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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.

The paper relates the 4-fold symmetric quandle homotopy (cocycle) invariants with topological objects. We show that the 4-fold symmetric quandle homotopy invariants are at least as powerful as the Dijkgraaf-Witten invariants. As an application, for an odd prime p, we show that the quandle cocycle invariant of a link in S 3 using the Mochizuki 3-cocycle is equivalent to the Dijkgraaf-Witten invariant with respect to Z/pZ of the double covering of S 3 branched along the link. We also reconstruct the Chern-Simons invariant of closed 3-manifolds as a quandle cocycle invariant via the extended Bloch group, in analogy to [IK].

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In this paper we determine the integral quandle homology groups of Alexander quandles of prime order. As a special case, this settles the delayed F ibonacci conjecture by M. Niebrzydowski and J. H. Przytycki in [7]. Moreover, we determine the cohomology group of the Alexander quandle and obtain relatively simple presentations of all higher degree cocycles which generate the cohomology group. Furthermore, we prove that the integral quandle homology of a finite connected Alexander quandle is annihilated by the order of the quandle.