Abstract

This chapter is mainly devoted to a proof, using groupoids and KK-theory, of Atiyah and Singer's index theorem on compact smooth manifolds. We first present an elementary introduction to groupoids, C*-algebras, KK-theory and pseudodifferential calculus on groupoids. We then show how the point of view adopted here generalizes to the case of conical pseudomanifolds.

Introduction

This chapter is meant to give the tools involved in our approach to index theory for singular spaces. The global framework adopted here is noncommutative geometry, with a particular focus on groupoids, C*-algebras and bivariant K-theory.

The idea of using C*-algebras to study spaces may be understood with the help of the Gelfand theorem, which asserts that Hausdorff locally compact spaces are in one-to-one correspondence with commutative C*-algebras. A starting point in noncommutative geometry is then to think of noncommutative C*-algebras as corresponding to a wider class of spaces, more singular than Hausdorff locally compact spaces. As a first consequence, given a geometrical or topological object which is badly behaved with respect to classical tools, noncommutative geometry suggests defining aC*-algebra encoding relevant information carried by the original object.

Refining this construction, one may try to define this C*-algebra as the C*- algebra of a groupoid [46, 47]. That is, one can try to build a groupoid directly, encoding the original object and regular enough to allow the construction of its C*-algebra. In the ideal case where the groupoid is smooth, one gets much more than a C*-algebra, which only reflects topological properties: the groupoid has a geometrical and analytical flavor enabling many applications.