Editing Atiyah–Singer index theorem (section)

== References ==
The papers by Atiyah are reprinted in volumes 3 and 4 of his collected works, {{harvs|last=Atiyah|year1=1988a|year2=1988b}}
{{refbegin|2}}
*{{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=free}}. 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=free}}
*{{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=free}}
*{{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. Anal.|year=1984|volume=57|pages=56–99|doi=10.1016/0022-1236(84)90101-0|doi-access=free}} 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 de l'Institut des Hautes Études Scientifiques|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=registration|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=free}}
*{{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=free}}
*{{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=free}}
*{{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=free}}
* {{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|publisher=CRC Press | 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}}
* {{cite arXiv|last=Kayani|first=U.|year=2020|title=Dynamical supersymmetry enhancement of black hole horizons|class=hep-th|eprint=1910.01080}}
* {{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=Robion Kirby|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=free}}
*{{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|first1=H. Blane|last1=Lawson|author1-link=H. Blaine Lawson|first2=Marie-Louise|last2=Michelsohn|author2-link=Marie-Louise Michelsohn|title=Spin Geometry|year=1989|isbn=0-691-08542-0|publisher=Princeton University Press}}
* {{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=Richard Palais |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|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=Isadore Singer|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=Dennis Sullivan|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=Dennis Sullivan|first2=N.|last2=Teleman|title=An analytic proof of Novikov's theorem on rational Pontrjagin classes|journal=Publications Mathématiques de l'Institut des Hautes Études Scientifiques|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 de l'Institut des Hautes Études Scientifiques|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=free}}
*{{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=René Thom|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=Edward Witten | 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=free }}
* {{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=First published in 2005|isbn=978-1571461377}} - Personal accounts on [[Michael Atiyah|Atiyah]], [[Raoul Bott|Bott]], [[Friedrich Hirzebruch|Hirzebruch]] and [[Isadore Singer|Singer]].
{{refend}}