An index obstruction to positive scalar curvature on fiber bundles over aspherical manifolds

TL;DR

This paper demonstrates that higher index invariants, specifically the Rosenberg index, serve as obstructions to positive scalar curvature on manifolds containing certain fibered submanifolds over aspherical bases, extending known geometric obstructions.

Contribution

It establishes a new link between the Rosenberg index and positive scalar curvature for fiber bundles over aspherical manifolds, introducing a novel multi-partitioned manifold index theorem.

Findings

Rosenberg index obstructs positive scalar curvature on ambient manifolds

New variant of multi-partitioned manifold index theorem introduced

Elementary obstructions via the $$-genus discussed

Abstract

We exhibit geometric situations, where higher indices of the spinor Dirac operator on a spin manifold $N$ are obstructions to positive scalar curvature on an ambient manifold $M$ that contains $N$ as a submanifold. In the main result of this note, we show that the Rosenberg index of $N$ is an obstruction to positive scalar curvature on $M$ if $N \hookrightarrow M \twoheadrightarrow B$ is a fiber bundle of spin manifolds with $B$ aspherical and $\pi_1(B)$ of finite asymptotic dimension. The proof is based on a new variant of the multi-partitioned manifold index theorem which might be of independent interest. Moreover, we present an analogous statement for codimension one submanifolds. We also discuss some elementary obstructions using the $\hat{A}$-genus of certain submanifolds.