Simplicial volume and essentiality of manifolds fibered over spheres

TL;DR

This paper investigates when manifolds fibered over spheres are rationally essential or have positive simplicial volume, revealing that certain high-dimensional mapping tori are often essential with positive volume, contrasting with fiber bundles over spheres.

Contribution

It demonstrates that mapping tori of manifolds in high odd dimensions with non-zero simplicial volume are common, and contrasts their properties with fiber bundles over spheres.

Findings

Mapping tori of high-dimensional manifolds often have positive simplicial volume.

Fiber bundles over spheres with dimension ≥ 3 are rationally inessential and have zero simplicial volume.

Results have implications for scalar curvature and characteristic classes.

Abstract

We study the question when a manifold that fibers over a sphere can be rationally essential, or even have positive simplicial volume. More concretely, we show that mapping tori of manifolds (whose fundamental groups can be quite arbitrary) of odd dimension at least 7 with non-zero simplicial volume are very common. This contrasts the case of fiber bundles over a sphere of dimension d > 1: we prove that their total spaces are rationally inessential if d is at least 3, and always have simplicial volume 0. Using a result by Dranishnikov, we also deduce a surprising property of macroscopic dimension, and we give two applications to positive scalar curvature and characteristic classes, respectively.