On finite derived quotients of 3–manifold groups

Cavendish, Will

doi:10.2140/agt.2015.15.3355

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On finite derived quotients of $3$–manifold groups

Will Cavendish

Algebraic & Geometric Topology 15 (2015) 3355–3369

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 $H_{1} {(M)}_{Tors} \otimes H_{1} {(M)}_{Tors} \to ℚ ∕ ℤ$ .

Keywords

finite sheeted covering spaces, 3–manifolds, first Betti number, linking pairing

Mathematical Subject Classification 2010

Primary: 57M10

Secondary: 57M60

References

Publication

Received: 30 July 2014

Revised: 13 April 2015

Accepted: 13 April 2015

Published: 12 January 2016

Authors

Will Cavendish
McKinsey and Company 110 Charles Street Toronto ON, M5S 1K9 Canada

Volume 15, issue 6 (2015)