Index theory and Groupoids

TL;DR

This paper provides an introduction to groupoids, $KK$-theory, and pseudodifferential calculus, demonstrating their application to a proof of the Atiyah-Singer index theorem and its generalization to conical pseudo-manifolds.

Contribution

It offers an elementary introduction to groupoids and $KK$-theory, and applies these tools to prove the Atiyah-Singer index theorem, extending the approach to conical pseudo-manifolds.

Findings

Groupoids and $KK$-theory provide a framework for index theory.

The approach generalizes to conical pseudo-manifolds.

Elementary introduction facilitates understanding of advanced topics.

Abstract

This paper collects the notes of a serie of lectures given by the two authors during the summer school "Geometric and topological methods for Quantum Field Theory" at Villa de Leyva, Colombia, summer 2007. These lecture notes are mainly devoted to a proof using groupoids and $KK$-theory of Atiyah-Singer index theorem on compact smooth manifolds. We will present an elementary introduction to groupoids, $C^*$-algebras, $KK$-theory and pseudodifferential calculus on groupoids. We will finish by showing that the point of view adopted here generalizes to the case of conical pseudo-manifolds.