Epimorphisms of 3-manifold groups

TL;DR

This paper investigates conditions under which a proper map between certain 3-manifolds, inducing an epimorphism on fundamental groups, is homotopic to a homeomorphism, focusing on properties like subgroup ranks and Heegaard genus.

Contribution

It establishes new criteria involving subgroup ranks and Heegaard genus for when such maps are homotopic to homeomorphisms, extending understanding of 3-manifold group epimorphisms.

Findings

Homotopy to homeomorphism under subgroup rank conditions

Homotopy to homeomorphism under Heegaard genus conditions

Applicable to aspherical compact orientable 3-manifolds with boundary

Abstract

Let $f\colon M\to N$ 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 $\tilde{N}$ of $N$ the Heegaard genus of $\tilde{N}$ and the Heegaard genus of the pull-back cover $\tilde{M}$ agree.