Algebraic & Geometric Topology

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

Rudolf Zeidler

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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 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.

Article information

Source
Algebr. Geom. Topol., Volume 17, Number 5 (2017), 3081-3094.

Dates
Received: 3 November 2016
Revised: 6 February 2017
Accepted: 26 February 2017
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1510841493

Digital Object Identifier
doi:10.2140/agt.2017.17.3081

Mathematical Reviews number (MathSciNet)
MR3704253

Zentralblatt MATH identifier
1377.58017

Subjects
Primary: 58J22: Exotic index theories [See also 19K56, 46L05, 46L10, 46L80, 46M20]
Secondary: 46L80: $K$-theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22] 53C23: Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces

Keywords
positive scalar curvature multipartitioned manifolds coarse index theory asymptotic dimension aspherical manifolds

Citation

Zeidler, Rudolf. An index obstruction to positive scalar curvature on fiber bundles over aspherical manifolds. Algebr. Geom. Topol. 17 (2017), no. 5, 3081--3094. doi:10.2140/agt.2017.17.3081. https://projecteuclid.org/euclid.agt/1510841493


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