Cobordism Invariance of the Index of Callias-Type Operators

TL;DR

This paper establishes that the index of Callias-type operators remains invariant under cobordism, introduces a gluing formula, and provides new proofs for existing index theorems, simplifying computations on non-compact manifolds.

Contribution

It introduces a cobordism invariance concept for Callias-type operators and derives a new gluing formula and proofs for index theorems, enhancing computational methods.

Findings

Index is preserved under cobordism of Callias-type operators.

A new gluing formula allows index computation via simpler manifolds.

Provides new proofs for the relative and Callias index theorems.

Abstract

We introduce a notion of cobordism of Callias-type operators over complete Riemannian manifolds and prove that the index is preserved by such a cobordism. As an application we prove a gluing formula for Callias-type index. In particular, a usual index of an elliptic operator on a compact manifold can be computed as a sum of indexes of Callias-type operators on two non-compact, but topologically simpler manifolds. As another application we give a new proof of the relative index theorem for Callias-type operators, which also leads to a new proof of the Callias index theorem.