An absolute version of the Gromov-Lawson relative index theorem

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.