Quandle coverings and their Galois correspondence
Volume 225 / 2014
Abstract
This article establishes the algebraic covering theory of quandles. For every connected quandle $Q$ with base point $q \in Q$, we explicitly construct a universal covering $p \colon (\tilde{Q},\tilde{q}) \to (Q,q)$. This in turn leads us to define the algebraic fundamental group $\pi_1(Q,q) := \mathop{\rm Aut}(p) = \{ g \in \mathop{\rm Adj}(Q)' \mid q^g = q \}$, where $\mathop{\rm Adj}(Q)$ is the adjoint group of $Q$. We then establish the Galois correspondence between connected coverings of $(Q,q)$ and subgroups of $\pi_1(Q,q)$. 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 $H_1(Q) \cong H^1(Q) \cong \mathbb{Z}[\pi_0(Q)]$, and we construct natural isomorphisms $H_2(Q) \cong \pi_1(Q,q)_\mathrm{ab}$ and $H^2(Q,A) \cong \mathop{\rm Ext}(Q,A) \cong \mathop{\rm Hom}(\pi_1(Q,q),A)$, reminiscent of the classical Hurewicz isomorphisms in degree $1$. This means that whenever $\pi_1(Q,q)$ is known, (co)homology calculations in degree $2$ become very easy.
