Quandle coverings and their Galois correspondence

Michael Eisermann. "Quandle coverings and their Galois correspondence." Fundamenta Mathematicae 225.0 (2014): 103-167. <http://eudml.org/doc/286640>.

@article{MichaelEisermann2014,
abstract = {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.},
author = {Michael Eisermann},
journal = {Fundamenta Mathematicae},
keywords = {quandle; covering; fundamental group; quandle cohomology},
language = {eng},
number = {0},
pages = {103-167},
title = {Quandle coverings and their Galois correspondence},
url = {http://eudml.org/doc/286640},
volume = {225},
year = {2014},
}

TY - JOUR
AU - Michael Eisermann
TI - Quandle coverings and their Galois correspondence
JO - Fundamenta Mathematicae
PY - 2014
VL - 225
IS - 0
SP - 103
EP - 167
AB - 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.
LA - eng
KW - quandle; covering; fundamental group; quandle cohomology
UR - http://eudml.org/doc/286640
ER -