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

TL;DR

This paper introduces a method to derive quandle cocycles from group cocycles, establishing an isomorphism between their third homologies and applying this to compute parts of Dijkgraaf-Witten invariants for certain 3-manifolds.

Contribution

It presents a new approach to connect group and quandle homologies, extending the Inoue-Kabaya map, and demonstrates applications to topological invariants.

Findings

All Mochizuki's quandle 3-cocycles are derived from non-abelian group cocycles.

The chain map induces an isomorphism between third homologies of universal central extended quandles.

Computed $ ext{Z}$-equivariant parts of Dijkgraaf-Witten invariants for cyclic branched coverings.

Abstract

We propose a simple method to produce quandle cocycles from group cocycles, as a modification of Inoue-Kabaya chain map. We further show that, in respect to "universal central extended quandles", the chain map induces an isomorphism between their third homologies. For example, all Mochizuki's quandle 3-cocycles are shown to be derived from group cocycles of some non-abelian group. 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.