SOME TOPOLOGICAL ASPECTS OF 4-FOLD SYMMETRIC QUANDLE INVARIANTS OF 3-MANIFOLDS

Abstract

The paper relates the 4-fold symmetric quandle homotopy (cocycle) invariants to 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 S3 constructed by the Mochizuki 3-cocycle is equivalent to the Dijkgraaf–Witten invariant with respect to ℤ/pℤ of the double covering of S3 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 [A. Inoue and Y. Kabaya, Quandle homology and complex volume, preprint(2010), arXiv:math/1012.2923].