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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

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 NM B is a fiber bundle of spin manifolds with B aspherical and π1(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 Â-genus of certain submanifolds.

positive scalar curvature, multipartitioned manifolds, coarse index theory, asymptotic dimension, aspherical manifolds
Mathematical Subject Classification 2010
Primary: 58J22
Secondary: 46L80, 53C23
Received: 3 November 2016
Revised: 6 February 2017
Accepted: 26 February 2017
Published: 19 September 2017
Rudolf Zeidler
Mathematisches Institut
Westfälische Wilhelms-Universität Münster