#### Volume 17, issue 5 (2017)

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

### Rudolf Zeidler

Algebraic & Geometric Topology 17 (2017) 3081–3094
##### 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 $N↪M↠B$ 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 $\stackrel{̂}{A}$-genus of certain submanifolds.

##### Keywords
positive scalar curvature, multipartitioned manifolds, coarse index theory, asymptotic dimension, aspherical manifolds
##### Mathematical Subject Classification 2010
Primary: 58J22
Secondary: 46L80, 53C23