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

  • Neculai S. Teleman
Chapter

Abstract

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.

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Authors and Affiliations

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

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