On finite derived quotients of 3-manifold groups

Abstract

This paper studies the set of finite groups appearing as ${\pi }_1(M)/{\pi }_1(M{)}^{(n)}$, where $M$ is a closed, orientable 3-manifold and ${\pi }_1(M{)}^{(n)}$ denotes the $n$-th term of the derived series of ${\pi }_1(M)$. Our main result is that if $M$ is a closed, orientable 3-manifold, $n\ge 2$, and $G\cong {\pi }_1(M)/{\pi }_1(M{)}^{(n)}$ is finite, then the cup product pairing $H^2(G)\otimes H^2(G)\to H^4(G)$ has cyclic image $C$, and the pairing $H^2(G)\otimes H^2(G)\stackrel{⌣}{⟶}C$ is isomorphic to the linking pairing $H_1(M{)}_{\text{Tors}}\otimes H_1(M{)}_{\text{Tors}}\to \mathbb{Q}/\mathbb{Z}$.