Quandle coverings and their Galois correspondence

• Autorzy: Michael Eisermann
• Języki publikacji: EN

Abstrakty

EN

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 $π₁(Q,q): = Aut(p) = {g ∈ Adj(Q)' | q^{g} = q}$, 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.

Kategorie tematyczne

• 18G50: Nonabelian homological algebra
• 57M25: Knots and links in S 3
• 20L05: Groupoids (i.e. small categories in which all morphisms are isomorphisms)
• 18B40: Groupoids, semigroupoids, semigroups, groups (viewed as categories)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2014
• Tom: 225
• Numer: 1
• Strony: 103-167

Daty

wydano

2014

Twórcy

autor

Michael Eisermann

• Institut für Geometrie und Topologie, Universität Stuttgart, Germany

Identyfikatory

DOI

10.4064/fm225-1-7