The relative index theorem and a characterization of Fredholm operators

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.