The class of a fibre in noncommutative geometry

This article studies the K-homology of a C*-algebra crossed product of a discrete group acting smoothly on a manifold, with the goal of better understanding its noncommutative geometry. The Baum–Connes apparatus is the main tool. Examples suggest that the correct notion of the ‘Dirac class’ of such a noncommutative space is the image under the equivalence determined by Baum–Connes of the fibre of the canonical fibration of the Borel space associated to the action, and a smooth model for the classifying space of the group. We give a systematic study of such fibre, or ‘Dirac classes,’ with applications to the construction of interesting spectral triples, and computation of their K-theory functionals, and we prove in particular that both the well-known deformation of the Dolbeault operator on the noncommutative torus, and the class of the boundary extension of a hyperbolic group, are both Dirac classes in this sense and therefore can be treated topologically in the same way.