Algebraic & Geometric Topology
- Algebr. Geom. Topol.
- Volume 17, Number 5 (2017), 3081-3094.
An index obstruction to positive scalar curvature on fiber bundles over aspherical manifolds
We exhibit geometric situations where higher indices of the spinor Dirac operator on a spin manifold are obstructions to positive scalar curvature on an ambient manifold that contains as a submanifold. In the main result of this note, we show that the Rosenberg index of is an obstruction to positive scalar curvature on if is a fiber bundle of spin manifolds with aspherical and 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 -genus of certain submanifolds.
Algebr. Geom. Topol., Volume 17, Number 5 (2017), 3081-3094.
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
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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
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