Foundations of topological racks and quandles

Abstract

We give a foundational account on topological racks and quandles. Specifically, we define the notions of ideals, kernels, units, and inner automorphism group in the context of topological racks. Further, we investigate topological rack modules and principal rack bundles. Central extensions of topological racks are then introduced providing a first step towards a general continuous cohomology theory for topological racks and quandles.

Continuous cohomology of topological quandles

“…For more informations about quandles, discrete and topological, the interested reader can consult [6,10,11,14,15,20].…”

“…We define the concept of extension of a quandle, in the topological context, following [3,5], see also [10]. Assume we are given a quandle X and an Alexander quandle (A, T ).…”

“…Let X be a topological quandle. Recall [1,10] that an X-module is a triple (A, η, τ ) where A is a topological abelian group, η is a family of continuous group automorphisms η x,y : A → A and τ is a family of continuous group morphisms τ x,y : A → A such that (1) η x * y,z η x,y = η x * z,y * z η x,z , (2) η x * y,z τ x,y = τ x * z,y * z η y,z , (3) τ x * y,z = η x * z,y * z τ x,z + τ x * z,y * z τ y,z , and (4) τ x,x + η x,x = id A . We consider the abelian groups Γ n (X, A), δ i 0 and δ i 1 as defined above in Section 3.…”

“…Since then quandles have been investigated for constructing knot and link invariants (e.g. [3,6,10]). Quandles have been also studied from different point of views such as the study of the set-theoretic Yang-Baxter equation [4] and pointed Hopf algebras [1].…”

“…Quandles have been also studied from different point of views such as the study of the set-theoretic Yang-Baxter equation [4] and pointed Hopf algebras [1]. Topological quandles were introduced in [20] and investigated further in [7,8,10]. Smooth quandles and their cohomology have been introduced by Nosaka [18].…”