The relative index theorem and a characterization of Fredholm operators

TL;DR

This paper generalizes the relative index theorem to hypoelliptic operators on non-compact manifolds and establishes a geometric characterization of Fredholm operators based on invertibility at infinity, enhancing index theory tools.

Contribution

It extends the relative index theorem to a broader class of hypoelliptic operators and provides a necessary and sufficient condition for Fredholmness based on invertibility at infinity.

Findings

Extended the relative index theorem to hypoelliptic operators of arbitrary order.

Established that invertibility at infinity is both necessary and sufficient for Fredholmness.

Connected Fredholmness to a model in unbounded KK-theory for non-compact spaces.

Abstract

We extend the relative index theorem on non-compact manifolds to encompass a wide variety of hypoelliptic differential operators of arbitrary order, demonstrating that the change in index when changing a differential operator locally can be calculated locally. We also show that the notion of invertibility at infinity (and coercive at infinity) is not only sufficient condition for an operator to be Fredholm but also necessary, resulting in a general geometric characterization of Fredholmness. This characterization connects to a model for unbounded \(KK\)-theory which assumes the operator to be Fredholm instead of having (locally) compact resolvent, and thus provides a convenient tool for index theory on non-compact spaces.