The relative index theorem for general first-order elliptic operators

TL;DR

This paper extends the relative index theorem to general first-order elliptic operators on measured manifolds, broadening its applicability beyond Dirac operators by employing boundary value problem techniques.

Contribution

It generalizes the relative index theorem to a wider class of elliptic operators using boundary value problem methods and graphical boundary condition decompositions.

Findings

Extended the relative index theorem to general first-order elliptic operators.

Developed boundary value problem framework for these operators.

Proved splitting, decomposition, and Phi-relative index theorems.

Abstract

The relative index theorem is proved for general first-order elliptic operators that are complete and coercive at infinity over measured manifolds. This extends the original result by Gromov-Lawson for generalised Dirac operators as well as the result of B\"ar-Ballmann for Dirac-type operators. The theorem is seen through the point of view of boundary value problems, using the graphical decomposition of elliptically regular boundary conditions for general first-order elliptic operators due to B\"ar-Bandara. Splitting, decomposition and the Phi-relative index theorem are proved on route to the relative index theorem.