On finite derived quotients of $3$–manifold groups

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 $H_1{\left(M\right)}_{\mathrm{\text{ Tors}}}\otimes H_1{\left(M\right)}_{\mathrm{\text{ Tors}}}\to \mathbb{Q}∕\mathbb{Z}$.