Quandle coverings and their Galois correspondence

Michael Eisermann

Fundamenta Mathematicae, Tome 227 (2014), p. 103-167 / Harvested from The Polish Digital Mathematics Library

Résumé

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,q̃̃) → (Q,q). This in turn leads us to define the algebraic fundamental group $π₁(Q,q): = Aut(p) = {g ∈ Adj(Q)^{\prime } | q^{g} = q}$, 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 H²(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.

Détail

Publié le : 2014-01-01

Zbl 1301.57006

EUDML-ID : urn:eudml:doc:286640