Homotopical interpretation of link invariants from finite quandles

Abstract

This paper demonstrates a topological meaning of quandle cocycle invariants of links with respect to finite connected quandles X, from a perspective of homotopy theory: Specifically, for any prime ℓ which does not divide the type of X, the ℓ-torsion of this invariants is equal to a sum of the colouring polynomial and a Z-equivariant part of the Dijkgraaf-Witten invariant of a cyclic branched covering space. Moreover, our homotopical approach involves application of computing some third homology groups and second homotopy groups of the classifying spaces of quandles, from results of group cohomology.

On third homologies of groups and of quandles via the Dijkgraaf–Witten invariant and Inoue–Kabaya map

“…Here I .L/ is the quandle cocycle invariant of links L [4] (see Remark 2.16 for some quandles satisfying the assumption on p Y ). While the equivalence of the two invariants was implied in the previous paper [22] by abstract nonsense and the proofs of (2) and (3) are based on some results in [22], the point of our results is that the cocycle is directly obtained from the chain map ẑ 3 .…”

“…It can easily be seen that if X is of type t X and of finite order, then so is z X . Furthermore, the quandle z X is connected [22,Lemma 6.8].…”

“…where OEM is the fundamental class in H 3 .M I Z/. Inspired by this analogue, for many quandles X , Nosaka [22] gave essentially topological meanings of the cocycle invariants, using the Dijkgraaf-Witten invariant and the homotopy group 2 .BX /.…”

“…For example, the second quandle homology has been extensively studied by Eisermann [7] on the basis of the first group homologies. In addition, Nosaka [22] roughly computed some third quandle homologies from the group homologies of 1 .BX / with some ambiguity. Furthermore, for any quandle X , Inoue and Kabaya [11] constructed a chain map ' IK from the quandle complex to a certain complex.…”

“…Compared with [22], we emphasize that our theorem serves to compute some parts of the Dijkgraaf-Witten invariants DW Z Ä . y C t L / via the right invariant I .L/.…”