Index theory and partitioning by enlargeable hypersurfaces

TL;DR

This paper proves a higher index theorem for odd-dimensional spin manifolds partitioned by hypersurfaces and uses it to relate enlargeability and scalar curvature constraints.

Contribution

It generalizes Higson-Roe's index theorem to higher dimensions and applies it to scalar curvature obstructions via enlargeable hypersurfaces.

Findings

Higher index theorem for partitioned spin manifolds

Obstruction to positive scalar curvature from enlargeable hypersurfaces

Extension of Higson-Roe theorem to higher dimensions

Abstract

In this paper we state and prove a higher index theorem for an odd-dimensional connected spin riemannian manifold $(M,g)$ which is partitioned by an oriented closed hypersurface $N$. This index theorem generalizes a theorem due to N. Higson and J. Roe in the context of Hilbert modules. Then we apply this theorem to prove that if $N$ is area-enlargeable and if there is a smooth map from $M$ into $N$ such that its restriction to $N$ has non-zero degree then the the scalar curvature of $g$ cannot be uniformly positive.