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Michel Boileau, Stefan Friedl, Epimorphisms of 3-manifold groups, The Quarterly Journal of Mathematics, Volume 69, Issue 3, September 2018, Pages 931–942, https://doi.org/10.1093/qmath/hay007
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Abstract
Let 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 of the ranks of and of agree, or for any finite cover of N the Heegaard genus of and the Heegaard genus of the pull-back cover agree.
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