Simplicial volume and essentiality of manifolds fibered over spheres
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.
Keywords
Cite
@article{arxiv.2107.05892,
title = {Simplicial volume and essentiality of manifolds fibered over spheres},
author = {Thorben Kastenholz and Jens Reinhold},
journal= {arXiv preprint arXiv:2107.05892},
year = {2022}
}
Comments
13 pages; comments welcome; V2: Restructured the paper and filled a gap in the proof of Theorem C (in the current version)