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

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 the trivial bundle. 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 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 of any dimension. Moreover, we obtain the first examples of manifolds (given by the aforementioned bundles with torsion Euler class) which do not admit self-maps of absolute degree greater than one, but admit maps of infinitely many different degrees from other manifolds.Comment: 12 pages; v2: results extended to circle bundles over aspherical manifolds with hyperbolic fundamental groups, title changed accordingl

Graph manifolds have virtually positive Seifert volume

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