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 nonzero 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 nontrivial covering and the bundle is virtually
trivial. As another application, we derive the first examples of nonvanishing
numerical invariants that are monotone with respect to the mapping degree on
nontrivial circle bundles over aspherical manifolds with hyperbolic fundamental
groups in any dimension.
Keywords
Hopf property, degree of self-map, homotopy equivalence,
aspherical manifold, circle bundle, fundamental group,
hyperbolic group
