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 $G = π_1(M^3)$ is the fundamental group of a compact irreducible 3-manifold $M^3$ and $ϕ\colon G \to \mathbb{Z}$ is a surjective homomorphism with finitely generated kernel, then $ϕ$ is induced by a topological fibration of $M^3$ over the circle.