Quandle coverings and their Galois correspondence.

*(English)*Zbl 1301.57006The author develops an algebraic theory of quandle covering. A surjective homomorphism between quandles \(p: (\widetilde{Q},\widetilde{*}) \rightarrow (Q,*)\) is a covering if \(p(x)=p(y)\) implies their quandle operations \(\widetilde{*}\,x\) and \(\widetilde{*}\,y\) on \(\widetilde{Q}\) are the same. The author introduces a notion of the fundamental group of a quandle \(Q\) as the subgroup of its adjoint group, and establishes a Galois correspondence between the fundamental group of quandles and quandles coverings: For each quandle there is a natural equivalence between the category of subgroups of its fundamental group and the category of its quandle coverings. As an application, he gives an interpretation of the 2nd quandle (co)homology in terms of its fundamental group.

Despite its great similarity to covering theory of topological spaces, the author notices that there are various subtleties and differences, and the theory of quandle covering is not a simple translation of the usual covering theory of spaces. The paper contains a lot of stimulating and illuminating examples that nicely explain the author’s idea and quandle covering theory.

Despite its great similarity to covering theory of topological spaces, the author notices that there are various subtleties and differences, and the theory of quandle covering is not a simple translation of the usual covering theory of spaces. The paper contains a lot of stimulating and illuminating examples that nicely explain the author’s idea and quandle covering theory.

Reviewer: Tetsuya Ito (Japan)