On third homologies of groups and of quandles via the Dijkgraaf-Witten invariant and Inoue-Kabaya map
aa r X i v : . [ m a t h . G T ] O c t On third homologies of groups and of quandlesvia the Dijkgraaf-Witten invariant and Inoue-Kabaya map
Takefumi Nosaka
Abstract
We propose a simple method to produce quandle cocycles from group cocycles, as a modification ofInoue-Kabaya chain map. We further show that, in respect to “universal central extended quandles”,the chain map induces an isomorphism between their third homologies. For example, all Mochizuki’squandle 3-cocycles are shown to be derived from group cocycles of some non-abelian group. As anapplication, we calculate some Z -equivariant parts of the Dijkgraaf-Witten invariants of some cyclicbranched covering spaces, via some cocycle invariant of links. Keywords quandle, group homology, 3-manifolds, link, branched covering, Massey product
A quandle, X , is a set with a binary operation whose definition was partially motivated fromknot theory. Fenn-Rourke-Sanderson [FRS1, FRS2] defined a space BX called rack space,in analogy to the classifying spaces of groups. Furthermore, Carter et. al [CJKLS, CKS]introduced quandle cohomologies H ∗ Q ( X ; A ) with local coefficients, by slightly modifying thecohomology of BX ; they further defined combinatorially a state-sum invariant I ψ ( L ) of links L constructed from a cocycle ψ ∈ H ∗ Q ( X ; A ). The construction can be seen as an analogueof the Dijkgraaf-Witten invariant [DW] of closed oriented 3-manifolds M constructed froma finite group G and a 3-cocycle κ ∈ H ( G ; A ): To be specific, the invariant is defined asthe formal sum of pairings expressed byDW κ ( M ) := X f ∈ Hom gr ( π ( M ) ,G ) h f ∗ ( κ ) , [ M ] i ∈ Z [ A ] , (1)where [ M ] is the fundamental class in H ( M ; Z ). Inspired by this analogue, for manyquandles X , the author [No3] gave essentially topological meanings of the cocycle invariantswith using the Dijkgraaf-Witten invariant and the homotopy group π ( BX ).We mainly focus on a relation between quandle homology and group one. There areseveral such studies. For example, the second quandle homology is well-studied by Eiser-mann [Eis] from first group homologies. In addition, the author [No3] roughly computedsome third quandle homologies from the group homologies of π ( BX ) with some ambigu-ity. Furthermore, for any quandle X , Inoue-Kabaya [IK] constructed a chain map ϕ IK fromthe quandle complex to a certain complex. Although the latter complex seems far fromsomething familiar, Kabaya [Kab] modified the ϕ IK mapping to a group homology under acertain strong condition of X . Furthermore, for certain special quandles, the author [No2]proposed a method to construct quandle cocycles from invariant theory via the chain map. his paper demonstrates a relation between third homologies of groups and those ofquandles via the Inoue-Kabaya map, with respect to a broad subclass of quandles. Herea quandle in the subclass is defined as a group G with an operation g ⊳ h := ρ ( gh − ) h for g, h ∈ G , where ρ : G → G is a fixed group isomorphism (Definition 2.1). Denotesuch a quandle by X = ( G, ρ ). Furthermore, denote by H gr n ( G ; Z ) Z a quotient of thegroup homology of G subject to the action by the ρ , called the Z -coinvariant. In § n , which induces a homomorphism(Φ n ) ∗ : H Qn ( X ; Z ) −→ H gr n ( G ; Z ) Z . Furthermore, we lift this map Φ n to being a chain map ϕ n from C Qn ( X ; Z ) to the usual grouphomology H gr n ( G ; Z ); see Proposition 2.6. As a corollary, if found a presentation of a group n -cocycle κ of G , we easily obtain that of the induced quandle n -cocycle ϕ ∗ n ( κ ). Hence, thisapproach is expected to be useful of computing the quandle cocycle invariant constructedfrom such a quandle ( G, ρ ).This paper moreover investigates properties of the chain map Φ n above. To begin with,we focus on a class of universal quandle coverings e Y , introduced by Eisermann [Eis]. Roughlyspeaking about this e Y , given a “connected” quandle Y of finite order, we can set the quandle e Y and an epimorphism p Y : e Y → Y (as a quandle covering); further the quandle e Y is of theform (Ker( ǫ Y ) , ρ ) for some group Ker( ǫ Y ); see Example 2.3 for details. We then show thatthe associated chain map e Φ induces an isomorphism( e Φ ) ∗ : H Q ( e Y ) ∼ = H gr3 (Ker( ǫ Y )) Z up to t Y -torsion , (2)where t Y ∈ N is the minimal satisfying ρ t Y = id (Theorem 2.11).Needless to say, the Φ is not always isomorphic for such quandles ( G, ρ ); However, bythe help of the universality of coverings, in some cases we can analyse the map Φ as follows.Next, we will demonstrate Mochizuki quandle 3-cocycles [Moc], which are most knownquandle 3-cocycles so far. Consider quandles Y of the forms ( F q , × ω ) with ω ∈ F q , calledAlexander quandle usually, where we regard the finite field F q as an additive group and thesymbol × ω is a ω -multiple of F q . He found all 3-cocycles of Y by solving a certain differentialequations over F q and his statement was a little complicated (see § ∗ n (see (20) and Lemma 4.6). Moreover, we show that the third quandle cohomology H Q ( Y ; F q ) is isomorphic to a sum of some group homologies via the maps Φ , Φ and e Φ (see Theorem 2.14 in details). In conclusion, all the Mochizuki 3-cocycles stems from somegroup 3-cocycles via the three maps.Furthermore, we propose a relation to a partial sum of some Dijkgraaf-Witten invariantsof b C tL , where b C tL denotes the t -fold cyclic covering space of S branched over a link L . See(9) for the detailed definition of the partial sum, and denote it by DW Z κ ( b C tL ) ∈ Z [ A ]. Tobe specific, we show (Theorem 2.16) that if the induced map p ∗ Y : H Q ( Y ; A ) → H Q ( e Y ; A )is surjective, and if Y is connected and of finite order, then any group 3-cocycle κ of the bove group Ker( ǫ Y ) admits some quandle 3-cocycle ψ of Y for which the equalityDW Z κ ( b C tL ) = I ψ ( L ) ∈ Z [ A ]holds. Here the right side I ψ ( L ) is the quandle cocycle invariant of links L [CKS] (seeRemark 2.17 for some quandles satisfying the assumption on p ∗ Y ). While the equivalence ofthe two invariants was implied in the previous paper [No3] by abstract nonsense, the pointis that the cocycle ψ is definitely obtained from the chain map e Φ .We here emphasize that our theorem serves as computing some parts of the Dijkgraaf-Witten invariants DW Z κ ( b C tL ) via the right invariant I ψ ( L ). A known standard way to com-pute the invariant is to find a fundamental class from a triangulation of M (see [DW, Wa]).However, presentations of group 3-cocycles are intricate in general. So, most known compu-tations of the Dijkgraaf-Witten invariants are those with respect to abelian groups. However,in computing them via the right invariant I ψ ( L ), we use no triangulation of M and manyquandle 3-cocycles are simpler than group ones (in our experience).In fact, in §
5, we succeed in some computations of the formal sums DW Z κ ( b C tL ) by usingthe Mochizuki 3-cocycles, which are derived from triple Massey products of a meta-abeliangroup G X (see Proposition 4.7). For example, we will calculate the cocycle invariants ofthe torus knots T ( m, n ) (see Theorem 5.1); hence we obtain the partial sum DW Z κ of theBrieskorn manifold Σ( m, n, t ), which is the covering space branched over the knot T ( m, n ).Furthermore, as a special case ω = −
1, we compute the cocycle invariant of some knots K ,and, hence, obtain some values DW Z κ ( b C tK ) for the double covering spaces branched along K (see Table 1 in § §
2, we introduce a lift of Inoue-Kabaya chain mapand state theorems. In §
3, we prove Theorems 2.11 and 2.16. In §
4, we show that Mochizuki3-cocycles are derived from some group 3-cocycles. In §
5, we calculate some partial sum ofthe Dijkgraaf-Witten invariants.
Notation and convention
A symbol F q is a finite field of characteristic p >
0. Denote H gr n ( G ) the group homology of a group G with trivial integral coefficients. Furthermore weassume that manifolds are smooth, connected, oriented. In § § We start by recalling basic concepts about quandles. A quandle , X , is a set with a binaryoperation ( x, y ) → x ⊳ y such that, for any x, y, z ∈ X, x ⊳ x = x , ( x ⊳ y ) ⊳ z = ( x ⊳ z ) ⊳ ( y ⊳ z )and there exists uniquely w ∈ X such that w ⊳ y = x . A quandle X is said to be of type t X , if t X > N number satisfying a = ( · · · ( a ⊳ b ) · · · ) ⊳ b [ N -times on the ight with b ] for any a, b ∈ X . Furthermore, the associated group of X , As( X ), is defined tobe the group presented byAs( X ) := h e x ( x ∈ X ) | e − x ⊳ y e − y e x e y ( x, y ∈ X ) i . The group As( X ) acts on X by the formula x · e y := x ⊳ y for x, y ∈ X . If the action istransitive, X is said to be connected . Furthermore, take a homomorphism ǫ X : As( X ) → Z sending e x to 1; so we have a group extension0 −→ Ker( ǫ X ) ι −→ As( X ) ǫ X −→ Z −→ . (3)Next, we introduce a subclass of quandles which we mainly use in this paper. Definition 2.1 ([Joy, § . Fix a group G and a group isomorphism ρ : G → G . Equip X = G with a quandle operation by setting g ⊳ h := ρ ( gh − ) h. (4)Note that the quandle ( G, ρ ) is of type t X , if and only if the t X -th power of ρ is the identity,i.e., ρ t X = id G .Although this class of such quandles ( G, ρ ) is a subclass of quandles, it includes interestingquandles as follows:
Example 2.2 (Alexander quandle) . Let X = G be an abelian group. Denoting ρ by T instead, we can regard X as a Z [ T ± ]-module. Then the quandle operation is rewritten in x ⊳ y := T x + (1 − T ) y, called Alexander quandle . Given a finite field F q and ω ∈ F ∗ q with ω = 1, the quandle oftype X = F q [ T ] / ( T − ω ) is called Alexander quandle on F q with ω .The type t X of X equals the minimal n satisfying T n = 1 in X . We easily check that X is connected if and only if (1 − T ) is invertible. Example 2.3 (Universal quandle covering) . Given a connected quandle X , consider thekernel G = Ker( ǫ X ) in (3). Fix a ∈ X . Using a group homomorphism ρ a : Ker( ǫ X ) → Ker( ǫ X ) define by ρ a ( g ) = e − a ge a , we have a quandle e X = (Ker( ǫ X ) , ρ a ), called extendedquandle of X . We easily see the independence of the choice of a ∈ X up to quandleisomorphisms.Considering the restriction of the action X x As( X ) to Ker( ǫ X ), a map p X : e X → X sending g to a · g is known to be a quandle homomorphism (see [Joy, Theorem 4.1]), andcalled (universal quandle) covering [Eis]. It can easily be seen that if X is of type t X andof finite order, so is e X . Furthermore, the e X is shown to be connected [No3, Lemma 6.8]),Finally, we briefly review the quandle complexes introduced by [CJKLS]. Let X be aquandle. Let us construct a complex by putting the free Z -module C Rn ( X ) spanned by( x , . . . , x n ) ∈ X n and letting its boundary ∂ Rn ( x , . . . , x n ) ∈ C Rn − ( X ) be X ≤ i ≤ n ( − i (cid:0) ( x , . . . , x i − , x i +1 , . . . , x n ) − ( x ⊳ x i , . . . , x i − ⊳ x i , x i +1 , . . . , x n ) (cid:1) . he composite ∂ Rn − ◦ ∂ Rn is zero. The pair ( C R ∗ ( X ) , ∂ R ∗ ) is called rack complex . Let C Dn ( X )be a submodule of C Rn ( X ) generated by n -tuples ( x , . . . , x n ) with x i = x i +1 for some i ∈ { , . . . , n − } if n ≥
2; otherwise, let C D ( X ) = 0. Since ∂ Rn ( C Dn ( X )) ⊂ C Dn − ( X ), wecan define a complex (cid:0) C Q ∗ ( X ) , ∂ ∗ (cid:1) by the quotient C Rn ( X ) /C Dn ( X ). The homology H Qn ( X )is called quandle homology of X . Dually, we can define the cohomologies H nR ( X ; A ) and H nQ ( X ; A ) with a commutative ring A .However, the second term of the differential ∂ Rn seems incomprehensible from the defini-tion. In the next subsection, for a quandle of the form ( G, ρ ), we give a simple formula ofthe ∂ Rn . We now construct a chain map (6) with respect to a class of quandles in Definition 2.1. Ourconstruction is a modification of Inoue-Kabaya map [IK, §
3] (see Remark 2.8).In this subsection, we often denote ρ ( x ) by x ρ and ρ n ( x ) by x nρ for short, respectively.For quandles X of the forms ( G, ρ ) in Definition 2.1, we will reformulate the rackcomplex C Rn ( X )( ∼ = Z h G n i ) in non-homogeneous coordinates. Define another differential ∂ R G n : C Rn ( X ) → C Rn ( X ) by setting ∂ R G n ( g , . . . , g n ) := X ≤ i ≤ n − ( − i (cid:0) ( g , . . . , g i − , g i g i +1 , g i +2 , . . . , g n ) − ( g ρ , . . . , g ρi − , g ρi g i +1 , g i +2 , . . . , g n ) (cid:1) . We easily check ∂ R G n − ◦ ∂ R G n = 0, and can further see Lemma 2.4.
Take a bijection G n → G n defined by setting ( x , . . . , x n ) ( x x − , x x − , . . . , x n − x − n , x n ) . (5) This map yields a chain isomorphism
Υ : ( C Rn ( X ) , ∂ Rn ) ∼ = ( C Rn ( X ) , ∂ R G n ) .Proof. By direct calculation.Furthermore, we define a subcomplex D n ( G ) generated by n -tuples ( g , . . . , g n ) such that g i = 1 for some i ≤ n −
1. We denote the quotient complex by C Q G n ( X ). By Lemma 2.4this homology, H Q G n ( X ), is isomorphic to the quandle homology H Qn ( X ) in § normalized chain complexes of groups, C gr n ( G ), in non- homogeneousterms (see, e.g. [Bro]) as follows: Let C gr n ( G ) denote the free Z -module generated by G n ,and let its boundary map ∂ gr n ( g , . . . , g n ) ∈ C gr n − ( G ) be( g , . . . , g n ) + X ≤ i ≤ n − ( − i ( g , . . . , g i − , g i g i +1 , g i +2 , . . . , g n ) + ( − n ( g , . . . , g n − ) . Furthermore, concerning the submodule D n ( G ) mentioned above, we easily check ∂ gr n (cid:0) D n ( G ) (cid:1) ⊂ D n − ( G ). We denote by C gr n ( G ) the quotient complex of C gr n ( G ) modulo D n ( G ). As is well-known, this homology coincides with the usual group homology of G (see [Bro, § I.5]).We next construct a chain map ϕ n from the complex C R G n ( X ) to another C gr n ( G ). efinition 2.5. Assume that a quandle X of the form ( G, ρ ) is of type t X . Take a set K n := { ( k , . . . , k n ) ∈ Z n | ≤ k i − k i − ≤ , ≤ k n ≤ t X − } . of order t X n − . We define a homomorphism ϕ n : C R G n ( X ) → C gr n ( G ) by setting ϕ n ( g , g , . . . , g n ) = X ( k ,...,k n ) ∈K n ( − k ( g k ρ , g k ρ , . . . , g k n ρn ) ∈ C gr n ( G ) . For example, when n = 3, the ϕ ( x, y, z ) is written in X ≤ i ≤ t X − ( x iρ , y iρ , z iρ ) − ( x ( i +1) ρ , y iρ , z iρ ) − ( x ( i +1) ρ , y ( i +1) ρ , z iρ ) + ( x ( i +2) ρ , y ( i +1) ρ , z iρ ) . (6) Proposition 2.6.
Let X be a quandle of the form ( G, ρ ) . If X is of type t X < ∞ , then thehomomorphism ϕ n : C Rn ( X ) → C gr n ( G ) is a chain map. Namely, ∂ gr n ◦ ϕ n = ϕ n − ◦ ∂ R G n .Furthermore the image of D n ( G ) is zero. In particular, the ϕ n induces a chain map fromthe quotient C Q G n ( X ) to C gr n ( G ) , and a homomorphism H Q G n ( X ) → H gr n ( G ) .Proof. The identity ∂ gr n ◦ ϕ n = ϕ n − ◦ ∂ R G n can be proven by direct calculation similar to [IK,Lemma 3.1] or [No2, Appendix], so we omit the details. It is not hard to check the latterpart directly.Accordingly, we obtain an easy method to quandle cocycles from group cocycles: Corollary 2.7.
Let a quandle X = ( G, ρ ) be of type t X . For a normalized group n -cocycle κ of G , then the pullback ϕ ∗ n ( κ ) is a quandle n -cocycle. Remark 2.8.
We now roughly compare our map ϕ n with a chain map ϕ IK introduced byInoue and Kabaya [IK]. For any quandle Q , they constructed a complex “ C ∆ n ( Q )” froma simplicial object, and formulated the map ϕ IK : C Rn ( Q ) → C ∆ n ( Q ) in its homogeneous coordinate system (see [IK, §
3] for details).To see this in some detail, we define a module, C gr n ( G ) Z , to be the quotient of C gr n ( G )modulo the relation ( g , . . . , g n ) = ( ρ ( g ) , . . . , ρ ( g n )), called Z -coinvariant of C gr n ( G ). Wedenote by π ρ the projection C gr n ( G ) → C gr n ( G ) Z . We can see that, if Q is a quandle of theform ( G, ρ ) and connected, then the above complex C ∆ n ( Q ) is isomorphic to the coinvariant C gr n ( G ) Z ; further, we can check the equality t X · ϕ IK = π ρ ◦ ϕ n . In summary, our map ϕ n isof a lift of the Inoue-Kabaya map ϕ IK in connected cases, and is relatively simple. So wefix a notation: Definition 2.9.
Let Φ n denote the composite chain map π ρ ◦ ϕ n : C Q G n ( X ) → C gr n ( G ) Z , i.e.,Φ n : C Q G n ( X ) ϕ n −→ C gr n ( G ) proj −−−→ C gr n ( G ) Z . Incidentally, we prepare a ‘reduced map’ of the Φ n , which is used temporarily in Theorem2.16. Consider a homomorphism P : C Rn ( X ) → C R G n − ( X ) derived from a map X n → X n − sending ( x , . . . , x n ) to ( x , . . . , x n − ). We discuss the composite Φ n − ◦ P as follows: roposition 2.10. Let X be a quandle ( G, ρ ) of type t X . The composite Φ n − ◦ P : C R G n ( X ) → C gr n − ( G ) Z is a chain map. Furthermore it induces a chain map from the quotient C Q G n ( X ) to C gr n − ( G ) Z .Proof. By direct calculation (cf. Proposition 2.6 and [IK, Lemma 3.1]). Φ In this paper, we study the chain map Φ n with n = 3 (Theorems 2.11, 2.14).We first study the maps Φ n with respect to extended quandles in Example 2.3. Theorem 2.11.
Let X be a connected quandle of type t X , and e X = (Ker( ǫ X ) , ρ a ) be theextended quandle in Example 2.3. Let e Φ n denote the chain map in Definition 2.9. Assumethat the H gr3 (As( X )) is finitely generated, e.g., X is of finite order. Then the induced map ( e Φ ) ∗ : H Q ( e X ) −→ H gr3 (Ker( ǫ X )) Z is an isomorphism modulo t X -torsion. Remark 2.12.
We here compare this theorem with the result [No3, Theorem 3.18] whichstated an existence of an isomorphism H Q ( e X ) ∼ = H gr3 (As( X )) modulo t X . So this theoremsays that the chain map ( e Φ ) ∗ gives an explicit presentation of this isomorphism. Indeed welate get a canonical isomorphism H gr3 (As( X )) ∼ = H gr3 (Ker( ǫ X )) Z modulo t X ; see Lemma 3.4.Next, as a special case, we focus on the Alexander quandles on F q in Example 2.2. Usingthe maps Φ n , we will characterize the third quandle cohomology from group homologies.Identifying X = F q with ( Z p ) h as an additive group, let ρ : F q → F q be the ω -multiple. Wethen have a chain map Φ ∗ n : C n gr (( Z p ) h ) Z → C nQ ( X ), and later show the following: Proposition 2.13.
Let X be an Alexander quandle on F q with ω in Example 2.2. Then theinduced map Φ ∗ : H (( Z p ) h ; F q ) Z → H Q ( X ; F q ) is injective.Furthermore, if H Q ( X ) vanishes, then this Φ ∗ is an isomorphism. In general, this Φ ∗ is not surjective. To solve the obstruction H Q ( X ), we consider thechain map e Φ n : C Qn ( e X ) → C gr n (Ker( ǫ X )) Z with respect to the extended quandle (Example2.3). We then obtain a commutative diagram H n gr (( Z p ) h ; F q ) Z Φ ∗ n / / Proj ∗ (cid:15) (cid:15) H nQ ( X ; F q ) p ∗ X (cid:15) (cid:15) H n gr (Ker( ǫ X ); F q ) Z e Φ ∗ n / / H nQ ( e X ; F q ) . Remark that, when n = 3, the bottom map e Φ ∗ is an isomorphism by Theorem 2.11. Denoteres( e Φ ∗ ) the isomorphism restricted on the cokernel Coker(Proj ∗ ). In addition, we take thechain map Φ n − ◦ P : C Q G n ( X ) → C gr n − ( G ) Z in Proposition 2.10,To summarize these homomorphisms, we characterize the third quandle cohomology of X : heorem 2.14. Let X be an Alexander quandle on F q . Let q be odd. Then there is asection s : H Q ( e X ; F q ) → H Q ( X ; F q ) of p ∗ X such that the following homomorphism is anisomorphism: (Φ ◦ P ) ∗ ⊕ Φ ∗ ⊕ (cid:0) s ◦ res( e Φ ∗ ) (cid:1) : H (( Z p ) h ; F q ) Z ⊕ H (( Z p ) h ; F q ) Z ⊕ Coker(Proj ∗ ) −→ H Q ( X ; F q ) . (7)The proof will appear in §
4. In conclusion, all the Mochizuki 3-cocycles are derived fromgroup 3-cocycles of ( Z p ) h and of Ker( ǫ X ) via the chain map Φ n .Incidentally, in higher degree, we now observe that the induced map ( ϕ n ) ∗ : H Qn ( X ) → H gr n ( G ) is far from injective and surjective. Example 2.15.
To see this, letting q = p , we observe the chain map ϕ n with respect toan Alexander quandle X on F p in Example 2.3. Then, we easily see Ker( ǫ X ) ∼ = Z p (cf.(18)). The cohomology H n gr ( Z p ; F p ) is F p for any n ∈ N . On the other hand, in [No1], theintegral quandle homology H Qn ( X ) was shown to be ( Z p ) b n , where b n ∈ Z is determined bythe recurrence formula b n +2 t = b n + b n +1 + b n +2 , b = b = · · · = b t − = 0 , and b t − = b t = 1 , and t > ω t = 1. In conclusion, since the b n is anexponentially growing, the map ( ϕ n ) ∗ is not bijective. Furthermore, we address a relation between shadow cocycle invariant [CKS] and the Dijkgraaf-Witten invariant [DW]. We now review the both invariants, and state Theorem 2.16.First, to describe the former invariant, we begin reviewing X -colorings. Given a quandle X , an X - coloring of an oriented link diagram D is a map C : { arcs of D } → X satisfyingthe condition in the left of Figure 1 at each crossing of D . Denote by Col X ( D ) the set of X -colorings of D . Note that two diagrams D and D related by Reidemeister moves admita 1:1-correspondence Col X ( D ) ↔ Col X ( D ); see [CJKLS, CKS] for details.We further define a shadow coloring to be a pair of an X -coloring C and a map λ fromthe complementary regions of D to X such that, if regions R and R ′ are separated by anarc α as shown in the right of Figure 1, the equality λ ( R ) ⊳ C ( α ) = λ ( R ′ ) holds. LetCol X ( D ) denote the set of shadow colorings of D . Given an X -coloring C , we put x ∈ X on the region containing a point at infinity. Then, by the rules in Figure 1, colors of otherregions are uniquely determined, and ensure a shadow coloring S denoted by ( C ; x ). Wethus obtain a bijection Col X ( D ) × X ≃ Col X ( D ) sending ( C , x ) to S = ( C ; x ).We briefly formulate (shadow) quandle cocycle invariants [CKS]. Let D be a diagram ofa link L , and S ∈
Col X ( D ) a shadow coloring. For a crossing τ shown in Figure 2, we definea weight of τ to be ǫ τ ( x, y, z ) ∈ C Q ( X ; Z ), where ǫ τ ∈ {± } is the sign of τ . Further the βγ C ( α ) ⊳ C ( β ) = C ( γ ) α RR ′ λ ( R ) ⊳ C ( α ) = λ ( R ′ ) Figure 1: The coloring conditions at each crossing and around arcs. fundamental class of S is defined to be [ S ] := P τ ǫ τ ( x, y, z ) ∈ C Q ( X ; Z ), and is known tobe a 3-cycle. For a quandle 3-cocycle ψ ∈ C Q ( X ; A ), we consider the pairing h ψ, [ S ] i ∈ A .The formal sum I ψ ( L ) := P S∈ Col X ( D ) Z {h ψ, [ S ] i} in the group ring Z [ A ] is called quandlecocycle invariant of L , where a symbol 1 Z { a } ∈ Z [ A ] means the basis represented by a ∈ A .By construction, in order to calculate the invariant concretely, it is important to find explicitformulas of quandle 3-cocycles. x y z x zy Figure 2: Positive and negative crossings with X -colors. On the other hand, we will briefly formulate a Dijkgraaf-Witten invariant below (9). Forthis, for a link L , denote by b C mL the m -fold cyclic covering space of S branched over L .Note that Z canonically acts on the space b C mL by the covering transformations. Accordingto [No3], when X is connected and of type t , for an X -coloring of L , we can construct a Z -equivariant homomorphism Γ C : π ( b C tL ) → Ker( ǫ X ), where Z acts on Ker( ǫ X ) via the splitsurjection (3); see § C . In summary, given a link-diagram D , wehave a mapΓ • : Col X ( D ) −→ Hom Z gr ( π ( b C tL ) , Ker( ǫ X )) , (8)where the right side is the set of the Z -equivariant group homomorphisms π ( b C tL ) → Ker( ǫ X ).Furthermore, consider the pushforward of the fundamental class [ b C tL ] ∈ H ( b C tL ) via thehomomorphism Γ C . Using this, with respect to a Z -invariant 3-cocycle κ of Ker( ǫ X ), wedefine a Z -equivariant part of Dijkgraaf-Witten invariant of b C tL by the formulaDW Z κ ( b C tL ) = X C∈ Col X ( D ) h κ, (Γ C ) ∗ ([ b C tL ]) i ∈ Z [ A ] . (9)We will show that, with respect to a connected quandle with a certain assumption, thetwo invariants explained above are equivalent (see § Theorem 2.16.
Let X be a finite connected quandle of type t X . Let an abelian group A contain no t X -torsion. Assume that the induced map p ∗ X : H Q ( X ; A ) → H Q ( e X ; A ) issurjective. Then any Z -invariant 3-cocycle κ of Ker( ǫ X ) admits a quandle -cocycle ψ of X and the equality I ψ ( L ) = | X | · DW Z κ ( b C t X L ) ∈ Z [ A ] . oreover, conversely, given a quandle -cocycle ψ of X , there is a Z -invariant group 3-cocycle κ of Ker( ǫ X ) for which the equality holds. Remark 2.17.
As is seen in the proof in § ψ in Theorem 2.16 concretely from a group 3-cocycle κ . For instance, if p X : e X → X is isomorphic, then the ψ is given by ϕ ∗ ( κ ). As another example, for Alexander quandleson F q , the relations between ψ and κ are given by explicit formulas (see § p ∗ X , the invariant DW Z κ ( b C tL ) constructed fromany Z -invariant 3-cocycle κ of Ker( ǫ X ) is can be computed from the quandle cocycle in-variants via link-diagrams. Fortunately, there are some quandles with the assumption ofthe surjectivity of p ∗ X . For example, connected Alexander quandles X which satisfy eitheroddness of | X | or evenness of t X ([No3, Lemma 9.15]), and “symplectic quandles X over F q ” with g = 1 ([No3, § X with g > p , as is shownin [No3, § H gr3 (Ker( ǫ X )) ∼ = H gr3 ( Sp (2 g ; F q )) ∼ = Z /q − H Q ( X ) ∼ = 0. Hence, ingeneral, the invariant DW Z κ ( b C t X L ) is not always interpreted from shadow cocycle invariants. Our objectivity in this section is to prove Theorems 2.11 and 2.16. This outline is, roughlyspeaking, a translation from some homotopical results in [No3] to terms of the group homol-ogy H gr3 (Ker( ǫ X )): Actually, using homotopy groups, the author studied the group homologyand a relation to Dijkgraaf-Witten invariant. So Section 3.1 reviews a group, Π ( X ), andtwo homomorphisms from the group. In § § § ∆ X,x and Θ X In order to construct the two homomorphisms in (10), (11), we first review the group Π ( X )defined in [FRS1, FRS2]. Consider the set of all X -colorings of all link-diagrams. Then wedefine Π ( X ) to be the quotient set subject to Reidemeister moves and concordance relationsillustrated in Figure 3. Then disjoint unions of X -colorings make Π ( X ) into an abeliangroup. For a connected quandle X of finite order, the group Π ( X ) was well-studied (seealso Theorem 3.1 below). aa aa a φ Figure 3: The concordance relations10 ext, we explain the homomrphism in (10) below. Recall from § X -colirng C and x ∈ X , we can construct a shadow coloring of the form ( C ; x ), and thefundamental class [( C ; x )] contained in H Q ( X ). We easily see that, if two X -colorings C , C ′ are related by Reidemeister moves and concordance relations, then the associated classes[( C ; x )], [( C ′ ; x )] are equal in H Q ( X ) by definition. Hence we obtain a homomorphism∆ X,x : Π ( X ) −→ H Q ( X ) , C 7−→ [( C ; x )] . (10)On the other hand, we will explain the homomorphism Θ X below (11). For this end,we first observe the fundamental group of the t -fold cyclic covering space b C tL . Given alink-diagram D of L , let γ , . . . , γ n be the arcs of D . Consider Wirtinger presentation of π ( S \ L ) generated by γ , . . . , γ n . For s ∈ Z , we take a copy γ i,s of the arc γ i . Then, byReidemeister-Schreier method (see, e.g., [Kab, §
3] for the details), the group π ( b C tL ) can bepresented bygenerators : γ i,s (0 ≤ i ≤ n, s ∈ Z ) , relations : γ k,s = γ − j,s − γ i,s − γ j,s for each crossings in the figure below , and γ i,s = γ i,s + t , γ ,s = 1 .γ i γ j γ k γ j γ k γ i Let X be a connected quandle of type t . Given an X -coloring C ∈
Col X ( D ), we denotethe color on the arc γ i by x i ∈ X . Define a group homomorphism Γ C : π ( b C tL ) → Ker( ǫ X ) bysetting Γ C ( γ i,s ) := e s − x e x i e − sx (see [No3, §
4] for the well-definedness). Furthermore, considerthe pushforward (Γ C ) ∗ ([ b C tL ]) ∈ H gr3 (Ker( ǫ X )), where [ b C tL ] is the fundamental class in H ( b C tL ).We thus obtain a map θ X,D : Col X ( D ) −→ H gr3 (Ker( ǫ X )) , C 7−→ (Γ C ) ∗ ([ b C tL ]) . As is shown [No3], if two X -colorings C , C ′ can be related by Reidemeister moves and theconcordance relations, then the identity θ X,D ( C ) = θ X,D ′ ( C ′ ) holds. Therefore the maps θ X,D with respect to all diagrams D yield a homomorphismΘ X : Π ( X ) −→ H gr3 (Ker( ǫ X )) . (11)This Θ X plays an important role to study the group Π ( X ) up to t -torsion. Indeed, Theorem 3.1 ([No3, Theorems 3.4 and 3.18] ) . Let X be a connected quandle of type t X .Put the inclusion ι : Ker( ǫ X ) → As( X ) in (3) . If the homology H gr3 (As( X )) is finitelygenerated, then the composite ι ∗ ◦ Θ X : Π ( X ) → H gr3 (As( X )) is a split surjection modulo t X -torsion , whose kernel is isomorphic to H Q ( X ) modulo t X -torsion.Furthermore, the induced map ( p X ) ∗ : Π ( e X ) → Π ( X ) is a split injection modulo t X -torsion, and this cokernel equals the kernel of the composite ι ∗ ◦ Θ X . In [No3], the maps Θ X , ι ∗ ◦ Θ X were denoted by Θ ΠΩ , Θ X , respectively. .2 A key lemma and Proofs of Theorems 2.11 and 2.16 For the proofs, we state a key lemma. We here fix a terminology: A (shadow) e X -coloringof D is said to be based , if an arc γ of D is colored by the identity 1 Ker( ǫ X ) ∈ e X . Lemma 3.2 (cf. [Kab, Theorem 9.1] ) . Let X be a connected quandle of type t X < ∞ ,and p X : e X → X the projection in Example 2.3. Let Υ : C R ( e X ) → C R ( e X ) be the chainisomorphism in (5) . Let S ∈
Col e X ( D ) be a based shadow coloring. Define an e X -coloring e C ∈
Col e X ( D ) to be the X -coloring as the restriction of S ∈
Col e X ( D ) . Then Θ X ([ p X ( e C )]) = ϕ ◦ Υ([ S ]) ∈ H gr3 (Ker( ǫ X )) . Before going to the proof, we will complete the proofs of Theorems 2.11 and 2.16.
Proof of Theorem 2.11.
As mentioned in Remark 2.12, there is an isomorphism H Q ( e X ) ∼ = H gr3 (As( X )) up to t X -torsion, as finitely generated Z -modules. Hence, in order to prove thatthe map ( e Φ ) ∗ is an isomorphism, it suffices to show the surjectivity.To this end, we set the composite of the three homomorphisms mentioned above:Π ( e X ) ( p X ) ∗ −−−−→ Π ( X ) Θ X −−−−→ H gr3 (Ker( ǫ X )) ι ∗ −−→ H gr3 (As( X )) . It follows from Theorem 3.1 above that the composite is an isomorphism up to t X -torsion.Note that, Lemma 3.4 below ensures an isomorphism ξ : H gr3 (As( X )) → H gr3 (Ker( ǫ X )) Z such that ξ ◦ ι ∗ = t X · ( π ρ ) ∗ , where π ρ is the projection C gr3 (Ker( ǫ X )) → C gr3 (Ker( ǫ X )) Z explained in § t X -torsion aswell: ( π ρ ) ∗ ◦ Θ X ◦ ( p X ) ∗ : Π ( e X ) −→ H gr3 (Ker( ǫ X )) Z . Therefore, for any 3-cycle
K ∈ H gr3 (Ker( ǫ X )) Z which is annihilated by t X , we choose somebased e X -coloring e C such that K = ( π ρ ) ∗ ◦ Θ X ◦ ( p X ) ∗ ([ e C ]). We here set a shadow coloring S of the form ( e C ; 1 e X ). Then, by the lemma 3.2, we notice the equalities e Φ (Υ([ S ])) = ( π ρ ) ∗ ◦ ( ϕ ◦ Υ)([ S ]) = ( π ρ ) ∗ ◦ Θ X ( p X ([ e C ])) = K . Noting the left element Υ([ S ]) ∈ H Q G ( e X ), we obtain the surjectivity of e Φ as required. Proof of Theorem 2.16.
We first construct two homomorphisms (12), (13). Let X be a finiteconnected quandle of type t X . Given a Z -invariant group 3-cocycle κ , consider a compositehomomorphism from Π ( X ):Π ( X ) Θ X −−−→ H gr3 (Ker( ǫ X )) h κ, •i −−−−→ A. (12)On the other hand, by the assumption of the surjectivity of p ∗ X : H Q ( X ; A ) → H Q ( e X ; A ),we can choose a quandle cocycle ψ ∈ H Q ( X ; A ) such that p ∗ X ( ψ ) = ( ϕ ◦ Υ) ∗ ( κ ). We thenset a composite homomorphismΠ ( X ) [ • ; x ] −−−−→ H Q ( X ) h ψ, •i −−−−→ A. (13) Kabaya [Kab] showed a similar statement under a certain strong condition of quandles. However, as seen in the proofs ofTheorems 2.11 and 2.16, in order to verify a relation to the Dijkgraaf-Witten invariant, we may deal with only the extendedquandle e X in Example 2.3, which may not satisfy the strong condition. e remark that this kernel contains the kernel of Θ X by Theorem 3.2, since A contains no t X -torsion by assumption.Next, we claim the equivalence of the two maps (12) and (13). For this, we put some e X -colorings e C , . . . , e C n , which generate the Π ( e X ); here we may assume that these coloringsare based by Lemma 3.3 below. Notice that, by Theorem 3.1, the group Π ( X ) is generatedby the kernel Ker( ι ∗ ◦ Θ X ) and the elements p X ( e C ) , . . . , p X ( e C n ). Therefore, the claimedequivalence results from the following equalities: h κ, Θ X ( p X ( e C i ) i = h κ, ϕ ◦ Υ([( e C i ; ˜ x )]) i = h p ∗ X ( ψ ) , [( e C i ; ˜ x )] i = h ψ, [ p X ( e C i ); p X (˜ x ))] i , where the first equality is obtained from Lemma 3.2.We will show the equivalence of the two invariants as stated in Theorem 2.16. Bydefinitions, these invariants are reformulated asDW Z κ ( b C t X L ) = X C∈ Col X ( D ) Z {h κ, Θ X ( C ) i} , I ψ ( L ) = X x ∈ X X C∈ Col X ( D ) Z {h ψ, [ C ; x ]) } ∈ Z [ A ] . Further, it is shown [IK, Theorem 4.3] that the right invariant I ψ ( L ) = | X | P C∈ Col X ( D ) Z {h ψ, [ C ; x ]) } for any x ∈ X . In conclusion, since the homomorphisms (12), (13) are equal as claimedabove, so are the two invariants.Finally, to prove the latter part of Theorem 2.16, recall that the map e Φ is an isomorphismafter tensoring A (Theorem 2.11). So, given a quandle 3-cocycle ψ , we define a Z -invariantgroup 3-cocycle κ of Ker( ǫ X ) to be (Υ ◦ e Φ ∗ ) − ( p ∗ X ( ψ )) . Hence, by a similar argument asabove, we have the desired equality I ψ ( L ) = | X | · DW Z κ ( b C t X L ). We will prove Lemma 3.2 as a modification of [Kab, Theorem 9.1].We will review descriptions in [Kab, § b C tL ] ∈ H ( b C tL ; Z ) of the branched covering space b C tL , Let N ( L ) ⊂ S denote a tubularneighborhood of S \ L . Let γ , . . . , γ n be the oriented arcs of the diagram D such as § γ i has boundaries. For each arc γ i , we can construct 4 tetrahedra T ( u ) i ⊂ b C tL with 1 ≤ u ≤
4, and further decompose S \ N ( L ) into these 4( n + 1) tetrahedraas a triangulation, which is commonly referred as to “a standard triangulation” (see, e.g.,[Wee] or [Kab, § t tetrahedra T ( u ) i,s in thebranched covering space b C tL , where 0 ≤ s ≤ t − ≤ u ≤ T ( u ) i,s corresponds with the s -th connectedcomponent of the preimage of the T ( u ) i ⊂ S via the branched covering b C tL → S . Let us fixthe orderings of T ( u ) i,s following Figure 8 in [Kab]. Then he showed the union S i,s,u T ( u ) i,s = b C tL and that the formal sum P i,s ǫ i ( T (1) i,s − T (2) i,s − T (3) i,s + T (4) i,s ) represents the orientation class[ b C tL ], where ǫ i ∈ {± } is the sign of the crossing at the endpoint of γ i .We next discuss labelings of the tetrahedron T ( u ) i,s by a group G . Put a map L : { T ( u ) i,s } i,s,u → G . Let I : { T ( u ) i,s } i,s,u → G be a constant map taking elements to the he identity of G . We would regard the product I × L as a labeling of vertices in T ( u ) i,s saccording to the ordering. Fix a homomorphism f : π ( b C tL ) → G and recall the generators γ i,s ∈ π ( b C tL ) in § §
4] that, if the labeling is globally compatible andthe L (cid:0) T ( u ) i,s (cid:1) s satisfy the following condition: • L ( T (1) i,s ) · f ( γ i,s ) = L ( T (3) i,s +1 ) , L ( T (2) i,s ) · f ( γ i,s ) = L ( T (4) i,s +1 ) ∈ G , (14)where the action · is diagonal, then the product I × L satisfies a group 1-cocycle conditionin the union S i,s,u T ( u ) i,s , and that the induced homomorphism coincides with the f . In thesequel, the pushforward f ∗ ([ b C tL ]) is represented by the formulaΥ (cid:0) X ≤ i ≤ n X ≤ s The proof will be shown by expressing the left side Θ X ( C ) in details;For this, given a shadow coloring S by e X = Ker( ǫ X ), we will define a labelling L compatible with the homomorphism Γ C : π ( b C tL ) → G with G = As( X ) as follows. Let( g, h, k ) ∈ e X = Ker( ǫ X ) be the weight of the endpoint of the arc γ i . Using the quandlestructure on e X , we then define a map L : { T ( u ) i,s } s,i,u → (Ker( ǫ X )) = e X by L ( T (1) i,s ) := ( g s − , h s − , k s − ) , L ( T (2) i,s ) := ( g s − ⊳ h s − , h s − , k s − ) , L ( T (3) i,s ) := ( g s ⊳ k s , h s ⊳ k s , k s ) , L ( T (4) i,s ) := (( g s ⊳ h s ) ⊳ k s , h s ⊳ k s , k s ) , where we put g s := e sa ge − sa ∈ Ker( ǫ X ) for short.We will verify the equalities (14) on L . From the definition of the action X x As( X ),we notice an equality e p X ( k ) = e a · k = k − e a k ∈ As( X ) for any k ∈ e X . In addition, we note( p X ) ∗ ( S ( γ )) = p X (1 e X ) = a ∈ X since the S is based by assumption. Hence, using thenotation γ i,s ∈ π ( b C tL ), we haveΓ C ( γ i,s ) = ( e a ) s − e a ·S ( γ i ) e − sa = ( e a ) s − e p X ( k ) e − sa = e s − a k − e a ke − sa . (16)Therefore, for any b ∈ X , using (16), we have the equality( e s − a be − sa ) · Γ C ( γ i,s ) = e sa ( b ⊳ k ) e − sa ∈ Ker( ǫ X ) . Consequently, applying b = g , b = h or b = g ⊳ h to this identity concludes the condition(14). Furthermore it is not hard to see that the labelling is globally compatible with all thetriangulations. Hence, the labeling L induces the homomorphism Γ C : π ( b C tL ) → As( X ).Finally, we discuss the push-forward of the orientation class (Γ C ) ∗ ([ b C tL ]) ∈ C gr3 (Ker( ǫ X ); Z ) . We first check that, for x, y, z ∈ e X = Ker( ǫ X ), the following equality holds:Υ − ◦ ϕ ◦ Υ( x, y, z ) = X ≤ s ≤ t ( x s , y s , z s ) − ( x s ⊳ y s , y s , z s ) − ( x s ⊳ z s , y s ⊳ z s , z s ) + (( x s ⊳ y s ) ⊳ z s , y s ⊳ z s , z s ) . ere this verification is easily obtained from recalling the definitions of ϕ in Definition 2.5and Υ in (5). Hence, compared with the map L , we have the equalityΥ − ◦ ϕ ◦ Υ([ S ]) = X i X s L (cid:0) ǫ i ( T (1) i,s − T (2) i,s − T (3) i,s + T (4) i,s ) (cid:1) ∈ C gr3 (Ker( ǫ X ))exactly. Since the right side is the push-forward Υ − (cid:0) (Γ C ) ∗ ([ b C tL ]) (cid:1) by (15), we conclude thedesired equality.We now provide proofs of two lemmas above. Lemma 3.3. Let X be a connected quandle. If an element in Π ( e X ) is represented by an e X -coloring of D , then the class equals some based e X -coloring of the D in Π ( e X ) .Proof. Let the arc γ be colored by h ∈ e X . Since the extended quandle e X is also connected[No3, Lemma 9.15], we have g , . . . , g n ∈ e X such that ( · · · ( h ⊳ g ) ⊳ · · · ) ⊳ g n = 1 e X . Thenby observing the following picture, we can change the C to another e X -coloring C ′ of D suchthat the arc γ is colored by h ⊳ g and that [ C ] = [ C ′ ] ∈ Π ( e X ). γ h = = = = ∈ Π ( X ) . C C h g C hg C ′ h ⊳ g g C ′ h ⊳ g Here the first and forth equalities are obtained from the concordance relation, and in thesecond (resp. third) equality the loop colored by g passes under (resp. over) the all arc of D . Note that we here only use Reidemeister moves. Hence, iterating this process, we havea based e X -coloring C ( n ) of the D such that [ C ] = [ C ( n ) ] ∈ Π ( e X ). Lemma 3.4. Let X be a connected quandle of type t X . Let ι : Ker( ǫ X ) → As( X ) be theinclusion (3) , and π ρ : C gr3 (Ker( ǫ X )) → C gr3 (Ker( ǫ X )) Z be the projection. Then there is anisomorphism ξ : H gr3 (As( X )) → H gr3 (Ker( ǫ X )) Z modulo t X -torsion such that ξ ◦ ι ∗ = t X · ( π ρ ) ∗ .Proof. Fix x ∈ X, and consider the subgroup h e nt X x i n ∈ Z of As( X ), which is contained in thecenter (see [No3, Lemma 4.1]). Put the quotient Q X := As( X ) / h e nt X x i n ∈ Z . By the Lyndon-Hochshild spectral sequence, the projection induces an isomorphism P ∗ : H gr3 (As( X )) ∼ = H gr3 ( Q X ) up to t X -torsion, since the H gr2 (As( X )) is shown to be annihilated by t X [No3,Corollary 6.4]. Furthermore, noting the group extension Ker( ǫ X ) → G X → Z /t X , thetransfer gives an isomorphism T : H gr3 ( Q X ) → H gr3 (Ker( ǫ X )) Z modulo t X ; see [Bro, § III.10].Hence, denoting T ◦ P ∗ by ξ , we have the equality ξ ◦ ι ∗ = t X · ( π ρ ) ∗ by construction. This section proves Proposition 2.13 and Theorem 2.14. The outline of the proof is asfollows. With respect to an Alexander quandle over F q , a basis of the third cohomologyover F q was found by Mochizuki [Moc], which we review in § o, we will construct group 3-cocycles of As( X ) as preimages of the basis via the chain mapsΦ and e Φ (see § X ) of a con-nected Alexander quandle X , shown by Clauwens [Cla]. Set a homomorphism µ X : X ⊗ X → X ⊗ X defined by µ X ( x ⊗ y ) = x ⊗ y − T y ⊗ x. Using this µ X , let us equip Z × X × Coker( µ X ) with a group operation given by( n, a, κ ) · ( m, b, ν ) = ( n + m, T m a + b, κ + ν + [ T m a ⊗ b ]) . (17)Then a homomorphism As( X ) → Z × X × Coker( µ X ) sending the generators e x to (1 , x, X ) are then described asAs( X ) ⊃ X × Coker( µ X ) ⊃ Coker( µ X ) ⊃ . (18)In particular, the kernel Ker( ǫ X ) in (3) is the subgroup on the set X × Coker( µ X ). Inciden-tally, an isomorphism H Q ( X ) ∼ = Coker( µ X ) is shown [Cla]. Notation Denote by G X the subgroup Ker( ǫ X ) on X × Coker( µ X ). From now on, in thissection, we let X be an Alexander quandle on F q with ω ∈ F q . Let X be of type t X . Thatis, t X is the minimal satisfying ω t X = 1. Note that t X is coprime to q since ω q − = 1. We will review Mochizuki 2-, 3-cocycles of X = F q . We here regard polynomials in the ring F q [ U , . . . , U n ] as functions from X n to F q , and as being in the complex C Q G n ( X ; F q ) in § Theorem 4.1 ([Moc, Lemma 3.7]) . The following set represents a basis of H Q ( X ; F q ) . { U q U q | ω q + q = 1 , ≤ q < q < q, and q i is a power of p. } . Next, we describe all the quandle 3-cocycles of X . To see this, recall the following threepolynomials over F q ([Moc, § χ ( U j , U j +1 ) := X ≤ i ≤ p − ( − i − i − U p − ij U ij +1 = (cid:0) ( U i − + U i ) p − U pi − − U pi (cid:1) /p,E ( a · p, b ) := (cid:0) χ ( ωU , U ) − χ ( U , U ) (cid:1) a · U b , E ( a, b · p ) := U a · (cid:0) χ ( U , U ) − χ ( ω − · U , U ) (cid:1) b . Define the following set I + q,ω consisting of the polynomials under some conditions: I + q,ω := { E ( q · p, q ) | ω p · q + q = 1 , q < q } ∪ { E ( q , q · p ) | ω q + p · q = 1 , q ≤ q }∪ { U q U q U q | ω q + q + q = 1 , q < q < q } . (19)Here the symbols q i range over powers of p with q i < q .Furthermore, we review polynomials denoted by Γ( q , q , q , q ). For this, we define a set Q q,ω ⊂ Z consisting of quadruples ( q , q , q , q ) such that q ≤ q , q < q , q < q , and ω q + q = ω q + q = 1 . Here, if p = 2, we omit q = q . • One of the following holds: Case 1 ω q + q = 1 . Case 2 ω q + q = 1, and q > q . Case 3 ( p = 2), ω q + q = 1, and q = q . Case 4 ( p = 2), ω q + q = 1, q ≤ q < q < q , ω q = ω q . Case 5 ( p = 2), ω q + q = 1, q < q < q < q , ω q = ω q .For ( q , q , q , q ) ∈ Q q,ω in each cases, the polynomial Γ( q , q , q , q ) is defined as follows : Case 1 Γ( q , q , q , q ) := U q U q + q U q . Case 2 Γ( q , q , q , q ) := U q U q + q U q − U q U q + q U q − ( ω q − − (1 − ω q + q )( U q U q U q + q − U q + q U q U q ) . Case 3 Γ( q , q , q , q ) := U q U q + q U q . Case 4 and Case 5 Γ( q , q , q , q ) := U q U q + q U q . Remark 4.2. The 3-cocycle in Case 3 (resp. 4 and 5) is obtained from that in Case 1 afterchanging the indices (1 , , , 4) to (1 , , , (cid:0) resp. to (3 , , , (cid:1) .We call the set Q q,ω Mochizuki quadruples . Then we state the main theorem in [Moc]: Theorem 4.3 ([Moc]) . The third cohomology H Q ( X ; F q ) is spanned by the following setcomposed of non-trivial 3-cocycles. Here q i means a power of p with q i < q . I + q,ω ∪ { Γ( q , q , q , q ) | ( q , q , q , q ) ∈ Q q,ω } ∪ { U q U q | ω q + q = 1 , q < q } . Remark 4.4. Unfortunately the original statement and his proof of this theorem containedslight errors, which had however been corrected by Mandemaker [Man]. First, to prove Proposition 2.13, we prepare a lemma for a study of the quandle 3-cocyclesin (19). Lemma 4.5. Let us identify G = ( Z p ) h with F q as an additive group. Then the secondgroup cohomology H ( G ; F q ) ∼ = ( F q ) h ( h +1)2 is generated by the following group 2-cocycles: { U q U q | ≤ q < q < q, where q i is a power of p. } . Furthermore, the third one H ( G ; F q ) ∼ = F q h ( h +1)( h +2)6 is spanned by the following 3-cocycles: { U q U q U q | q < q < q } ∪ { (( U + U ) q − U q − U q ) · U q /p | q < q }∪ { U q (( U + U ) q − U q − U q ) /p | q ≤ q } , where q , q , q run over powers of p with ≤ q j < q. Moreover, regarding the multiplicationof ω ∈ F q as an action of Z on F q , the Z -invariant parts H i gr ( G ; F q ) Z are generated by theabove polynomials of degree d satisfying ω d = 1 . Here i = 2 , . In Cases 3, 4 and 5, we change the forms of Γ( q , q , q , q ) in [Moc]; however, our Γ are cohomlogous to the original ones. he group cohomologies of abelian groups can be calculated in many ways, e.g., by similarcalculations to [Bro, V. § 6] or [Moc]; So we omit proving Lemma 4.5.Returning to our subject, we apply these generators in Lemma 4.5 to the pullback of thechain map ϕ (see Definition 2.5). Then easy computations show the identities ϕ ∗ ( U q U q U q ) = t X (1 − ω q )(1 − ω q + q ) · F ( q , q , q ) ,ϕ ∗ (cid:0) ( U + U ) q − U q − U q ) U q /p (cid:1) = t X (1 − ω q ) · E ( q , q ) ,ϕ ∗ (cid:0) U q (( U + U ) q − U q − U q ) /p (cid:1) = t X (1 − ω q ) · E ( q , q ) ∈ C Q ( X ; F q ) . (20)Compared with the way in [Moc] that the right quandle 3-cocycles were found as solutions ofa differential equation over F q , the three identities via the map ϕ ∗ are simple and miraculous.Using the identities we will prove Proposition 2.13 as follows: Proof of Proposition 2.13. The injectivity of Φ ∗ = ( π ρ ◦ ϕ ) ∗ follows from that this Φ ∗ gives a1:1 correspondence between a basis of H (( Z p ) h ; F q ) Z and a basis of a subspace of H Q ( X ; F q )because of the previous three identities (compare Theorem 4.3 with Lemma 4.5).Next, assume H Q ( X ; F q ) = 0. Then, Theorem 4.1 implies that no pair ( q , q ) satisfies ω q + q = 1 and q < q < q . Hence, by observing Theorem 4.3 carefully, the H Q ( X ; F q ) isgenerated by the image of Φ ∗ . Therefore Φ ∗ is an isomorphism as desired.Next, we will prove Theorem 2.14. To this end, we now observe the cokernel Coker(Φ ∗ ).To begin, we study the chain map (Φ ◦ P ) ∗ : H (( Z p ) h ; F q ) Z → H Q ( X ) stated inProposition 2.10. Recall from Lemma 4.5 that this domain is generated by polynomials ofthe form U q U q . So, recalling the composite Φ ◦ P from Proposition 2.10, we easily see(Φ ◦ P ) ∗ ( U q U q ) = t X (1 − ω q ) U q U q ∈ C Q G ( X ; F q ) . Hence, the third term in Theorem 4.3 is spanned by the image of this map (Φ ◦ P ) ∗ .Furthermore, we will discuss the cokernel of Φ ∗ ⊕ (Φ ◦ P ) ∗ . By observing Theorem 4.3carefully, we see that a basis of the cokernel consists of the polynomials Γ’s coming fromthe Mochizuki quadruples Q q,ω . Let us denote a quadruple ( q , q , q , q ) ∈ Q q,ω by q forshort. Case by case, we now introduce a map θ q Γ : ( G X ) → F q by setting the values of θ q Γ at ( x, a ⊗ b, y, c ⊗ d, z, e ⊗ f ) ∈ ( X × Coker( µ X )) as follows. In Case 1, θ q Γ is defined by theformula(1 − ω ) − q (cid:0) x q y q + q + x q + q y q − (1 − ω ) − q ( ω q a q b q + a q b q − x q + q ) y q +(1 − ω ) − q ( a q b q + ω q a q b q − x q + q ) y q (cid:1) z q . (21)In Case 2, the value of θ q Γ is given by the formula(1 − ω ) − q − q (cid:0) x q ( y q + q z q + y q z q + q ) − ( x q + q y q + x q y q + q ) z q +(1 − ω ) − q ( x q + q − ω q a q b q − a q b q ) y q z q − (1 − ω ) − q ( x q + q − ω q a q b q − a q b q ) y q z q (cid:1) . urthermore, for Case 3 (resp. 4 and 5), the value is defined to be that of Case 1 by changingthe indices (1 , , , 4) to (1 , , , (cid:0) resp. to (3 , , , (cid:1) , according to Remark 4.2. Lemma 4.6. For q = ( q , q , q , q ) ∈ Q q,ω , the map θ q Γ from ( G X ) to F q is a Z -invariantgroup -cocycle of G X .Moreover, using the map e Φ , the pullback e Φ ∗ ( θ q Γ ) equals t X · p ∗ X (Γ( q )) in C Q ( e X ; F q ) .Proof. Note that a map θ : ( G X ) → A is a Z -invariant group 3-cocycle, by definition, ifand only if it satisfies the two equalities θ ( b , c , d ) − θ ( ab , c , d ) + θ ( a , bc , d ) − θ ( a , b , cd ) + θ ( a , b , c ) = 0 ,θ (( ωa, α ) , ( ωb, β ) , ( ωc, γ )) = θ (( a, α ) , ( b, β ) , ( c, γ )) , for any a = ( a, α ) , b = ( b, β ) , c = ( c, γ ) , d = ( d, δ ) ∈ G X = X × Coker( µ X ). Then, byelementary and direct computations, it can be seen that the maps θ q Γ are Z -invariant group3-cocycles of G X . Also, the desired equality e Φ ∗ ( θ q Γ ) = t X · p ∗ X (Γ( q )) can be obtained by adirect calculation. Proof of Theorem 2.14. Let q be odd. As is known [No3, Lemma 9.15], the induced map p ∗ X : H Q ( X ; F q ) → H Q ( e X ; F q ) is surjective. Hence, according to Lemma 4.6, there existsa section s : H Q ( e X ; F q ) → H Q ( X ; F q ) such that s ( e Φ ∗ ( θ q Γ )) = Γ( q ) for any q ∈ Q q,ω . Tosummarize the above discussion, the sum ((Φ ◦ P ) ∗ ⊕ Φ ∗ ) ⊕ (cid:0) s ◦ res( e Φ ∗ ) (cid:1) in (7) is anisomorphism to H Q ( X ; F q ).Incidentally, we will show that the group 3-cocycles θ q Γ above except Case 2 are presentedby Massey products. To see this, we consider a group homomorphism f q i : G X → F q ; ( x, α ) x q i , which is a group 1-cocycle of G X . For group 1-cocycles f, g and h , we denote by f ∧ g thecup product; further, if f ∧ g = g ∧ h = 0 ∈ H ( G X ; F q ), we denote by < f, g, h > the tripleMassey product in H ( G X ; F q ) as usual (see, e.g., [Kra] for the definition). Proposition 4.7. Let e = 2 . Let ( q , q , q , q ) ∈ Q q,ω satisfy Case e in § θ Γ described above is of the followings form in the cohomology H ( G X ; F q ) . H ( G X ; F q ) ∋ θ Γ = (1 − ω q ) − < f q , f q , f q > ∧ f q for e = 1 , (1 − ω q ) − < f q , f q , f q > ∧ f q for e = 3 , (1 − ω q ) − < f q , f q , f q > ∧ f q for e = 4 or 5 . Proof. We use notation ( x, a ⊗ b, y, c ⊗ d, z, e ⊗ f ) ∈ ( X × Coker( µ X )) as above. Noticefirst that the cup product f q ∧ f q is the usual product x q y q (see [Bro, V. § Case1 , we now calculate the Massey product < f q , f q , f q > . We easily check two equalities x q y q = (1 − ω ) − q δ (cid:0) a q b q + ω q a q b q − x q + q (cid:1) , q y q = (1 − ω ) − q δ (cid:0) ω q a q b q + a q b q − x q + q (cid:1) . Hence, from the definition of Massey products, < f q , f q , f q > is represented by(1 − ω ) − q ( a q b q + ω q a q b q − x q + q ) y q + (1 − ω ) − q x q ( ω q c q d q + c q d q − y q + q ) . Furthermore, we set a group 2-cocycle defined by F := (1 − ω ) − q (cid:0) < f q , f q , f q > +(1 − ω ) − q δ ( ω q x q a q b q + x q a q b q − x q + q + q ) (cid:1) . A direct calculation then shows the equality F · z q = θ q Γ by definitions, i.e., < f q , f q , f q > ∧ f q = θ q Γ ∈ H ( G X ; F q ) as desired.Similarly, the same calculation holds for Cases 3, 4, 5 according to Remark 4.2.However, a geometric meaning of the cocycle θ q Γ with Case 2 remains to be open. As an application of Theorem 2.16, we will compute some Z -equivariant parts of Dijkgraaf-Witten invariants, which is equivalent to a shadow cocycle invariant. In this section, weconfirm ourselves to Alexander quandles on F q with ω ∈ F q . Recall from Lemma 4.6 that thequandle 3-cocycles Γ( q , q , q , q ) found by Mochizuki (see § G X .So we focus on the cocycles, and fix some notation. Let q denote a Mochizuki quadruple( q , q , q , q ) in Q q,ω for short, and replace Γ( q , q , q , q ) by Γ( q ) e , if q satisfies Case e in § e ≤ X -colorings was well-studied. In fact, if D is a diagram of a knot K , then there is a bijectionCol X ( D ) ←→ X ⊕ M i =1 F q [ T ] / ( T − ω, ∆ i ( T ) / ∆ i +1 ( T )) , (22)where ∆ i ( T ) is the i -th Alexander polynomial of K (see [Ino]). Therefore, we shall studyweights in the cocycles invariants. Γ( q , q , q , q )This subsection considers the torus knots T ( m, n ). We here remark that m and n arerelatively prime and the isotopy T ( m, n ) ≃ T ( n, m ); thereby n may be relatively prime to p without loss of generality. We determine all of the values of the invariants for T ( m, n ) asfollows: Theorem 5.1. Let q be relatively prime to n . Let T ( m, n ) be the torus knot. Let q ∈ Q q,ω be a Mochizuki quadruple, and Γ( q ) e be the associated quandle 3-cocycle. Then the quandlecocycle invariant I Γ( q ) e ( T ( m, n )) is expressed by one of the following formulas: i) If e = 1 , ω mn = 1 , ω m = 1 and ω n = 1 , then I Γ( q ) (cid:0) T ( m, n ) (cid:1) = q X a ∈ F q Z {− mn ( ζ − ω ) q + q ω q (1 − ζ ) q + q · a q + q + q + q } ∈ Z [ F q ] , (23) where ζ is the n -th primitive root of unity satisfying ω m = ζ m . Furthermore, if e = 3 (resp. or ), then the value of I Γ( q ) e is obtained from the above value I Γ( q ) afterchanging the indices (1 , , , to (1 , , , (cid:0) resp. to (3 , , , (cid:1) such as Remark 4.2.(ii) Let p = 2 or , and let e = 1 . If ω n = 1 and if m is divisible by p , then I Γ( q ) ( T ( m, n )) = q X a ∈ F q Z (cid:8) mnp (1 − ω ) q + q a q + q + q + q (cid:9) ∈ Z [ F q ] . (24) Furthermore, if e = 3 (resp. or ), then the value I Γ( q ) e is obtained from the value I Γ( q ) after changing the indices (1 , , , to (1 , , , (cid:0) resp. to (3 , , , (cid:1) , similarly.(iii) Let e = 2 . If p = 2 , ω n = 1 and if m is divisible by , then I Γ( q ) ( T ( m, n )) is equal to q P a,δ ∈ F q Z { mn E ( a, δ ) / } ∈ Z [ F q ] . Here E ( a, δ ) ∈ F q is temporarily defined by a q + q (cid:0) (1 + ω q ) a q δ q + (1 + ω q ) a q δ q (cid:1) + a q + q (cid:0) (1 + ω q ) a q δ q + (1 + ω q ) a q δ q (cid:1) . (iv) Otherwise, the invariant is trivial. Namely, I Γ( q ) e ( T ( m, n )) ∈ Z . This is proved in § e = 2, the invariant is non-trivial in only the case (iii). Remark 5.2. Asami and Kuga [AK, § I Γ( q ) e ( T ( m, n ))in the only case F q = F and n = 3, by the help of computer.We consider the t -fold cyclic covering of S branched over T ( m, n ). This is the Brieskornmanifold Σ( m, n, t ); see [Mil]. Hence, we obtain a Z -equivariant part of the Dijkgraaf-Witteninvariant of Σ( m, n, t ). Corollary 5.3. Let m, n be coprime integers. Let X be of type t . Let a Mochizuki quadruple ( q , q , q , q ) ∈ Q q,ω satisfy Case 1, and θ Γ ∈ H ( G X ; F q ) be the group 3-cocycle in Lemma4.6. Let p > be coprime to n and to t . If ω mn = 1 , ω n = 1 and ω m = 1 , then DW Z θ Γ (cid:0) Σ( m, n, t ) (cid:1) = X a ∈ F q Z {− tmn ( ζ − ω ) q + q ω q (1 − ζ ) q + q a q + q + q + q } ∈ Z [ F q ] . Proposition 4.7 says that the cocycle θ Γ forms a Massey product; Hence we clarify partiallythe Massey product structure of some Brieskorn manifolds. Here remark that there are afew methods to compute Massey products with Z /p -coefficients, in a comparison with thosewith Q -coefficients viewed from rational homotopy theory.Finally, we comment on the interesting result in Theorem 5.1 (ii). For finite nilpotentgroups G , the Massey products in H ( G ; F q ) with p = 2 , q = p , the group G X is isomorphic to the group “ P (3)” in [Le]. See [Le, Theorems 6and 7] for exceptional phenomenon of the cohomology ring H ∗ gr ( G X ; F p ) with p = 2 , .2 Further examples in the case ω = − I Γ( q ) e ( K ) of knots, although it is elementary.We now consider the simplest case ω = − 1; hence the quandle X is of type 2. Further-more, note that, for any Mochizuki quadruple q = ( q , q , q , q ), the associated 3-cocycleforms U q U q + q U q by definition; it is not hard to compute the cocycle invariant. However,for many knots whose colorings satisfy Col X ( D ) ∼ = ( F q ) , the invariants are frequently ofthe form q P a ∈ F q a q + q + q + q up to constant multiples in computer experiments. In orderto avoid the case Col X ( D ) ∼ = ( F q ) , recall the bijection (22). Accordingly we shall deal withsome knots having non-trivial second Alexander polynomials as follows: Example 5.4. Let ω = − 1. The knots K in Table 1 are those whose crossing numbersare < 11 satisfying Col X ( D ) ∼ = ( F q ) with p > 3. We only list results of the invariantswithout the proofs. Here note that, according to Theorem 2.16 and Proposition 4.7, thecocycle invariant stems from triple Massey products of double branched covering spaces.Refer to the tables in [Kaw, Appendix F] for the correspondences between knots K anddouble coverings of S branched over K . K p I Γ( q ) ( K )9 G ( q ; 1 , G ( q ; 3 , G ( q ; 3 , K p I Γ( q ) ( K )10 G ( q ; 2 , q G ( q )10 G ( q ; 1 , Table 1: The values of I Γ( q ) ( K ). Here, q ∈ Z and q ∈ Q q,ω are arbitrary, and, for n, m ∈ Z , the symbols G ( q ; n, m ) and G ( q ) are polynomials expressed by G ( q ; n, m ) := q X a,b ∈ F q Z (cid:8) n ( a q + q b q + q + a q + q b q + q + a q + q b q + q + a q + q b q + q )+ m ( a q + q b q + q + a q + q b q + q ) (cid:9) ∈ Z [ F q ] , G ( q ) := q X a,b ∈ F q Z (cid:8) a q + q + q + q + a q b q + q + q ) + ( a q + q + q b q + a q + q b q + q )+2( a q + q + q b q + a q + q + q b q + a q + q b q + q + a q + q b q + q ) (cid:9) ∈ Z [ F q ] . For the proof, we first recall a slight reduction of the cocycle invariant by [IK, Theorem 4.3],that is, we may consider only shadow colorings of the forms S = ( C ; 0). More precisely, I ψ ( L ) = q · X C∈ Col X ( D ) Z {h ψ, [( C ; 0)] i} ∈ Z [ A ] . (25) e establish terminologies on the torus knot T ( m, n ). Regard T ( m, n ) as the closureof a braid ∆ m , where ∆ := σ n − · · · σ ∈ B n . Let α , . . . , α n be the top arcs of ∆ m . For1 ≤ i ≤ m , We let x i, , . . . , x i,n − be the crossings in the i -th ∆; see Figure 4. · · · · · ·· · · · · ·· · · · · ·· · · · · ·· · · · · · α α α j α n x i, x i, x i,j x i,n − Figure 4: The arcs α j and crossing points x i,j on the diagram of the torus knot. Proof of Theorem 5.1. Although Asami and Kuga [AK] formulated explicitly X -colorings of T ( m, n ), we will give a reformulation of them appropriate to the 3-cocycle Γ( q ) e . If given an X -coloring C of T ( m, n ), we define a j := C ( α j ), and put a vector a = ( a , . . . , a n ) ∈ ( F q ) n ;Notice that it satisfies the equation a = a P m , where P is given by a companion matrix P := ω · · · ω · · · · · · ω − ω − ω · · · − ω − ω ∈ Mat( n × n ; F q ) . Remark that the characteristic polynomial is ( λ − λ n − ω n ) / ( λ − ω ), and that the rootsare λ = ζ k ω and 1, where 1 ≤ k < n and ζ means an n -th primitive root of unity in thealgebraic closure F p . Therefore, the proof comes down to the following two cases: Case I ω n = 1. Namely, the roots are distinct. Case II ω n = 1. Then, λ = 1 is a unique double root of the characteristic polynomial.We will calculate the weights coming from such X -colorings case by case. While thestatement (i) will be derived from Case I, those (ii) and (iii) will come from Case II.( Case I ) Let ω n = 1. We will study the solusions of a = a P m . We easily see that, if( ζ − k ω ) m = 1 for some k , then the solution is of the form a j +1 = a (cid:0) (1 − ζ kj ) / (1 − ζ ) (cid:1) + a (cid:0) ζ kj / (1 − ω ) (cid:1) + δ for some a, δ ∈ F p ; conversely, if the equation a = a P m has a non-trivial solution, thenthere is a unique k satisfying ( ζ − k ω ) m = 1 and 0 < k < n . It is further verified that such asolution gives rise to an X -coloring C if and only if a, δ, ζ are contained in F q . In assumary,we may assume that a, δ, ζ ∈ F q and ( ζ − ω ) m = 1 with ζ = ω . Indeed this assumptionjustifies a shadow coloring S of the form ( C ; 0). emark 5.5. We give a remark on this assumption. Notice that, for s ∈ Z , two equalities ω m = ζ m and ω s = 1 imply ζ ms = 1 and, hence, ζ s = 1, since m and n are coprime. Inparticular, considering special cases of s = q + q and s = q + q , we have ζ q + q = ζ q + q = 1.Similarly we notice that, if ω q + q = 1, then ζ q + q = 1.We will present the weights of [ S ] = [( C ; 0)], where C is the X -coloring as the solusionmentioned above. We then can easily check the color of every regions in the link-diagram.After a tedious calculation, the weight of x i,j is consequently given by (cid:16) aζ − i ω i − ζ j − − ζ + (1 − ω j − ) δ, aζ − i ω i ( 1 − ζ j − − ζ + ζ j − − ω ) + δ, aζ − i − ω i +1 − ω + δ (cid:17) ∈ C Q ( X ) . We next compute the pairing h Γ( q ) e , [ S ] i ∈ F q in turn. To begin with the case e = 1,recalling Γ( q ) e = U q U q + q U q = ( x − x ) q ( x − x ) q + q x q , we describe the paring as X i ≤ m, j ≤ n − ( aζ − i ω i ζ j − − ω − ω j − δ (cid:1) q (cid:0) aζ − i − ω i ( ω − ζ )( ζ j − − ω )(1 − ζ ) (cid:1) q + q (cid:0) aζ − i − ω i +1 − ω + δ (cid:1) q ∈ F q . Here we note P mi =1 ( ζ − ω ) si = 0 unless ζ − s ω s = 1. Therefore, several terms in this formulavanish by Remark 5.5 above. It is easily seen that the non-vanishing term in h Γ( q ) , [ S ] i is a q + q + q + q ( ζ − ω ) q + q ω q (1 − ω ) q + q + q + q (1 − ζ ) q + q X i ≤ m, j ≤ n − ( ζ − ω ) i ( q + q + q + q ) ζ jq (1 − ζ j ) q + q ∈ F q . (26)Here, by Remark 5.5 again, we notice two equalities( ζ − ω ) q + q + q + q = 1 , ζ jq (1 − ζ j ) q + q = ζ jq + ζ jq − . Therefore, since P n − j =1 ζ jq = P n − j =1 ζ jq = − 1, the sum in the formula (26) equals − nm .By (25), we hence obtain the required formula (23).Further, by Remark 4.2, the same calculations hold for the cases 3 ≤ e ≤ e = 2. For the shadow coloring S = ( C ; 0), we claim h Γ( q ) , [ S ] i = 0.To see this, by a similar calculation to (26), we reduce the paring h Γ( q ) , [ S ] i to h Γ( q ) , [ S ] i = − nma q + q + q + q · A q , where A q is temporarily defined by the formula( ζ − ω ) q + q ω q (1 − ζ ) q + q − ( ζ − ω ) q + q ω q (1 − ζ ) q + q + 1 − ω q + q − ω q (cid:0) ( ζ − ω ) q ω q + q (1 − ζ ) q − ( ζ − ω ) q ω q (1 − ζ ) q (cid:1) . We now assert that the last term in this formula A q is zero. Indeed, noting (1 − ζ ) − q = ζ q (1 − ζ ) − q ζ − q = ζ q ( ζ − − q by Remark 5.5, we easily have( ζ − ω ) q ω q + q (1 − ζ ) q − ( ζ − ω ) q ω q (1 − ζ ) q = ( ζ − ω ) q ω q + q + ( ζ − ω ) q ζ q ω q (1 − ζ ) q = 0 . Similarly we easily see an equality (1 − ζ ) − q − q = ζ q + q (1 − ζ ) − q − q ; therefore the firstand second terms in A q are canceled. Hence A q = 0 as claimed. In conclusion, the cocycleinvariants using Γ( q ) are trivial as desired. Case II ) We next consider another case of ω n = 1. Notice that the matrix P − E n is ofrank n − 1. Hence, if the above equation a = a P m has a non-trivial solution, then m mustbe divisible by p (consider the Jordan-block of P ). For such an m , we can verify that thesolution is of the form a j = aω − aω j + δ for some a, δ ∈ F q , which provides an X -coloring C . Put a shadow coloring of the form S = ( C ; 0). The weight of the crossing x i,j is thengiven by (cid:16) a (cid:0) − j )(1 − ω ) ω j − + ai ( ω − − ω j − )+( a + δ )(1 − ω j − ) , a (1 − ω j + iω − i )+ δ, ai ( ω − δ (cid:17) . Let us calculate the pairings h Γ( q ) e , [ S ] i . To begin, when e = 1, the h Γ( q ) , [ S ] i equals X i ≤ m, j ≤ n − (cid:0) ( aj ( ω − − ai ( ω − − δ ) ω j − (cid:1) q (cid:0) a − aω j (cid:1) q + q (cid:0) ai ( ω − 1) + δ (cid:1) q . (27)We here consider the sum on i . However, we notice that P i ≤ m i q + q = P i ≤ m i = m ( m +1)(2 m + 1) / 6. Hence, since m is divisible by p , the pairing vanishes unless p = 2 , e , the pairings are zero unless p = 2 , 3. Wetherefore may devote to the cases p = 2 , p = 3 and e = 1. Note that the non-vanishing term in (27) is only thecoefficients of P i q + q , and that P i ≤ m i q + q = − m/ 3. Then the pairing (27) is reduced tobe a q + q + q + q (1 − ω ) q + q X ≤ j ≤ n − ω q ( j − (1 − ω j ) q + q X ≤ i ≤ m i q + q = mn a q + q + q + q (1 − ω ) q + q , where P ω q ( j − (1 − ω j ) q + q = 2 nω − q in this equality follows from ω n = 1. Hence, byrunning over all shadow colorings, we obtain the required formula (24). Similarly, when p = 2 and e = 1, a calculation using Lemma 5.6(I) below can show the formula (24).Furthermore, the same calculation holds for the cases 3 ≤ e ≤ p = 2 , 3. Actually,it is done by changing the quadruple ( q , q , q , q ) in the previous calculation in Case 1, asa routine reason for the cases.At last, it is enough for the proof to work out the remaining case e = 2 and p = 2 , q ) , the pairing is reduced to h Γ( q ) , [ S ] i = h U q U q + q U q , [ S ] i − h U q U q + q U q , [ S ] i . (28)We claim that if p = 3, h Γ( q ) , [ S ] i = 0. The first term is reduced to 2 mna q + q + q + q (1 − ω ) q + q / 3, by a similar calculation to (24). The second term is obtained by changing theindices (1 , , , 4) in the first term to (2 , , , h Γ( q ) , [ S ] i vanishes.To complete the proofs, we let p = 2. The explicit formula of the first term in (28) followsfrom Lemma 5.6 (III) below. Furthermore, by the previous change of the indices, we knowthe second term. In summary, we conclude the desired formula in (iii).The following lemma used in the above proof can be obtained from the definitions andelementary calculations, although they are a little complicated. emma 5.6. Let S = ( C ; 0) be the shadow coloring in Case II as above.(I) If p = 2 and ω q + q = 1 , then the pairing h U q U q + q U q , [ S ] i is equal to (1 + ω ) q + q a q + q + q + q mn/ .(II) If ω q + q = 1 , and if p = 2 or , then h U q U q U q + q − U q + q U q U q , [ S ] i = 0 . (III) If p = 2 and ω q + q = 1 , then h U q U q + q U q , [ S ] i is equal to mn (cid:16) a q + q (cid:0) (1 + ω q ) a q δ q + (1 + ω q ) a q δ q (cid:1) + 1 + ω − q + ω − q + ω q + q ω q + q a q + q + q + q (cid:17) . Acknowledgment The author expresses his gratitude to Tomotada Ohtsuki and Michihisa Wakui for valuablecomments on group cohomologies and 3-manifolds. He is particularly grateful to YuichiKabaya for useful discussions and making several suggestions for improvement. References [AK] S. 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