Stallings's Fibring Theorem and $\mathrm{PD}^3$-pairs
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866909454659747840 |
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| author | Bridson, Martin R. Kielak, Dawid Kudlinska, Monika |
| author_facet | Bridson, Martin R. Kielak, Dawid Kudlinska, Monika |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_10725 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Stallings's Fibring Theorem and $\mathrm{PD}^3$-pairs Bridson, Martin R. Kielak, Dawid Kudlinska, Monika Geometric Topology Group Theory 20J05, 57K30 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. |
| title | Stallings's Fibring Theorem and $\mathrm{PD}^3$-pairs |
| topic | Geometric Topology Group Theory 20J05, 57K30 |
| url | https://arxiv.org/abs/2307.10725 |