K-homology and Fredholm Operators II: Elliptic Operators

TL;DR

This paper provides an expository proof of the Atiyah-Singer index theorem for elliptic operators, clarifying foundational points in index theory and computing the geometric K-cycle associated with the analytic K-cycle.

Contribution

It offers a detailed proof of the Atiyah-Singer index theorem for elliptic operators and clarifies fundamental aspects of index theory that lacked published proofs.

Findings

Computed the geometric K-cycle corresponding to the analytic K-cycle.

Clarified basic points in index theory with rigorous proofs.

Supported the solution of the Heisenberg-elliptic index problem.

Abstract

This is an expository paper which gives a proof of the Atiyah-Singer index theorem for elliptic operators. Specifcally, we compute the geometric K-cycle that corresponds to the analytic K-cycle determined by the operator. This paper and its companion ("K-homology and index theory II: Dirac 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.