Teorema de l'índex d'Atiyah-Singer

En geometria diferencial, el teorema de l'índex d'Atiyah–Singer, demostrat per Michael Atiyah i Isadore Singer (1963), afirma que per un operador diferencial el·líptic en una varietat compacta, l'índex analític (relacionat amb la dimensió de l'espai de solucions) és igual a l'índex topològic (definit en termes d'algunes dades topològiques). Inclou molts altres teoremes, com ara el teorema de Chern–Gauss–Bonnet i el teorema de Riemann–Roch, com a casos especials, i té aplicacions en la física teòrica.

Bibliografia

Els articles d'Atiyah van ser reimpresos en els volums 3 i 4 de la recopilació de les seves obres, (Atiyah 1988a, 1988b)

• Atiyah, M. F. (1970), "Global Theory of Elliptic Operators", Proc. Int. Conf. on Functional Analysis and Related Topics (Tokyo, 1969), University of Tokio
• Atiyah, M. F. (1976), "Elliptic operators, discrete groups and von Neumann algebras", Colloque "Analyse et Topologie" en l'Honneur de Henri Cartan (Orsay, 1974), vol. 32–33, Asterisque, Soc. Math. France, Paris, pàg. 43–72
• Atiyah, M. F. & Segal, G. B. (1968), "The Index of Elliptic Operators: II", Annals of Mathematics, Second Series 87 (3): 531–545, 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 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, 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, 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, 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, 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, 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, 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, DOI 10.1007/BF01425417. Atiyah, M.; Bott, R. & Patodi, V. K. (1975), "Errata", Invent. Math. 28 (3): 277–280, 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, DOI 10.1007/BF01389783, Atiyah, Michael & Schmid, Wilfried (1979), "Erratum", Invent. Math. 54 (2): 189–192, 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, <https://books.google.com/books?isbn=0198532776>
• 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
• Baum, P.; Fulton, W. & Macpherson, R. (1979), "Riemann-Roch for singular varieties", Acta Mathematica 143: 155–191, doi:10.1007/BF02684299, <http://www.numdam.org/item/PMIHES_1975__45__101_0/>
• 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
• Connes, A. (1986), "Non-commutative differential geometry", Publications Mathématiques 62: 257–360, doi:10.1007/BF02698807, <http://www.numdam.org/item/PMIHES_1985__62__41_0/>
• Connes, A. (1994), Noncommutative Geometry, San Diego: Academic Press, ISBN 978-0-12-185860-5, <https://archive.org/details/noncommutativege0000conn>
• Connes, A. & Moscovici, H. (1990), "Cyclic cohomology, the Novikov conjecture and hyperbolic groups", Topology 29 (3): 345–388, doi:10.1016/0040-9383(90)90003-3, <http://www.alainconnes.org/docs/novikov.pdf>
• Connes, A.; Sullivan, D. & Teleman, N. (1994), "Quasiconformal mappings, operators on Hilbert space and local formulae for characteristic classes", Topology 33 (4): 663–681, DOI 10.1016/0040-9383(94)90003-5
• Donaldson, S.K. & Sullivan, D. (1989), "Quasiconformal 4-manifolds", Acta Mathematica 163: 181–252, DOI 10.1007/BF02392736
• Gel'fand, I. M. (1960), "On elliptic equations", Russ. Math. Surv. 15 (3): 113–123, 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, doi:10.1007/BF01210843, <http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.cmp/1103940796>
• 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, <http://www.emis.de/monographs/gilkey/> 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", arΧiv: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, 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, 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, vol. 88, Annals of Mathematics Studies in Mathematics, 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, <http://www-math.mit.edu/~rbm/book.html> Free online textbook.
• Novikov, S.P. (1965), "Topological invariance of the rational Pontrjagin classes", Doklady Akademii Nauk SSSR 163: 298–300, <http://www.mi.ras.ru/~snovikov/21.pdf>
• Palais, Richard S. (1965), Seminar on the Atiyah–Singer Index Theorem, vol. 57, Annals of Mathematics Studies, S.l.: Princeton Univ Press, ISBN 978-0-691-08031-4, <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.)
• Shanahan, P. (1978), The Atiyah–Singer index theorem: an introduction, vol. 638, Lecture Notes in Mathematics, Springer, ISBN 978-0-387-08660-6, DOI 10.1007/BFb0068264
• Singer, I.M. (1971), "Future extensions of index theory and elliptic operators", Prospects in Mathematics, vol. 70, Annals of Mathematics Studies in Mathematics, pàg. 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, pàg. 543–595, ISBN 978-0-12-158860-1
• Sullivan, D. & Teleman, N. (1983), "An analytic proof of Novikov's theorem on rational Pontrjagin classes", Publications Mathématiques (Paris) 58: 291–293, doi:10.1007/BF02953773, <http://www.numdam.org/item/PMIHES_1983__58__79_0/>
• Teleman, N. (1980), "Combinatorial Hodge theory and signature operator", Inventiones Mathematicae 61 (3): 227–249, DOI 10.1007/BF01390066
• Teleman, N. (1983), "The index of signature operators on Lipschitz manifolds", Publications Mathématiques 58: 251–290, doi:10.1007/BF02953772, <http://www.numdam.org/item/PMIHES_1983__58__39_0/>
• Teleman, N. (1984), "The index theorem on topological manifolds", Acta Mathematica 153: 117–152, DOI 10.1007/BF02392376
• Teleman, N. (1985), "Transversality and the index theorem", Integral Equations and Operator Theory 8 (5): 693–719, DOI 10.1007/BF01201710
• Thom, R. (1956), "Les classes caractéristiques de Pontrjagin de variétés triangulées", Symp. Int. Top. Alg. Mexico, pàg. 54–67
• Witten, Edward (1982), "Supersymmetry and Morse theory", J. Diff. Geom. 17 (4): 661–692, DOI 10.4310/jdg/1214437492
• Shing-Tung Yau, ed. (2009), The Founders of Index Theory (2nd ed.), Somerville, Mass.: International Press of Boston, ISBN 978-1571461377 - Personal accounts on Atiyah, Bott, Hirzebruch and Singer.