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

Abstract

On third homologies of groups and of quandles via the Dijkgraaf-Witten invariant and Inoue-Kabaya map TAKEFUMI NOSAKA We propose a simple method for producing quandle cocycles from group cocycles by a modification of the Inoue-Kabaya chain map. Further, we show that, with respect to "universal extension of quandles", the chain map induces an isomorphism between third homologies (modulo some torsion). For example, all Mochizuki's quandle 3-cocycles are shown to be derived from group cocycles. As an application, we calculate some Z-equivariant parts of the Dijkgraaf-Witten invariants of some cyclic branched covering spaces, via some cocycle invariant of links.

On the fundamental 3-classes of knot group representations

“…Next, regarding the middle term in the zigzag sequence (5), this section reviews the quandle homology [CJKLS] and Inoue-Kabaya chain map [IK]. As seen in [CKS,IK,No3,No1], quandle theory is useful for reducing some 3-dimensional discussions to diagrammatic objects.…”

“…Hence, the map ϕ 3 with n = 3 induces a homomorphism (ϕ 3 ) * : H Q 3 (X) −→ H ∆ 3 (X; Z). We refer the reader to several studies on the chain map; see [IK,Kab,No1,No2,No3].…”

“…First, we will observe some triple Massey products. This example is essentially due to [No1,§4.2]. Regard the finite field Fq as the abelian group (Z/p) m , where q = p m and p = 2.…”

“…The cohomology H 3 (G; Fq ) is complicated. In fact, the cohomology has 3-cocycles θ Γ , which are derived from triple Massey product (see [No1,Proposition 4.8]). However, the author [No1,Lemma 4.7] showed that the pullback ϕ * 3 θ Γ : X 3 → Fq is formulated as…”

“…Let us roughly explain our approach to the theorem. As seen in [CKS,IK,No3,No1], quandle theory [Joy] and homology [CKS] have advantages to some diagrammatic computation in knot theory. Thus, inspired by the works [IK, AS], we will construct a bridge between the quandle and group homologies using chain maps, in order to reduce the 3-class [M, ∂M] to a quandle 3-class.…”