A short proof of an index theorem, II

TL;DR

This paper presents a modified equivariant KK-theory to provide a KK-theoretical proof of an index theorem for Dirac-Schrodinger operators on non-compact manifolds with nowhere positive curvature, extending previous results.

Contribution

It introduces a slight modification of equivariant KK-theory and applies it to prove an index theorem for Dirac-Schrodinger operators on non-compact manifolds.

Findings

Established a KK-theoretical proof of the index theorem for Dirac-Schrodinger operators.

Showed that the boundary of Dirac is Dirac, generalizing earlier work.

Extended results of Higson and Roe to a broader setting.

Abstract

We introduce a slight modification of the usual equivariant $KK$-theory. We use this to give a $KK$-theoretical proof of an equivariant index theorem for Dirac-Schrodinger operators on a non-compact manifold of nowhere positive curvature. We incidentally show that the boundary of Dirac is Dirac; generalizing earlier work of Baum and coworkers, and a result of Higson and Roe.