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) = [ (Q)], and we construct natural isomorphisms H_2(Q) = (Q,q)_ ab and H^2(Q,A) = Ext(Q,A) = Hom( (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.
@cite_18: Quandle cocycles are constructed from extensions of quandles. The theory is parallel to that of group cohomology and group extensions. An interpretation of quandle cocycle invariants as obstructions to extending knot colorings is given, and is extended to links component-wise.
@cite_7: 
@cite_0: Given a knot K in the 3-sphere, let QK be its fundamental quan- dle as introduced by D.Joyce. Its first homology group is easily seen to be H1(QK) = Z. We prove that H2(QK) = 0 if and only if K is trivial, and H2(QK) = Z whenever K is non-trivial. An analogous result holds for links, thus characterizing trivial components. More detailed information can be derived from the conjugation quandle: let Q K be the conjugacy class of a meridian in the knot group 1(S 3 rK). We show that H2(Q K) = Z p , where p is the number of prime summands in a connected sum decomposition of K.
@cite_16: Abstract A fundamental step in the classification of finite-dimensional complex pointed Hopf algebras is the determination of all finite-dimensional Nichols algebras of braided vector spaces arising from groups. The most important class of braided vector spaces arising from groups is the class of braided vector spaces ( C X,c q ) , where X is a rack and q is a 2-cocycle on X with values in C × . Racks and cohomology of racks appeared also in the work of topologists. This leads us to the study of the structure of racks, their cohomology groups and the corresponding Nichols algebras. We will show advances in these three directions. We classify simple racks in group-theoretical terms; we describe projections of racks in terms of general cocycles; we introduce a general cohomology theory of racks containing properly the existing ones. We introduce a “Fourier transform” on racks of certain type; finally, we compute some new examples of finite-dimensional Nichols algebras.
@cite_10: Geometric representations of cycles in quandle homology theory are given in terms of colored knot diagrams. Abstract knot diagrams are generalized to diagrams with exceptional points which, when colored, correspond to degenerate cycles. Bounding chains are realized, and used to obtain equivalence moves for homologous cycles. The methods are applied to prove that boundary homomorphisms in a homology exact sequence vanish.
@cite_20: Trunks are objects loosely analogous to categories. Like a category, a trunk has vertices and edges (analogous to objects and morphisms), but instead of composition (which can be regarded as given by preferred triangles of morphisms) it has preferred squares of edges. A trunk has a natural cubical nerve, analogous to the simplicial nerve of a category. The classifying space of the trunk is the realisation of this nerve. Trunks are important in the theory of racks [8]. A rackX gives rise to a trunkT (X) which has a single vertex and the setX as set of edges. Therack spaceBX ofX is the realisation of the nerveNT (X) ofT(X). The connection between the nerve of a trunk and the usual (cubical) nerve of a category determines in particular a natural mapBX ↦BAs(X) whereBAs(X) is the classifying space of the associated group ofX. There is an extension to give a classifying space for an augmented rack, which has a natural map to the loop space of the Brown-Higgins classifying space of the associated crossed module [8, Section 2] and [3].
As it could be expected, these notions are closely related to quandle extensions and cohomology, which have both been intensively studied in recent years. The subject of rack cohomology originated in the work of R. ,Fenn, C. ,Rourke, and B. ,Sanderson @cite_20 , who constructed a classifying topological space @math for every rack @math . The corresponding quandle (co)homology theory was taken up by J.S. ,Carter and his collaborators, in order to construct knot invariants (see for example @cite_10 @cite_18 @cite_7 ). Quandle coverings were introduced and applied to knot quandles in @cite_0 . They have also appeared in the context of non-abelian extensions, explored by N. ,Andruskiewitsch and M. ,Gra na @cite_16 , where a corresponding non-abelian cohomology theory was proposed. This generalized cohomology, in turn, has been taken up and applied to knot invariants in @cite_7 .