An explicit description of the second cohomology group of a quandle

TL;DR

This paper provides an explicit description of the second cohomology group of finite indecomposable quandles using the inflation-restriction sequence and results on rack cohomology, with multiple applications.

Contribution

It introduces a new explicit method for computing 2-cocycles of finite indecomposable quandles, advancing the understanding of quandle cohomology.

Findings

Explicit description of 2-cocycles for finite indecomposable quandles

Application of inflation-restriction sequence in quandle cohomology

New insights into the structure of rack cohomology

Abstract

We use the inflation-restriction sequence and a result of Etingof and Gra\~na on the rack cohomology to give a explicit description of 2-cocycles of finite indecomposable quandles with values in an abelian group. Several applications are given.