K-homology and Fredholm operators I: Dirac Operators

TL;DR

This paper provides an accessible proof of the Atiyah-Singer index theorem for Dirac operators, framing it as a computation of K-homology of a point, and aims to clarify foundational aspects of index theory.

Contribution

It offers a clear, detailed proof of the Atiyah-Singer index theorem for Dirac operators, addressing gaps in published proofs and clarifying key points in index theory.

Findings

Proof of the Atiyah-Singer index theorem for Dirac operators

Computation of K-homology of a point

Clarification of foundational index theory points

Abstract

This is an expository paper which gives a proof of the Atiyah-Singer index theorem for Dirac operators, presenting the theorem as a computation of the K-homology of a point. This paper and its follow up ("K-homology and index theory II: Elliptic Operators") was written to clear up basic points about index theory that are generally accepted as valid, but for which no proof has been published. Some of these points are needed for the solution of the Heisenberg-elliptic index problem in our paper "K-homology and index theory on contact manifolds" Acta. Math. 2014.