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

TL;DR

The paper provides a homological proof linking algebraic fibering of groups within $ ext{PD}^3$-pairs to the topological fibering of 3-manifolds, extending Stallings's classical theorem.

Contribution

It offers a new homological proof of Stallings's theorem connecting algebraic and topological fibering in 3-manifolds.

Findings

Algebraic fibering in $ ext{PD}^3$-pairs implies topological fibering of 3-manifolds.

Provides a self-contained proof of Stallings's theorem using homological methods.

Establishes a correspondence between algebraic properties of groups and geometric structures.

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 $\phi \colon G \to \mathbb{Z}$ is a surjective homomorphism with finitely generated kernel, then $\phi$ is induced by a topological fibration of $M^3$ over the circle.