  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 $\ell$ which does not divide the
type of $X$, the $\ell$-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.
