A cohomological formula for the Atiyah-Patodi-Singer index on manifolds with boundary

TL;DR

This paper derives a cohomological formula for the index of elliptic operators on manifolds with boundary, extending classical index theory using noncommutative geometry and the tangent groupoid approach.

Contribution

It introduces a new cohomological index formula for boundary manifolds using noncommutative symbols and Connes' tangent groupoid, generalizing Atiyah-Patodi-Singer theory.

Findings

Provides a cohomological index formula involving a noncommutative symbol

Expresses the index as an integral over the singular normal bundle

Extends Atiyah-Patodi-Singer theorem using noncommutative geometry

Abstract

We give a cohomological formula for the index of a fully elliptic pseudodifferential operator on a manifold with boundary. As in the classic case of Atiyah-Singer, we use an embedding into an euclidean space to express the index as the integral of a cohomology class depending in this case on a noncommutative symbol, the integral being over a $C^\infty$-manifold called the singular normal bundle associated to the embedding. The formula is based on a K-theoretical Atiyah-Patodi-Singer theorem for manifolds with boundary that is drawn from Connes' tangent groupoid approach.