## Algebraic & Geometric Topology

### On finite derived quotients of $3$–manifold groups

Will Cavendish

#### Abstract

This paper studies the set of finite groups appearing as $π1(M)∕π1(M)(n)$, where $M$ is a closed, orientable $3$–manifold and $π1(M)(n)$ denotes the $nth$ term of the derived series of $π1(M)$. Our main result is that if $M$ is a closed, orientable $3$–manifold, $n ≥ 2$, and $G≅π1(M)∕π1(M)(n)$ is finite, then the cup-product pairing $H2(G) ⊗ H2(G) → H4(G)$ has cyclic image $C$, and the pairing $H2(G) ⊗ H2(G)→⌣C$ is isomorphic to the linking pairing .

#### Article information

Source
Algebr. Geom. Topol., Volume 15, Number 6 (2015), 3355-3369.

Dates
Revised: 13 April 2015
Accepted: 13 April 2015
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.agt/1510841070

Digital Object Identifier
doi:10.2140/agt.2015.15.3355

Mathematical Reviews number (MathSciNet)
MR3450764

Zentralblatt MATH identifier
1334.57001

Subjects
Primary: 57M10: Covering spaces
Secondary: 57M60: Group actions in low dimensions

#### Citation

Cavendish, Will. On finite derived quotients of $3$–manifold groups. Algebr. Geom. Topol. 15 (2015), no. 6, 3355--3369. doi:10.2140/agt.2015.15.3355. https://projecteuclid.org/euclid.agt/1510841070

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