Title: Aspherical circle bundles and a problem of Hopf
Speaker: Christoforos Neofytidis (University of Geneva)
Abstract: A long standing question of Hopf asks whether every self-map of absolute degree one of a closed oriented manifold is a homotopy equivalence. This question gave rise to several other problems, most notably whether the fundamental groups of aspherical manifolds are Hopfian, i.e. any surjective endomorphism is an isomorphism. Recall that the Borel conjecture states that any homotopy equivalence between two closed aspherical manifolds is homotopic to a homeomorphism. In this talk, we verify a strong version of Hopf's problem for certain aspherical manifolds. Namely, we show that a self-map of a circle bundle over a closed oriented negatively curved manifold is either homotopic to a homeomorphism or homotopic to a non-trivial covering and the bundle is trivial. Our main result is that a non-trivial circle bundle over a closed oriented negatively curved manifold does not admit self-maps of absolute degree greater than one. This extends in all dimensions the case of circle bundles over closed hyperbolic surfaces (which was shown by Brooks and Goldman in their study of the Seifert volume) and provides the first examples of non-vanishing semi-norms on the fundamental classes of circle bundles over negatively curved manifolds in all dimensions.
Seminar URL: https://research.math.osu.edu/ggt/
