Algebraic & Geometric Topology

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

Takefumi Nosaka

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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 –equivariant parts of the Dijkgraaf–Witten invariants of some cyclic branched covering spaces, via some cocycle invariant of links.

Article information

Algebr. Geom. Topol., Volume 14, Number 5 (2014), 2655-2692.

Received: 9 February 2013
Revised: 8 October 2013
Accepted: 14 October 2013
First available in Project Euclid: 19 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20J06: Cohomology of groups 57M12: Special coverings, e.g. branched
Secondary: 57M27: Invariants of knots and 3-manifolds 57N65: Algebraic topology of manifolds

quandle group homology $3$–manifolds link branched covering Massey product


Nosaka, Takefumi. On third homologies of groups and of quandles via the Dijkgraaf–Witten invariant and Inoue–Kabaya map. Algebr. Geom. Topol. 14 (2014), no. 5, 2655--2692. doi:10.2140/agt.2014.14.2655.

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