On finite derived quotients of $3$–manifold groups

Will Cavendish

doi:10.2140/agt.2015.15.3355

2015 On finite derived quotients of $3$–manifold groups

Will Cavendish

Algebr. Geom. Topol. 15(6): 3355-3369 (2015). DOI: 10.2140/agt.2015.15.3355

Abstract

This paper studies the set of finite groups appearing as $π_{1} (M) ∕ π_{1} {(M)}^{(n)}$ , where $M$ is a closed, orientable $3$ –manifold and $π_{1} {(M)}^{(n)}$ denotes the $n^{t h}$ term of the derived series of $π_{1} (M)$ . Our main result is that if $M$ is a closed, orientable $3$ –manifold, $n \geq 2$ , and $G ≅ π_{1} (M) ∕ π_{1} {(M)}^{(n)}$ 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 $H^{2} (G) \otimes H^{2} (G) \overset{⌣}{\to} C$ is isomorphic to the linking pairing $H_{1} {(M)}_{Tors} \otimes H_{1} {(M)}_{Tors} \to ℚ ∕ ℤ$ .

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Citation Download Citation

Will Cavendish "On finite derived quotients of $3$–manifold groups," Algebraic & Geometric Topology 15(6), 3355-3369, (2015). https://doi.org/10.2140/agt.2015.15.3355

Received: 30 July 2014; Accepted: 13 April 2015; Published: 2015

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