Simplicial volume of fiber bundles with nonpositively curved fibers

TL;DR

This paper proves that the simplicial volume of certain fiber bundles with nonpositively curved fibers is zero, establishing a precise relationship between the simplicial volumes of the total space and the base space.

Contribution

It demonstrates that the simplicial volume of fiber bundles with nonpositively curved fibers is zero under specific curvature conditions, and characterizes when the total space's simplicial volume vanishes.

Findings

Simplicial volume of the total space is zero for bundles with nonpositively curved fibers.

The total space's simplicial volume is zero iff the base's simplicial volume is zero for higher-dimensional negatively curved fibers.

Provides conditions linking curvature properties to simplicial volume in fiber bundles.

Abstract

We prove the simplicial volume of the total space of a smooth fiber bundle with fiber being an oriented closed connected (occ) manifold of nonpositive curvature and negative Ricci curvature over an occ manifold with a closed universal covering is zero. Furthermore, if the fiber is an occ negatively curved manifold with dimension more than $2$, the simplicial volume of the total space is zero if and only if the simplicial volume of the base space is zero.