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

Nosaka, Takefumi

doi:10.2140/agt.2014.14.2655

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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

DOI: 10.2140/agt.2014.14.2655

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

References

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

Volume 14, issue 5 (2014)