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

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 { A , ρ , F } $$\{ \mathcal {A}, \rho , F \}$$ consisting of an associative, not necessarily commutative or topological algebra A $$\mathcal {A}$$ , a Fredholm operator F acting on a separable Hilbert space H and ρ : A → L ( H ) $$\rho : \mathcal {A} \longrightarrow \mathit {L}(H)$$ a representation of the algebra A $$\mathcal {A}$$ 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.