Quandle coverings and their Galois correspondence

Abstract

This article establishes the algebraic covering theory of quandles. For every connected quandle we explicitly construct a universal covering, which in turn leads us to define the algebraic fundamental group as the automorphism group of the universal covering. We then establish the Galois correspondence between connected coverings and subgroups of the fundamental group. Quandle coverings are thus formally analogous to coverings of topological spaces, and resemble Kervaire's algebraic covering theory of perfect groups. A detailed investigation also reveals some crucial differences, which we illustrate by numerous examples. As an application we obtain a simple formula for the second (co)homology group of a quandle Q. It has long been known that H_1(Q) = H^1(Q) = \Z[\pi_0(Q)], and we construct natural isomorphisms H_2(Q) = \pi_1(Q,q)_{ab} and H^2(Q,A) = Ext(Q,A) = Hom(\pi_1(Q,q),A), reminiscent of the classical Hurewicz isomorphisms in degree 1. This means that whenever the fundamental group is known, (co)homology calculations in degree 2 become very easy.