Topology Seminar

Seminal work of Thurston shows that one can construct a plethora of finite volume, hyperbolic 3-manifolds using mapping tori of surface homeomorphisms. In fact, Thurston shows that such a mapping torus is hyperbolizable precisely when the corresponding bundle is atoroidal which is equivalent to the associated mapping class being pseudo-Anosov.

Motivated by the desire to push this construction into dimension 4, it has been an open question as to whether one can similarly construct atoroidal surface bundles over surfaces. This was finally resolved this year by Kent and Leininger, who produced infinitely many such bundles. It is a folklore conjecture that, despite the analogy, such manifolds should not be hyperbolizable and one hope was to prove this by showing the non-vanishing of a certain obstruction to hyperbolicity called the signature. In this talk we will show that, on the contrary, the signature of these bundles vanishes and hence they retain the potential to be hyperbolic 4-manifolds. This is joint work with Jean-Francois Lafont and Lorenzo Ruffoni.