2021/07/13 by Thorben Kastenholz, Kastenholz, Thorben, Jens Reinhold +1
Mathematics · #20J05 #57M07 #Algebraic Topology (math.AT) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology #primary: 53C23 secondary: 57N65
paper · pdf · doi:10.48550/arxiv.2107.05892
openalex publication_date 2021/07/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.