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

Martin R. Bridson, Dawid Kielak, Monika Kudlinska

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.

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

Abstract

We give a relatively self-contained proof that if a group fibres algebraically and is part of a -pair, then is the fundamental group of a fibred compact aspherical 3-manifold. This yields a homological proof of a classical theorem of Stallings: if is the fundamental group of a compact irreducible 3-manifold and is a surjective homomorphism with finitely generated kernel, then is induced by a topological fibration of over the circle.
Paper Structure (1 section, 2 theorems, 18 equations)

This paper contains 1 section, 2 theorems, 18 equations.

Table of Contents

  1. Acknowledgements

Key Result

Theorem 1

Let $M^3$ be a compact 3-manifold with aspherical (possibly empty) boundary. Suppose that there exists a short exact sequence with $K$ finitely generated. Then $M^3$ is the total space of a fibre bundle with $\Phi$ inducing $\phi$.

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • proof : Proof of \ref{['PD3_group']}
  • proof : Proof of \ref{['main']}