On finite derived quotients of 3-manifold groups

TL;DR

This paper characterizes the structure of finite quotients of the derived series of 3-manifold groups, linking their cohomology pairings to the linking pairing of the manifold's torsion first homology.

Contribution

It establishes a precise relationship between the cup product pairing in these finite quotients and the linking pairing of the manifold's torsion homology, revealing new structural insights.

Findings

The cup product pairing has cyclic image in finite derived quotients.

The cup product pairing is isomorphic to the linking pairing on torsion homology.

Finite derived quotients encode topological linking information.

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 $H^2(G)\otimes H^2(G)\stackrel{\smile}{\longrightarrow} C$ is isomorphic to the linking pairing $H_1(M)_{\textrm{Tors}}\otimes H_1(M)_{\textrm{Tors}}\to \mathbb{Q}/\mathbb{Z}$.