## 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

#### 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 $N\hookrightarrow M\twoheadrightarrow B$ 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 $\widehat{A}$-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