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

Zeidler, Rudolf

doi:10.2140/agt.2017.17.3081

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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

DOI: 10.2140/agt.2017.17.3081

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 $π_{1} (B)$ 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 $\hat{A}$ -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

References

Publication

Received: 3 November 2016

Revised: 6 February 2017

Accepted: 26 February 2017

Published: 19 September 2017

Authors

Rudolf Zeidler
Mathematisches Institut Westfälische Wilhelms-Universität Münster Münster Germany
http://wwwmath.uni-muenster.de/u/rudolf.zeidler/

Volume 17, issue 5 (2017)