An example of a closed 5-manifold of nonpositive curvature that fibers over a circle

TL;DR

This paper constructs a specific 5-dimensional closed aspherical manifold with nonpositive curvature that fibers over a circle, illustrating complex relationships between curvature, fiber structure, and relative hyperbolicity of fundamental groups.

Contribution

It provides an explicit example of a 5-manifold with nonpositive curvature that fibers over a circle, with a fiber whose fundamental group exhibits different relative hyperbolicity properties.

Findings

Constructs a 5-manifold with nonpositive curvature that fibers over a circle.

Shows the fiber's fundamental group is not relatively hyperbolic despite the total space's properties.

Demonstrates complex interactions between curvature, fibration, and group hyperbolicity.

Abstract

We exhibit a closed aspherical 5-manifold of nonpositive curvature that fibers over a circle whose fundamental group is hyperbolic relative to abelian subgroups such that the fiber is a closed aspherical 4-manifold whose fundamental group is not hyperbolic relative to abelian subgroups.