On a problem of Hopf for circle bundles over aspherical manifolds with hyperbolic fundamental groups

TL;DR

This paper proves that circle bundles over aspherical manifolds with hyperbolic fundamental groups admit non-trivial self-maps only if they are virtually trivial, extending known results and providing new invariants related to mapping degree.

Contribution

It generalizes the Hopf problem for a broad class of aspherical manifolds with hyperbolic fundamental groups and introduces new monotone invariants for these bundles.

Findings

Self-maps of degree greater than one exist only for virtually trivial bundles.

Every non-zero degree self-map is homotopic to a homeomorphism or a non-trivial covering.

First examples of non-vanishing monotone invariants with respect to degree in this setting.

Abstract

We prove that a circle bundle over a closed oriented aspherical manifold with hyperbolic fundamental group admits a self-map of absolute degree greater than one if and only if it is virtually trivial. This generalizes in every dimension the case of circle bundles over hyperbolic surfaces, for which the result was known by the work of Brooks and Goldman on the Seifert volume. As a consequence, we verify the following strong version of a problem of Hopf for the above class of manifolds: Every self-map of non-zero degree of a circle bundle over a closed oriented aspherical manifold with hyperbolic fundamental group is either homotopic to a homeomorphism or homotopic to a non-trivial covering and the bundle is virtually trivial. As another application, we derive the first examples of non-vanishing numerical invariants that are monotone with respect to the mapping degree on non-trivial circle bundles over aspherical manifolds with hyperbolic fundamental groups in any dimension.