Essentiality and simplicial volume 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 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.

Spaces of positive scalar curvature metrics on totally nonspin manifolds with spin boundary

“…In the case where M is nonspin, but its universal cover is Spin, there are cases where R + (M) is not connected or has even infinitely many path components, see [2,7,8,15,16,23,25]. For totally nonspin manifolds, Kastenholz-Reinhold give an example of a closed, totally nonspin manifold of dimension 6 whose space of psc-metrics has infinitely many components in [21].…”

On the simplicial volume and the Euler characteristic of (aspherical) manifolds

“…(5) Oriented closed connected manifolds that are the boundary of an oriented compact connected manifolds with zero simplicial volume (Remark 2.3); (6) Oriented closed connected n-manifolds that admit a smooth non-trivial S 1 -action [104]. More generally, manifolds admitting an F -structure also have zero simplicial volume [24,91]; (7) Oriented closed aspherical manifolds supporting an affine structure whose holonomy map is injective and contains a pure translation [14]; (8) Oriented closed connected smooth manifolds with zero minimal volume [8,51] or zero minimal volume entropy [51, p. 37] [4]; (9) Oriented closed connected graph 3-manifolds [51,99]; (10) All mapping tori of oriented closed connected 3-manifolds [16]; however, the general behaviour of simplicial volume of general mapping tori is very diverse [65].…”

Spaces of Positive Scalar Curvature metrics on totally nonspin Manifolds with spin boundary

“…In the case where M is nonspin, but its universal cover is Spin, there are cases where R + (M ) is not connected or has even infinitely many path components, see [BG96; Rei20; DG21; Wer20; Goo20; Des21; GW22]. For totally nonspin manifolds, Kastenholz-Reinhold give an example of a closed, totally nonspin manifold of dimension 6 whose space of psc-metrics has infinitely many components in [KR21].…”