The class of a fibre in Noncommutative Geometry

TL;DR

This paper explores the K-homology of crossed product spaces in noncommutative geometry, identifying the Dirac class via Baum-Connes and applying it to spectral triples and K-theory computations.

Contribution

It introduces a systematic approach to defining and analyzing Dirac classes in noncommutative spaces arising from group actions, linking them to Baum-Connes and spectral triples.

Findings

The deformation of the Dolbeault operator on the noncommutative torus is a Dirac class.

The boundary extension class of a hyperbolic group is a Dirac class.

The approach unifies topological treatment of different noncommutative geometric examples.

Abstract

This paper studies the K-homology of a crossed product of a discrete group acting smoothly on a manifold, with a better understanding of the noncommutative geometry of the crossed-product as the primary goal, and the Baum-Connes apparatus as 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 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.