Algebraic & Geometric Topology

On finite derived quotients of $3$–manifold groups

Will Cavendish

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This paper studies the set of finite groups appearing as π1(M)π1(M)(n), where M is a closed, orientable 3–manifold and π1(M)(n) denotes the nth term of the derived series of π1(M). Our main result is that if M is a closed, orientable 3–manifold, n 2, and Gπ1(M)π1(M)(n) is finite, then the cup-product pairing H2(G) H2(G) H4(G) has cyclic image C, and the pairing H2(G) H2(G)C is isomorphic to the linking pairing H1(M) Tors H1(M) Tors .

Article information

Algebr. Geom. Topol., Volume 15, Number 6 (2015), 3355-3369.

Received: 30 July 2014
Revised: 13 April 2015
Accepted: 13 April 2015
First available in Project Euclid: 16 November 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M10: Covering spaces
Secondary: 57M60: Group actions in low dimensions

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


Cavendish, Will. On finite derived quotients of $3$–manifold groups. Algebr. Geom. Topol. 15 (2015), no. 6, 3355--3369. doi:10.2140/agt.2015.15.3355.

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