Skip to main content

Index Theorems and Noncommutative Topology

  • Chapter
  • First Online:
Geometric and Topological Methods for Quantum Field Theory

Part of the book series: Lecture Notes in Physics ((LNP,volume 668))

  • 703 Accesses

Abstract

These lecture notes are mainly devoted to a K-theory proof of the Atiyah-Singer index theorem. Some applications of the K-theory to noncommutative topology are also given.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M.F.Atiyah. KTheory. Benjamin Press, New-York. 1967.

    Google Scholar 

  2. M.F.Atiyah and I.M.Singer. The index of elliptic operators IV. Ann. of Math. 93 (1971), 119–138.

    Google Scholar 

  3. A.Connes. Sur la théorie non commutative de l’intégration. In Lectures Notes in Mathematics. 725. Editor Pierre de la Harpe (1979), 19–143.

    Google Scholar 

  4. A. Connes. A survey of foliations and operator algebras. Operator algebras and apllications, Part 1. Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, R.I. 38 (1982), 521–628.

    Google Scholar 

  5. A.Connes. An analogue of the Thom isomorphism for cross products of a C*-algebra by an action of R. Adv. in Math. 39 (1981), 31–55.

    Article  Google Scholar 

  6. A.Connes and G.Skandalis. The longitudinal index theorem for foliations. Publ. RIMS, 20 (No 6), 1984, 1139–1183.

    Google Scholar 

  7. T. Fack and G. Skandalis. Connes’ Analogue of the Thom Isomorphism of the Kasparov Groups. Invent. Math. 64 (1981), 7–14.

    Article  Google Scholar 

  8. P. B. Gilkey. Invariance Theory, The Heat Equation, And the Atiyah-Singer Index Theorem. Mathematics Lectures Series, 11, Publish or Perish, Inc. 1984.

    Google Scholar 

  9. D. Husemoller. Fibre bundles.2d Edition. Graduate Texts in Mathematics (20). Springer-Verlag. New York - Heidelberg - Berlin, 1975.

    Google Scholar 

  10. M. Karoubi. K-Theory. An introduction. Grundlehren der mathematischen Wissenschaften, 226 – Springer-Verlag. Berlin-Heidelberg – New York, 1978.

    Google Scholar 

  11. .G.Kasparov. Hilbert C*-modules. Theorems of Stinespring and Voiculescu. J. Op. Theory, 4 (1980), 133–150.

    Google Scholar 

  12. G.G.Kasparov. The operator K-functor and extensions of C*-algebras. Math. U.S.S.R. – Izv. 16 (1981), 513–572.

    Google Scholar 

  13. G.G.Kasparov. Equivariant KK-theory and the Novikov conjecture. Invent. Math. 91 (1988), 147–201.

    Article  Google Scholar 

  14. N.Kuiper. The homotopy type of the unitary group of Hilbert space. Topology 3 (1965), 19–30.

    Article  Google Scholar 

  15. S.Lang, Real Analysis. Addison-Wesley Publishing Company. 1983.

    Google Scholar 

  16. H.B.Lawson, M-L.Michelson. Spin Geometry. Princeton Mathematical Series, 38. Princeton University Press. 1989.

    Google Scholar 

  17. G.K.Pedersen.C*-algebras and their Automorphism groupsLondon Mathematical Society. Monographs no 14. Academic Press. London – New York – San Francisco. 1979.

    Google Scholar 

  18. J.Roe. Elliptic operators, topology and asymptotic methods. Pitman Research Notes in Mathematics Series (179). Longman Scientific and Technical. Longman Group UK Limited. 1988.

    Google Scholar 

  19. B.Simon. Trace ideals and their applications. London Math. Soc. Lecture Note Series, 35. Cambridge University Press. 1979.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Hernán Ocampo Sylvie Paycha Andrés Vargas

Rights and permissions

Reprints and permissions

About this chapter

Cite this chapter

Fack, T. Index Theorems and Noncommutative Topology. In: Ocampo, H., Paycha, S., Vargas, A. (eds) Geometric and Topological Methods for Quantum Field Theory. Lecture Notes in Physics, vol 668. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11374060_4

Download citation

Publish with us

Policies and ethics