K-Homology and Noncommutative Geometry

This chapter deals with some of the key aspects of the geometric part of the subject of Noncommutative Geometry. Connes’ program for a Noncommutative (Riemannian) Geometry is based on using ideas from classical Index Theory to endow potentially noncommutative C*-algebras with further geometric structure. For example, the C*-algebra \(C(M)\) for a compact smooth manifold M contains the dense and holomorphically closed subalgebra \(C^\infty (M)\) , and the geometry of M leads to many interesting functionals on \(C^\infty (M)\) , like \(\tau (f, g) = \int _\gamma fdg,\) where \(\gamma \) is a closed curve in M. Such \(\gamma \) defines a closed 1-current: a continuous linear functional \(\Omega ^1(M) \to \mathbb {C}\) .