Simplicial volume and essentiality of manifolds fibered over spheres

Abstract

AbstractWe study the question when a manifold that fibers over a sphere can be rationally essential, or have positive simplicial volume. More concretely, we show that mapping tori of manifolds (whose fundamental groups can be quite arbitrary) of dimension with non‐zero simplicial volume are very common. This contrasts the case of fiber bundles over a sphere of dimension : we prove that their total spaces are rationally inessential if , 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.