  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.
