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 Gamma of pi(1)(N) the ranks of Gamma and of f(*)(-1) (Gamma) agree, or for any finite cover (N) over tilde of N the Heegaard genus of (N) over tilde and the Heegaard genus of the pull-back cover (M) over tilde agree.