On homotopy groups of quandle spaces and the quandle homotopy invariant of links
type: Item
instance of: scholarly article
title: On homotopy groups of quandle spaces and the quandle homotopy invariant of links
zbMATH Open document ID: 1227.57020
author: Takefumi Nosaka
published in: Topology and its Applications
publication date: 2011-05-16T00:00:00Z
review text: Let \(L\) be an oriented link in the \(3\)-sphere \(S^3\), \(Q_L\) the link quandle of \(L\) which is the conjugacy class of \(\pi_1(S^3\setminus L)\) including the meridians of \(L\), and \(X\) a quandle. The author studies the quandle homotopy invariant and the quandle cocycle invariants related with \(X\)-colorings, \(\text{Hom}(Q_L, X)\), of \(L\). Following the definition of the rack space due to [\textit{R.~Fenn}, \textit{C.~Rourke} and \textit{B.~Sanderson}, Appl. Categ. Struct. 3, No. 4, 321--356 (1995; Zbl 0853.55021)], the author defines the quandle space \(BX\) of \(X\), the invariant \({\Xi}_X(L; C)\) of an \(X\)-coloring \(C\) valued in \(\pi_2(BX)\), and the quandle homotopy invariant \({\Xi}_X(L)\) as the formal sum of \(\{{\Xi}_X(L; C)\}\) running over all \(C\). When \(X\) is a finite quandle, \(BX\) is a path connected CW complex and \({\Xi}_X(L)\) is valued in \(\mathbb{Z}[\pi_2(BX)]\) (i.e.\ \(\text{Hom}(Q_L, X)\) is a finite set). By using universality of the quandle homotopy invariant, an alternative definition of the quandle cocycle invariant due to [\textit{J.~S.~Carter}, \textit{D.~Jelsovski}, \textit{S.~Kamada}, \textit{L.~Langford} and \textit{M.~Saito}, Trans. Am. Math. Soc. 355, No.~10, 3947--3989 (2003; Zbl 1028.57003)] is given. In Section 3, the structure of \(\pi_2(BX)\) is studied, and it is applied to obtain some properties of the quandle cocycle invariants. Suppose that \(X\) is finite. Then \(\pi_2(BX)\) is finitely generated (Theorem 3.1), and its rank and the order of its torsion part are estimated (Theorem 3.6). In particular, it is finite if and only if \(X\) is connected (i.e.\ For any \(x, y\in X\), there is an element \(z\in X\) such that \(x=y\ast z\)). In Section 4, \(\pi_2(BX)\) and its generator are determined concretely for \(X\) such as the dihedral quandle of odd prime order, the tetrahedral quandle, the conjugacy quandle of a conjugacy class of the \(4\)-th symmetric group, and the knot quandle. For some cases, the…
reviewed by: Teruhisa Kadokami
Mathematics Subject Classification ID: 57M27
zbMATH DE Number: 5888202
zbMATH Keywords: link
zbMATH Keywords: rack
zbMATH Keywords: quandle
zbMATH Keywords: rack space
zbMATH Keywords: homotopy group
zbMATH Keywords: Postnikov tower
zbMATH Keywords: 2-cocycle invariant
zbMATH Keywords: Hurewicz homomorphism
zbMATH Keywords: group homology
zbMATH Keywords: transfer
MaRDI profile type: Publication
full work available at URL: j.topol.2011.02.006
OpenAlex ID: W1991528311
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DOI: 10.1016/J.TOPOL.2011.02.006
description: scientific article; zbMATH DE number 5888202