Quandle Coverings and Their Galois Correspondence

This article establishes the algebraic covering theory of quandles. For every connected quandle Q with base point q ∈ Q, we explicitly construct a universal covering p : (Q̃, q̃) → (Q,q). This in turn leads us to define the algebraic fundamental group π1(Q,q) := Aut(p) = {g ∈ Adj(Q)′ | qg = q}, where Adj(Q) is the adjoint group of Q. We then establish the Galois correspondence between connected coverings of (Q,q) and subgroups of π1(Q,q). 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 H1(Q) ∼= H1(Q) ∼= Z[π0(Q)], and we construct natural isomorphisms H2(Q)∼= π1(Q,q)ab and H(Q,A)∼= Ext(Q,A)∼= Hom(π1(Q,q),A), reminiscent of the classical Hurewicz isomorphisms in degree 1. This means that whenever π1(Q,q) is known, (co)homology calculations in degree 2 become very easy.