Quandle cohomology, extensions and automorphisms

TL;DR

This paper explores the deep connections between quandle cohomology, extensions, and automorphisms, establishing exact sequences and functorial properties that enhance understanding of quandle structures and their applications.

Contribution

It introduces new exact sequences relating quandle cohomology and automorphisms, and demonstrates how group extensions translate into quandle extensions, advancing the theoretical framework.

Findings

Derived a four-term exact sequence involving quandle cocycles and automorphisms.

Established a non-abelian counterpart with dynamical cohomology classes.

Constructed homomorphisms linking group cohomology to quandle cohomology.

Abstract

The paper establishes new relationship between cohomology, extensions and automorphisms of quandles. We derive a four term exact sequence relating quandle 1-cocycles, second quandle cohomology and certain group of automorphisms of an abelian extension of quandles. A non-abelian counterpart of this sequence involving dynamical cohomology classes is also established, and some applications to lifting of quandle automorphisms are given. Viewing the construction of the conjugation, the core and the generalised Alexander quandle of a group as an adjoint functor of some appropriate functor from the category of quandles to the category of groups, we prove that these functors map extensions of groups to extensions of quandles. Finally, we construct some natural group homomorphisms from the second cohomology of a group to the second cohomology of its core and conjugation quandles.