Epimorphisms of 3-manifold groups

Abstract

Let f : M → N be a proper map between two aspherical compact orientable 3-manifolds with empty or toroidal boundary. We assume that N is not a closed graph-manifold. Suppose that f induces an epimorphism on fundamental groups. We show that f is homotopic to a homeomorphism if one of the following holds: either for any finite-index subgroup Γ of π 1 (N ) the ranks of Γ and of f −1 * (Γ) agree, or for any finite cover N of N the Heegaard genus of N and the Heegaard genus of the pull-back cover M agree.2010 Mathematics Subject Classification. 57M10, 57M27.

On topological rigidity of Alexandrov $3$-spaces

“…Consequences of Agol's virtually fibering Theorem. The following theorem was proved by Boileau and Friedl, [2].…”

“…This problem has been studied using simplicial volume, specifically degree theorems, and more recently, Agol's solution to the virtually fibering conjecture by Boileau and Friedl [2].…”