Talbot Workshop 2010 Talk 2: K-Theory and Index Theory

TL;DR

This paper provides an accessible overview of index theory through topological K-theory, explaining key concepts like the Atiyah-Singer index theorem, orientations, and higher index theory with minimal background assumptions.

Contribution

It offers a clear, introductory perspective on index theory within topological K-theory, connecting analytical and topological viewpoints for newcomers.

Findings

Explanation of the Atiyah-Singer index theorem using Gysin maps

Description of orientations in complex K-theory via spin^c structures

Discussion of higher index theory with Cl_k modules

Abstract

These are notes from a talk at the 2010 Talbot Workshop on Twisted K-theory and Loop Groups. This particular talk is an overview of index theory from the point of view of topological K-theory. Assuming little background in analysis, but some familiarity with complex K-theory, the talk covers the (families) Atiyah-Singer index theorem from the point of view of Gysin maps for fibrations, and orientations for complex K-theory in terms of spin^c structures, Clifford algebras and Dirac operators. Higher index theory is also discussed, in terms of Cl_k modules and Cl_k-linear operators. It is meant to be a readable introduction to the subject, with references to the literature.