#### Volume 15, issue 6 (2015)

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On finite derived quotients of $3$–manifold groups

### Will Cavendish

Algebraic & Geometric Topology 15 (2015) 3355–3369
##### Abstract

This paper studies the set of finite groups appearing as ${\pi }_{1}\left(M\right)∕{\pi }_{1}{\left(M\right)}^{\left(n\right)}$, where $M$ is a closed, orientable $3$–manifold and ${\pi }_{1}{\left(M\right)}^{\left(n\right)}$ denotes the ${n}^{th}$ term of the derived series of ${\pi }_{1}\left(M\right)$. Our main result is that if $M$ is a closed, orientable $3$–manifold, $n\ge 2$, and $G\cong {\pi }_{1}\left(M\right)∕{\pi }_{1}{\left(M\right)}^{\left(n\right)}$ is finite, then the cup-product pairing ${H}^{2}\left(G\right)\otimes {H}^{2}\left(G\right)\to {H}^{4}\left(G\right)$ has cyclic image $C$, and the pairing ${H}^{2}\left(G\right)\otimes {H}^{2}\left(G\right)\stackrel{⌣}{\to }C$ is isomorphic to the linking pairing .

##### Keywords
finite sheeted covering spaces, 3–manifolds, first Betti number, linking pairing
Primary: 57M10
Secondary: 57M60