Simplicial volume and essentiality of manifolds fibered over spheres

We 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 $2n+1⩾7$ with non-zero simplicial volume are very common. This contrasts the case of fiber bundles over a sphere of dimension $d⩾2$: we prove that their total spaces are rationally inessential if $d⩾3$, 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.