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

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 $\mathbb{Z}$–equivariant parts of the Dijkgraaf–Witten invariants of some cyclic branched covering spaces, via some cocycle invariant of links.