Simply connected latin quandles

TL;DR

This paper introduces a combinatorial method to study constant cohomology of connected quandles and proves that certain classes of these quandles are simply connected, meaning they have no nontrivial coverings.

Contribution

It develops a combinatorial approach to constant cohomology and proves that affine cyclic and finite doubly transitive quandles are simply connected.

Findings

Connected affine quandles over cyclic groups are simply connected.

Finite doubly transitive quandles (except order 4) are simply connected.

Provides a new proof extending Graña's result for prime-sized quandles.

Abstract

A (left) quandle is connected if its left multiplication group acts transitively. In 2014, Eisermann introduced the concept of quandle coverings, corresponding to so-called constant quandle cocycles that form a subset of quandle cocycles. A connected quandle is said to be \emph{simply connected} if it has no nontrivial coverings, or, equivalently, if its second constant cohomology groups are trivial. In this paper we develop a combinatorial approach to constant cohomology. Upon applying our theory, we prove that connected quandles that are affine over cyclic groups are simply connected (extending a result of Gra\~{n}a for quandles of prime size) and that finite doubly transitive quandles of order different from $4$ are simply connected.