Spaces, Bundles, Homology/Cohomology and Characteristic Classes in Non-commutative Geometry

  • Neculai S. Teleman


In Chap.  1 we recalled some of the basic notions and results which are commonly used in differential geometry. We had presented them with the intent of showing how they pass into non-commutative geometry. By definition, a non-commutative space is a spectral triple\(\{ \mathcal {A}, \rho , F \}\) consisting of an associative, not necessarily commutative or topological algebra \(\mathcal {A}\), a Fredholm operator F acting on a separable Hilbert space H and \(\rho : \mathcal {A} \longrightarrow \mathit {L}(H)\) a representation of the algebra \(\mathcal {A}\) onto the Hilbert space H, subject to additional conditions. Such a structure codifies an abstract elliptic operator defined by Atiyah (K-Theory, Benjamin, 1967). While in differential geometry elliptic operators are defined after multiple structures are summed up, in non-commutative geometry this process is reversed. The study of non-commutative geometry consists of finding the hidden mathematical structures codified by a spectral triple. In Chap. 2 we show how the notions of space, bundles, homology/cohomology and characteristic classes can be extracted from spectral triples. We stress that non-commutative geometry objects are defined in such a way that (1) the locality and (2) the commutativity assumptions, used in the differentiable geometry counterparts, are not postulated.


  1. 3.
    Gelfand I. M., Naimark M. A. (1943). On the imbedding of normed rings into the ring of operators on a Hilbert space. Math. Sbornik 12 (2): 197–217 (1943).Google Scholar
  2. 4.
    Hochschild G.: On the cohomology groups of an associative algebra. Annals of Mathematics 46 (1945), 58–67MathSciNetCrossRefGoogle Scholar
  3. 5.
    Chevalley C., Eilenberg S. : Cohomology theory of Lie groups and Lie algebras. Trans. Amer. Math. Soc. 63 (1948) 85–124.MathSciNetCrossRefGoogle Scholar
  4. 28.
    Rickart C. E.: Banach Algebras. Robert E. Krieger Publishing Company, 1960.zbMATHGoogle Scholar
  5. 35.
    Gerstenhaber M.: The cohomology structure of an associative ring. Ann. of Math. 78 (1963) 267–288.MathSciNetCrossRefGoogle Scholar
  6. 40.
    Dixmier J.: Les C -alg\(\grave {e}\)bras et leur repr\(\acute {e}\)sentations. Cahier Scientifiques, Fasc. XXIX, Gautier-Villars, Paris, 1964.Google Scholar
  7. 42.
    Palais R.: Seminar on the Atiyah - Singer Index Theorem. Annals of Mathematics Studies 57, Princeton University Press, 1965Google Scholar
  8. 45.
    Spanier E. H.: Algebraic Topology, McGraw - Hill Series in Higher Mathematics, New York, 1966.zbMATHGoogle Scholar
  9. 62.
    Teleman N. : A characteristic ring of a Lie algebra extension. Note I., Note II Atti Accad. Nazionale Lincei Serie VIII, Vol. LII, pp.498–320. Atti Accad. Nazionale Lincei Serie VIII, Vol. LII, 1972, pp. 498–320, 1972Google Scholar
  10. 68.
    Mac Lane S.: Homology, Third Ed., Grundlehren der mathematischen Wissenschaften in Einzeldarstellung Band 114, Springer Verlag, Heidelberg, 1975.Google Scholar
  11. 74.
    H\(\ddot {o}\)rmander L., The Weyl calculus of pseudodifferential operators. Comm. Pure Appl. Math. 32, pp. 359–443, 1979.Google Scholar
  12. 80.
    Arveson W.: An Invitation to C -Algebras. Springer Verlag, 1981.zbMATHGoogle Scholar
  13. 82.
    Connes A.: Non-Commutative differential geometry. Publications Math\(\acute {e}\)matiques I.H.E.S. Vol. 62, pp.256–35, 1985.CrossRefGoogle Scholar
  14. 83.
    Teleman N.: The Index of Signature Operators on Lipschitz Manifolds. Publ. Math. Paris, IHES, Vol. 58, pp. 251–290, 1983.zbMATHGoogle Scholar
  15. 85.
    Loday J.-L. Loday, Quillen D.: Cyclic homology and Lie algebra homology of matrices. Comm. Math. Helvetici, 59, pp. 565–591, 1984.CrossRefGoogle Scholar
  16. 89.
    Karoubi M., Homologie cyclique et K-Th\(\acute {e}\)orie. Ast\(\acute {e}\)risque No. 149 (1987). Soc. Math. de France.Google Scholar
  17. 91.
    Quillen D.: Algebra cochains and cyclic cohomology. Publications Math\(\acute {e}\)matiques I.H.E.S. Vol. 68, pp.139–174, 1988.MathSciNetCrossRefGoogle Scholar
  18. 96.
    Connes A., Moscovici H.: Cyclic Cohomology, the Novikov Conjecture and Hyperbolic Groups, Topology Vol. 29, pp.345–388, 1990.MathSciNetCrossRefGoogle Scholar
  19. 97.
    Loday J.-L.: Cyclic Homology, Grundlehren in mathematischen Wissenschaften 301, Springer Verlag, Berlin Heidelberg, 1992.Google Scholar
  20. 99.
    Connes A.: Noncommutative Geometry, Academic Press, 1994.zbMATHGoogle Scholar
  21. 101.
    Connes A., Sullivan D., Teleman N.: Quasiconformal Mappings, Operators on Hilbert Space and Local Formulae for Characteristic Classes, Topology Vol. 33, Nr. 4, pp. 663–681, 1994.Google Scholar
  22. 104.
    Cuntz J.; Quillen D., Operators on noncommutative differential forms and cyclic homology, Geometry, Topology and Physics, 77–111, International Press, Cambridge, MA, 1995.zbMATHGoogle Scholar
  23. 115.
    Cuntz J., Cyclic Theory, Bivariant K-Theory and the Bivariant Chern-Connes Character, Operator Algebras and Non-Commutative Geometry II, Encyclopedia of Mathematical Sciences, Vol. 121, 1–71, Springer Verlag, 2004.Google Scholar
  24. 117.
    Kubarski J., Teleman N. Linear Direct Connections, Proceedings 7th Conference on “Geometry and Topology of Manifolds - The Mathematical Legacy of Charles Ehresmann”, Betlewo, May 2005.Google Scholar
  25. 121.
    Teleman N.: Direct connections and Chern character, Singularity Theory, Proceedings of the Marseille Singularity School and Conference CIRM, Marseilles, 24 January - 25 February 2005, Eds. D. Cheniot, N. Duterte, C. Murolo, D. Trotman, A. Pichon. World Scientific Publishing Company, 2007.Google Scholar
  26. 122.
    Teleman N.: Modified Hochschild and periodic cyclic homology. C -Algebras and Elliptic Theory II, Trends in Mathematics, pp. 251–265, Birkh\(\ddot {a}\)user Verlag, Basel, Switzerland, 2008Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Neculai S. Teleman
    • 1
  1. 1.Dipartimento di Scienze MatematicheUniversità Politecnica delle MarcheAnconaItaly

Personalised recommendations