619-3. On page 120 Gel'fand suggests that the index of an elliptic operator should be expressible in terms of topological data.
Getzler, E. (1983), «Pseudodifferential operators on supermanifolds and the Atiyah–Singer index theorem», Commun. Math. Phys. 92 (2): 163-178, Bibcode:1983CMaPh..92..163G, S2CID 55438589, doi:10.1007/BF01210843 .
Getzler, E. (1988), «A short proof of the local Atiyah–Singer index theorem», Topology 25: 111-117, doi:10.1016/0040-9383(86)90008-X .
Gilkey, Peter B. (1994), Invariance Theory, the Heat Equation, and the Atiyah–Singer Theorem, ISBN 978-0-8493-7874-4 . Free online textbook that proves the Atiyah–Singer theorem with a heat equation approach
Hamilton, M. J. D. (2020). «The Higgs boson for mathematicians. Lecture notes on gauge theory and symmetry breaking». arXiv:1512.02632

[math.DG].

Higson, Nigel; Roe, John (2000), Analytic K-homology, Oxford University Press, ISBN 9780191589201 .
Hilsum, M. (1999), «Structures riemaniennes L^p et K-homologie», Annals of Mathematics 149 (3): 1007-1022, JSTOR 121079, S2CID 119708566, arXiv:math/9905210, doi:10.2307/121079 .
Kasparov, G.G. (1972), «Topological invariance of elliptic operators, I: K-homology», Math. USSR Izvestija (Engl. Transl.) 9 (4): 751-792, Bibcode:1975IzMat...9..751K, doi:10.1070/IM1975v009n04ABEH001497 .
Kirby, R.; Siebenmann, L.C. (1969), «On the triangulation of manifolds and the Hauptvermutung», Bull. Amer. Math. Soc. 75 (4): 742-749, doi:10.1090/S0002-9904-1969-12271-8 .
Kirby, R.; Siebenmann, L.C. (1977), Foundational Essays on Topological Manifolds, Smoothings and Triangulations, Annals of Mathematics Studies in Mathematics 88, Princeton: Princeton University Press and Tokio University Press .
Melrose, Richard B. (1993), The Atiyah–Patodi–Singer Index Theorem, Wellesley, Mass.: Peters, ISBN 978-1-56881-002-7 . Free online textbook.
Novikov, S.P. (1965), «Topological invariance of the rational Pontrjagin classes», Doklady Akademii Nauk SSSR 163: 298-300 .
Palais, Richard S. (1965), Seminar on the Atiyah–Singer Index Theorem, Annals of Mathematics Studies 57, S.l.: Princeton Univ Press, ISBN 978-0-691-08031-4 . This describes the original proof of the theorem (Atiyah and Singer never published their original proof themselves, but only improved versions of it.)
Shanahan, P. (1978), The Atiyah–Singer index theorem: an introduction, Lecture Notes in Mathematics 638, Springer, ISBN 978-0-387-08660-6, doi:10.1007/BFb0068264 Parámetro desconocido |citeseerx= ignorado (ayuda).
Singer, I.M. (1971), «Future extensions of index theory and elliptic operators», Prospects in Mathematics, Annals of Mathematics Studies in Mathematics 70, pp. 171-185 .
Sullivan, D. (1979), «Hyperbolic geometry and homeomorphisms», J.C. Candrell, "Geometric Topology", Proc. Georgia Topology Conf. Athens, Georgia, 1977, New York: Academic Press, pp. 543-595, ISBN 978-0-12-158860-1, Zbl 0478.57007 .
Sullivan, D.; Teleman, N. (1983), «An analytic proof of Novikov's theorem on rational Pontrjagin classes», Publications Mathématiques (Paris) 58: 291-293, S2CID 8348213, Zbl 0531.58045, doi:10.1007/BF02953773 .
Teleman, N. (1980), «Combinatorial Hodge theory and signature operator», Inventiones Mathematicae 61 (3): 227-249, Bibcode:1980InMat..61..227T, S2CID 122247909, doi:10.1007/BF01390066 .
Teleman, N. (1983), «The index of signature operators on Lipschitz manifolds», Publications Mathématiques 58: 251-290, S2CID 121497293, Zbl 0531.58044, doi:10.1007/BF02953772 .
Teleman, N. (1984), «The index theorem on topological manifolds», Acta Mathematica 153: 117-152, Zbl 0547.58036, doi:10.1007/BF02392376 .
Teleman, N. (1985), «Transversality and the index theorem», Integral Equations and Operator Theory 8 (5): 693-719, S2CID 121137053, doi:10.1007/BF01201710 .
Thom, R. (1956), «Les classes caractéristiques de Pontrjagin de variétés triangulées», Symp. Int. Top. Alg. Mexico, pp. 54-67 .
Witten, Edward (1982), «Supersymmetry and Morse theory», J. Diff. Geom. 17 (4): 661-692, MR 0683171, doi:10.4310/jdg/1214437492 .
Shing-Tung Yau, ed. (2009), The Founders of Index Theory (2nd edición), Somerville, Mass.: International Press of Boston, ISBN 978-1571461377 Texto «orig-year» ignorado (ayuda). - Personal accounts on Atiyah, Bott, Hirzebruch and Singer.

↑ ,.

[FOOTNOTEAtiyahSinger1963-1] Atiyah y Singer, 1963.

[FOOTNOTEHamilton202011-2] Hamilton, 2020, p. 11.

[FOOTNOTEGel'fand1960-3] Gel'fand, 1960.

[FOOTNOTEPalais1965-4] Palais, 1965.

[FOOTNOTECartan-Schwartz1965-5] Cartan-Schwartz, 1965.

[FOOTNOTEAtiyahSinger1968a-6] Atiyah y Singer, 1968a.

[FOOTNOTEAtiyah_&_Singer_(1968a);_Atiyah_&_Singer_(1968b);_Atiyah_&_Singer_(1971a);_Atiyah_&_Singer_(1971b).-7] Atiyah & Singer (1968a); Atiyah & Singer (1968b); Atiyah & Singer (1971a); Atiyah & Singer (1971b).,.

[FOOTNOTENovikov1965-8] Novikov, 1965.

[FOOTNOTEKirbySiebenmann1969-9] Kirby y Siebenmann, 1969.

[FOOTNOTEAtiyah1970-10] Atiyah, 1970.

[FOOTNOTESinger1971-11] Singer, 1971.

[FOOTNOTEAtiyahBottPatodi1973-12] Atiyah, Bott y Patodi, 1973.

[FOOTNOTEMelrose1993-13] Melrose, 1993.

[FOOTNOTESullivan1979-14] Sullivan, 1979.

[FOOTNOTEGetzler1983-15] Getzler, 1983.

[FOOTNOTEWitten1982-16] Witten, 1982.

[FOOTNOTEConnes1986-17] Connes, 1986.

[FOOTNOTEDonaldsonSullivan1989-18] Donaldson y Sullivan, 1989.

[FOOTNOTE-19] ,.

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@@ Línea 29: / Línea 29: @@
 == Referencias ==
 {{listaref|4}}
+== Bibliografía ==
+Los trabajos de Atiyah se reimprimen en los volúmenes 3 y 4 de sus obras recopiladas, {{harvnp|last=Atiyah|year1=1988a|year2=1988b}}
+*{{citation|last=Atiyah|first= M. F.|author-link=Michael Atiyah
+|chapter=Global Theory of Elliptic Operators|title= Proc. Int. Conf. on Functional Analysis and Related Topics (Tokyo, 1969)|year= 1970|publisher=University of Tokio|zbl=0193.43601}}
+*{{citation|last=Atiyah|first= M. F.|author-link=Michael Atiyah
+|chapter=Elliptic operators, discrete groups and von Neumann algebras|title= Colloque "Analyse et Topologie" en l'Honneur de Henri Cartan (Orsay, 1974)|pages=  43–72|series= Asterisque|volume= 32–33|publisher= Soc. Math. France, Paris|year= 1976|mr=0420729}}
+*{{citation |first1=M. F. |last1=Atiyah|author1-link=Michael Atiyah|first2= G. B.|last2= Segal |author2-link=Graeme Segal|title=The Index of Elliptic Operators: II|journal= Annals of Mathematics | series = Second Series |volume= 87 |issue=3|year= 1968|pages= 531–545 |doi=10.2307/1970716 |jstor=1970716}} This reformulates the result as a sort of Lefschetz fixed-point theorem, using equivariant K-theory.
+*{{citation|last1= Atiyah|first1= Michael F. |author1-link=Michael Atiyah|last2=Singer|first2= Isadore M. |author2-link=Isadore Singer|title=The Index of Elliptic Operators on Compact Manifolds|journal= Bull. Amer. Math. Soc.|volume= 69|pages= 422–433|year= 1963 |doi= 10.1090/S0002-9904-1963-10957-X|issue= 3|doi-access= free}} An announcement of the index theorem.
+*{{citation|last1= Atiyah|first1= Michael F. |author1-link=Michael Atiyah|last2=Singer|first2= Isadore M. |author2-link=Isadore Singer|title=The Index of Elliptic Operators I|journal= Annals of Mathematics |volume=87|pages= 484–530|year= 1968a|doi= 10.2307/1970715|issue= 3|jstor=1970715}} This gives a proof using K-theory instead of cohomology.
+*{{citation|last1= Atiyah|first1= Michael F. |author1-link=Michael Atiyah|last2=Singer|first2= Isadore M. |author2-link=Isadore Singer |title=The Index of Elliptic Operators III|journal=[[Annals of Mathematics]] | series = Second Series |volume= 87|issue=3|year= 1968b|pages= 546–604|doi= 10.2307/1970717|jstor= 1970717}} This paper shows how to convert from the K-theory version to a version using cohomology.
+*{{citation|last1= Atiyah|first1= Michael F. |author1-link=Michael Atiyah|last2=Singer|first2= Isadore M. |author2-link=Isadore Singer|title=The Index of Elliptic Operators IV|journal= [[Annals of Mathematics]] | series = Second Series |volume= 93|issue=1|year= 1971a|pages=  119–138|doi= 10.2307/1970756 |jstor=1970756}} This paper studies families of elliptic operators, where the index is now an element of the K-theory of the space parametrizing the family.
+*{{citation|last1= Atiyah|first1= Michael F. |author1-link=Michael Atiyah|last2=Singer|first2= Isadore M. |author2-link=Isadore Singer|title=The Index of Elliptic Operators V|journal=[[Annals of Mathematics]] | series = Second Series |volume= 93|issue= 1|year= 1971b|pages= 139–149|doi= 10.2307/1970757 |jstor=1970757}}. This studies families of real (rather than complex) elliptic operators, when one can sometimes squeeze out a little extra information.
+*{{citation|first1=M. F.|last1= Atiyah|author1-link=Michael Atiyah|first2= R.|last2= Bott|author2-link=Raoul Bott |title=A Lefschetz Fixed Point Formula for Elliptic Differential Operators|journal= Bull. Am. Math. Soc. |volume=72 |year=1966|pages= 245–50|doi=10.1090/S0002-9904-1966-11483-0|issue=2 |doi-access=}}. This states a theorem calculating the Lefschetz number of an endomorphism of an elliptic complex.
+*{{citation|first1=M. F.|last1= Atiyah|author1-link=Michael Atiyah|first2= R.|last2= Bott|author2-link=Raoul Bott |title=A Lefschetz Fixed Point Formula for Elliptic Complexes: I |journal=Annals of Mathematics | series = Second series |volume= 86|issue=2 |year= 1967|pages= 374–407|doi=10.2307/1970694 |jstor=1970694}} and {{citation|first1=M. F.|last1= Atiyah|first2= R.|last2= Bott |title=A Lefschetz Fixed Point Formula for Elliptic Complexes: II. Applications
+|journal=[[Annals of Mathematics]] | series = Second Series |volume=88|issue=3|year= 1968|pages=451–491|doi=10.2307/1970721|jstor=1970721}}  These give the proofs and some applications of the results announced in the previous paper.
+*{{citation|last1=Atiyah|first1= M.|author1-link=Michael Atiyah|last2= Bott|first2= R.|author2-link=Raoul Bott|last3= Patodi|first3= V. K.|author3-link=Vijay Kumar Patodi|title= On the heat equation and the index theorem|journal= Invent. Math.|volume= 19 |year=1973|pages= 279–330|	doi=10.1007/BF01425417|mr=0650828|issue=4|bibcode=1973InMat..19..279A|s2cid= 115700319}}. {{citation|title= Errata |journal=Invent. Math.|volume=  28  |year=1975|pages= 277–280	|doi=10.1007/BF01425562|mr= 0650829|issue= 3|bibcode=1975InMat..28..277A|last1=Atiyah|first1=M.|last2=Bott|first2=R.|last3=Patodi|first3=V. K.|doi-access=}}
+*{{citation|last1= Atiyah|first1= Michael|author1-link=Michael Atiyah|last2= Schmid|first2= Wilfried|author2-link=Wilfried Schmid |title=A geometric construction of the discrete series for semisimple Lie groups|journal= Invent. Math.|volume= 42 |year=1977|pages= 1–62|doi=10.1007/BF01389783|mr= 0463358|bibcode=1977InMat..42....1A|s2cid= 189831012}}, {{citation|title= Erratum|journal=  Invent. Math. |volume= 54  |year=1979|issue= 2|pages=189–192|doi=10.1007/BF01408936|author= Atiyah, Michael|last2=Schmid|first2=Wilfried|mr= 0550183|bibcode=1979InMat..54..189A|doi-access=}}
+*{{citation|last= Atiyah|first= Michael|author-link=Michael Atiyah|title= Collected works. Vol. 3. Index theory: 1 |series=Oxford Science Publications|publisher= The Clarendon Press, Oxford University Press|location=New York|year= 1988a| isbn= 978-0-19-853277-4 |url=https://books.google.com/books?isbn=0198532776|mr= 0951894}}
+*{{citation|last= Atiyah|first= Michael|author-link=Michael Atiyah|title= Collected works. Vol. 4. Index theory: 2 |series=Oxford Science Publications|publisher= The Clarendon Press, Oxford University Press|location=New York|year= 1988b| isbn= 978-0-19-853278-1|mr= 0951895 }}
+*{{citation|first1=P.|last1=Baum|author1-link=Paul Baum (mathematician)|first2=W.|last2=Fulton|author2-link=William Fulton (mathematician)|first3=R.|last3=Macpherson|author3-link=Robert MacPherson (mathematician)|title=Riemann-Roch for singular varieties|journal=Acta Mathematica|volume=143|year=1979|pages=155–191|doi=10.1007/BF02684299|zbl=0332.14003|s2cid=83458307|url=http://www.numdam.org/item/PMIHES_1975__45__101_0/}}
+*{{citation|first1=Nicole |last1=Berline|first2= Ezra |last2=Getzler|first3= Michèle|last3= Vergne|title= Heat Kernels and Dirac Operators|year=1992|isbn= 978-3-540-53340-5|publisher=Springer|location=Berlin}} This gives an elementary proof of the index theorem for the Dirac operator, using the heat equation and supersymmetry.
+*{{citation|author-link=Jean-Michel Bismut|first=Jean-Michel|last=Bismut|title=The Atiyah–Singer Theorems: A Probabilistic Approach. I. The index theorem|journal=J. Funct. Analysis|year=1984|volume=57|pages=56–99|doi=10.1016/0022-1236(84)90101-0|doi-access=}} Bismut proves the theorem for elliptic complexes using probabilistic methods, rather than heat equation methods.
+*{{citation|last=Cartan-Schwartz|author-link=Cartan-Schwartz|title=Séminaire Henri Cartan. Théoreme d'Atiyah-Singer sur l'indice d'un opérateur différentiel elliptique. 16 annee: 1963/64 dirigee par Henri Cartan et Laurent Schwartz. Fasc. 1; Fasc. 2. (French)|publisher= École Normale Supérieure, Secrétariat mathématique, Paris|year=1965|zbl=0149.41102}}
+*{{citation|first=A.|last=Connes|author-link=Alain Connes|title=Non-commutative differential geometry|journal=Publications Mathématiques|volume=62|year=1986|doi=10.1007/BF02698807|pages=257–360|zbl=0592.46056|s2cid=122740195|url=http://www.numdam.org/item/PMIHES_1985__62__41_0/}}
+*{{citation|first=A.|last=Connes|author-link=Alain Connes|title=Noncommutative Geometry|publisher=Academic Press|isbn=978-0-12-185860-5|location=San Diego|year=1994|zbl=0818.46076|url-access=|url=https://archive.org/details/noncommutativege0000conn}}
+*{{citation|first1=A.|last1=Connes|author1-link=Alain Connes|first2=H.|last2=Moscovici|title=Cyclic cohomology, the Novikov conjecture and hyperbolic groups|journal=Topology|volume=29|year=1990|pages=345–388|url=http://www.alainconnes.org/docs/novikov.pdf|zbl=0759.58047|doi=10.1016/0040-9383(90)90003-3|issue=3|doi-access=}}
+*{{citation|first1=A.|last1=Connes|author1-link=Alain Connes|first2=D.|last2=Sullivan|author2-link=Dennis Sullivan|first3=N.|last3=Teleman|title=Quasiconformal mappings, operators on Hilbert space and local formulae for characteristic classes|journal=Topology|volume=33|issue=4|year=1994|pages=663–681|zbl=0840.57013|doi=10.1016/0040-9383(94)90003-5|doi-access=}}
+*{{citation|first1=S.K.|last1=Donaldson|author1-link=Simon Donaldson|first2=D.|last2=Sullivan|author2-link=Dennis Sullivan|title=Quasiconformal 4-manifolds|journal=Acta Mathematica|volume=163|year=1989|pages=181–252|doi=10.1007/BF02392736|zbl=0704.57008|doi-access=}}
+*{{citation|first=I. M. |last=Gel'fand|author-link=Israel Gelfand|title=On elliptic equations|journal=Russ. Math. Surv.|volume= 15 |issue=3|year=1960|pages=  113–123|doi=10.1070/rm1960v015n03ABEH004094|bibcode=1960RuMaS..15..113G}} reprinted in volume 1 of his collected works, p.&nbsp;65–75, {{ISBN|0-387-13619-3}}. On page 120 Gel'fand suggests that the index of an elliptic operator should be expressible in terms of topological data.
+*{{citation|first=E.|last= Getzler|author-link=Ezra Getzler|url=http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.cmp/1103940796 |title=Pseudodifferential operators on supermanifolds and the Atiyah–Singer index theorem |journal=Commun. Math. Phys.|volume= 92 |year=1983|pages=  163–178 |doi=10.1007/BF01210843|issue=2|bibcode = 1983CMaPh..92..163G |s2cid= 55438589}}
+*{{citation|first=E. |last=Getzler|doi=10.1016/0040-9383(86)90008-X |title=A short proof of the local Atiyah–Singer index theorem|journal=Topology|volume= 25 |year=1988|pages= 111–117|doi-access=}}
+* {{citation|first=Peter B. |last=Gilkey|year=1994|url= http://www.emis.de/monographs/gilkey/|title=Invariance Theory, the Heat Equation, and the Atiyah–Singer Theorem| isbn=978-0-8493-7874-4}} Free online textbook that proves the Atiyah–Singer theorem with a heat equation approach
+* {{cite arxiv|last=Hamilton|first=M. J. D.|year=2020|title=The Higgs boson for mathematicians. Lecture notes on gauge theory and symmetry breaking|class=math.DG|eprint=1512.02632}}
+* {{citation|first1=Nigel |last1=Higson|first2=John |last2=Roe|title=Analytic K-homology |publisher= Oxford University Press |year=2000 |isbn=9780191589201}}
+*{{citation|first1=M.|last1=Hilsum|title=Structures riemaniennes ''L''<sup>p</sup> et ''K''-homologie|journal= Annals of Mathematics|volume=149|issue=3|year=1999|pages=1007–1022|doi=10.2307/121079|arxiv=math/9905210|jstor=121079|s2cid=119708566}}
+*{{citation|first=G.G.|last=Kasparov|title=Topological invariance of elliptic operators, I: K-homology|journal=Math. USSR Izvestija (Engl. Transl.)|volume=9|year=1972|issue=4|pages=751–792|doi=10.1070/IM1975v009n04ABEH001497|bibcode = 1975IzMat...9..751K }}
+*{{citation|first1=R.|last1=Kirby|author1-link=|first2=L.C.|last2=Siebenmann|author2-link=Laurent C. Siebenmann|title=On the triangulation of manifolds and the Hauptvermutung|journal=Bull. Amer. Math. Soc.|volume=75|year=1969|pages=742–749|doi=10.1090/S0002-9904-1969-12271-8|issue=4|doi-access=}}
+*{{citation|first1=R.|last1=Kirby|author1-link=Robion Kirby|first2=L.C.|last2=Siebenmann|author2-link=Laurent C. Siebenmann|title=Foundational Essays on Topological Manifolds, Smoothings and Triangulations|publisher=Princeton University Press and Tokio University Press|location=Princeton|series=Annals of Mathematics Studies in Mathematics|volume=88|year=1977}}
+* {{citation|first=Richard B. |last=Melrose|year=1993|url=http://www-math.mit.edu/~rbm/book.html |title=The Atiyah–Patodi–Singer Index Theorem| isbn=978-1-56881-002-7|publisher=Peters|location=Wellesley, Mass.}} Free online textbook.
+*{{citation|first=S.P.|last=Novikov|author-link=Sergei Novikov (mathematician)|title=Topological invariance of the rational Pontrjagin classes|journal=[[Doklady Akademii Nauk SSSR]]|volume=163|year=1965|url=http://www.mi.ras.ru/~snovikov/21.pdf|pages=298–300}}
+*{{citation|first=Richard S. |last=Palais|author-link= |year=1965|title=Seminar on the Atiyah–Singer Index Theorem| series=Annals of Mathematics Studies|volume=57| isbn=978-0-691-08031-4|publisher=Princeton Univ Press|location=S.l.|url=https://books.google.com/books?isbn=0-691-08031-3}} This describes the original proof of the theorem (Atiyah and Singer never published their original proof themselves, but only improved versions of it.)
+*{{citation|last=Shanahan|first= P.|title=The Atiyah–Singer index theorem: an introduction|isbn=978-0-387-08660-6 |series=Lecture Notes in Mathematics |volume=638|publisher= Springer|year= 1978|doi=10.1007/BFb0068264|citeseerx= 10.1.1.193.9222}}
+*{{citation|first=I.M.|last=Singer|author-link=|title=Prospects in Mathematics|chapter=Future extensions of index theory and elliptic operators|series=Annals of Mathematics Studies in Mathematics|volume=70|year=1971|pages=171–185}}
+*{{citation|first=D.|last=Sullivan|author-link=|chapter=Hyperbolic geometry and homeomorphisms|title=J.C. Candrell, "Geometric Topology", Proc. Georgia Topology Conf. Athens, Georgia, 1977|publisher=Academic Press|location=New York|year=1979|pages=543–595|isbn=978-0-12-158860-1|zbl=0478.57007}}
+*{{citation|first1=D.|last1=Sullivan|author1-link=|first2=N.|last2=Teleman|title=An analytic proof of Novikov's theorem on rational Pontrjagin classes|journal=Publications Mathématiques|location=Paris|volume=58|year=1983|pages=291–293|doi=10.1007/BF02953773|zbl=0531.58045|s2cid=8348213|url=http://www.numdam.org/item/PMIHES_1983__58__79_0/}}
+*{{citation|first=N.|last=Teleman|title=Combinatorial Hodge theory and signature operator|journal=Inventiones Mathematicae|volume=61|year=1980|pages=227–249|doi=10.1007/BF01390066|issue=3|bibcode = 1980InMat..61..227T |s2cid=122247909}}
+*{{citation|first=N.|last=Teleman|title=The index of signature operators on Lipschitz manifolds|journal=Publications Mathématiques|volume=58|year=1983|pages=251–290|doi=10.1007/BF02953772|zbl=0531.58044|s2cid=121497293|url=http://www.numdam.org/item/PMIHES_1983__58__39_0/}}
+*{{citation|first=N.|last=Teleman|title=The index theorem on topological manifolds|journal=Acta Mathematica|volume=153|year=1984|pages=117–152|doi=10.1007/BF02392376|zbl=0547.58036|doi-access=}}
+*{{citation|first=N.|last=Teleman|title=Transversality and the index theorem|journal=Integral Equations and Operator Theory|volume=8|year=1985|pages=693–719|doi=10.1007/BF01201710|issue=5|s2cid=121137053}}
+*{{citation|first=R.|last=Thom|author-link=|chapter=Les classes caractéristiques de Pontrjagin de variétés triangulées|title=Symp. Int. Top. Alg. Mexico|year=1956| pages=54–67}}
+*{{Citation | last1=Witten | first1=Edward | author1-link=| title=Supersymmetry and Morse theory | year=1982 | journal=J. Diff. Geom. | volume=17 | issue=4 | pages=661–692 | mr=0683171| doi=10.4310/jdg/1214437492 | doi-access= }}
+* {{citation|editor-last=Shing-Tung Yau|editor-link=Shing-Tung Yau|title=The Founders of Index Theory|year=2009|publisher=International Press of Boston|place=Somerville, Mass.|edition=2nd|orig-year|isbn=978-1571461377}} - Personal accounts on [[Michael Atiyah|Atiyah]], [[Raoul Bott|Bott]], [[Friedrich Hirzebruch|Hirzebruch]] and [[Isadore Singer|Singer]].