#### Volume 14, issue 5 (2014)

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On third homologies of groups and of quandles via the Dijkgraaf–Witten invariant and Inoue–Kabaya map

### Takefumi Nosaka

Algebraic & Geometric Topology 14 (2014) 2655–2692
##### Abstract

We propose a simple method for producing quandle cocycles from group cocycles by a modification of the Inoue–Kabaya chain map. Further, we show that, with respect to “universal extension of quandles”, the chain map induces an isomorphism between third homologies (modulo some torsion). For example, all Mochizuki’s quandle $3$–cocycles are shown to be derived from group cocycles. As an application, we calculate some $ℤ$–equivariant parts of the Dijkgraaf–Witten invariants of some cyclic branched covering spaces, via some cocycle invariant of links.

##### Keywords
quandle, group homology, $3$–manifolds, link, branched covering, Massey product
##### Mathematical Subject Classification 2010
Primary: 20J06, 57M12
Secondary: 57M27, 57N65
##### Publication
Received: 9 February 2013
Revised: 8 October 2013
Accepted: 14 October 2013
Published: 5 November 2014
##### Authors
 Takefumi Nosaka Faculty of Mathematics Kyushu University 744, Motooka, Nishi-ku Fukuoka 819-0395 Japan