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

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 ${\pi }_1\left(B\right)$ of finite asymptotic dimension. The proof is based on a new variant of the multipartitioned 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 $\widehat{A}$-genus of certain submanifolds.