An absolute version of the Gromov-Lawson relative index theorem

TL;DR

This paper extends the Gromov-Lawson relative index theorem to an absolute setting for Dirac operators on complete manifolds with group actions, expressing the index as contributions from inside and outside a compact set.

Contribution

It introduces an absolute version of the Gromov-Lawson relative index theorem for Dirac operators with warped product structures at infinity, incorporating equivariant index formulas.

Findings

Derived an equivariant index formula combining interior and exterior contributions.

Extended the relative index theorem to an absolute setting for manifolds with warped product ends.

Connected the results to Atiyah-Patodi-Singer index theorem in the equivariant context.

Abstract

A Dirac operator on a complete manifold is Fredholm if it is invertible outside a compact set. Assuming a compact group to act on all relevant structure, and the manifold to have a warped product structure outside such a compact set, we express the equivariant index of such a Dirac operator as an Atiyah-Segal-Singer type contribution from inside this compact set, and a contribution from outside this set. Consequences include equivariant versions of the relative index theorem of Gromov and Lawson, in the case of manifolds with warped product structures at infinity, and the Atiyah-Patodi-Singer index theorem.