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 ${\pi }_1(B)$ 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.

12 pages; v2: Changed title, minor revision following the referee's suggestions. To appear in Algebr. Geom. Topol

References in corpus (3)

• The strong Novikov conjecture for low degree cohomology
• Codimension two index obstructions to positive scalar curvature
• Large scale index of multi-partitioned manifolds

Cited by in corpus (4)

• Width, Largeness and Index Theory
• Nonnegative scalar curvature on manifolds with at least two ends
• Curvature-Dimension Condition Meets Gromov's $n$-Volumic Scalar Curvature
• Band width and the Rosenberg index