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 π1(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.