A partitioned manifold index theorem for noncompact hypersurfaces

TL;DR

This paper generalizes Roe's partitioned manifold index theorem to noncompact hypersurfaces, establishing a $K$-theoretic equality that extends the applicability of index obstructions to positive scalar curvature.

Contribution

It extends Roe's theorem to noncompact hypersurfaces with specific embedding conditions, linking indices via $K$-theory of Roe algebras.

Findings

Established a $K$-theoretic equality for noncompact hypersurfaces.

Extended obstructions to positive scalar curvature to broader settings.

Connected to recent related results by other researchers.

Abstract

Roe's partitioned manifold index theorem applies when a complete Riemannian manifold $M$ is cut into two pieces along a compact hypersurface $N$. It states that a version of the index of a Dirac operator on $M$ localized to $N$ equals the index of the corresponding Dirac operator on $N$. This yields obstructions to positive scalar curvature, and implies cobordism invariance of the index of Dirac operators on compact manifolds. We generalize this result to cases where $N$ may be noncompact, under assumptions on the way it is embedded into $M$. This results in an equality between two classes in the $K$-theory of the Roe algebra of $N$. Bunke and Ludewig, and Engel and Wulff, have recently obtained related results based on different approaches.