Stallings's Fibring Theorem and ${\mathrm{PD}}^3$-pairs

Abstract

We give a relatively self-contained proof that if a group $G$ fibres algebraically and is part of a ${\mathrm{PD}}^3$-pair, then $G$ is the fundamental group of a fibred compact aspherical 3-manifold. This yields a homological proof of a classical theorem of Stallings: if $G={\pi }_1(M^3)$ is the fundamental group of a compact irreducible 3-manifold $M^3$ and $\varphi :G\to \mathbb{Z}$ is a surjective homomorphism with finitely generated kernel, then $\varphi$ is induced by a topological fibration of $M^3$ over the circle.

arXiv comment7 pages. v2, added further background to the introduction and a clarification in the proof of Theorem 2. To appear in "Topology at Infinity of Discrete Groups", Contemporary Mathematics series