Twisted cohomology pairings of knots III; triple cup products
aa r X i v : . [ m a t h . G T ] A ug Twisted cohomology pairings of knots III; triple cup products
Takefumi Nosaka
Abstract
Given a representation of a link group, we introduce a trilinear form, as a topological invariant. We showthat, if the link is either hyperbolic or a knot with malnormality, then the trilinear form equals the pairingof the (twisted) triple cup product and the fundamental relative 3-class. Further, we give some examplesof the computation.
Keywords
Cup product, Bilinear form, knot, twisted Alexander polynomial, group homology, quandle
Contents m, m )-torus link T m,m . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.3 Examples of Theorem 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 This paper examines topological invariants of trilinear forms, while the previous papers [N2]in this series discussed of bilinear forms. In general, bilinear form arising from Poincar´eduality is a powerful method, as in algebraic surgery theory and classification theorems ofsome manifolds. In contrast, there are not so many studies of trilinear forms. However,some 3-forms and trilinear cup products appear in 3-dimensional geometry together withtopological information (see, e.g., [CGO, M, L, S, Tur1]). For example, we mention interestingobservations from the Chern-Simons invariant (or φ -theory) of the form k π Z M tr( A ∧ dA + 23 A ∧ A ∧ A ) . We give the definition of trilinear pairing (see (1)) in a general situation where the coeffi-cients are arbitrary. Let Y be a compact 3-manifold with toroidal boundary with orientation -class [ Y, ∂Y ] ∈ H ( Y, ∂Y ; Z ) ∼ = Z . Choose a group homomorphism π ( Y ) → G , a right G -module M , and a G -invariant trilinear function ψ : M → A over a ring A . Then, we candefine the composite map H ( Y, ∂Y ; M ) ⊗ ⌣ −−−→ H ( Y, ∂Y ; M ⊗ ) h• , [ Y,∂Y ] i −−−−−−−−→ M ⊗ ψ, −−−→ A. (1)Here M is regarded as the local coefficient of Y via f , and the first map ⌣ is the cup product,and the second (resp. third) is defined by the pairing with [ Y, ∂Y ] (resp. ψ ). In contrast tothis definition, the 3-form (1) is considered to be something uncomputable. Actually, it seemshard to concretely deal with the 3-class [ Y, ∂Y ] and the cup products.This paper addresses the link case where Y is the 3-manifold which is obtained from the3-sphere by removing an open tubular neighborhood of a link L , i.e., Y = S \ νL. In fact, if L is a hyperbolic link, we obtain a diagrammatic method of computing the trilinear pairings.To be precise, in Section 2.2, starting from a link diagram, we define invariants of trilinearforms, and show (Theorem 2.5) that the invariant is equal to (1), if L is a hyperbolic link. Inaddition, we also show a similar theorem in the torus case (see Theorem 2.6). The point inthe theorem is that, in the computation, we do not need describing [ Y, ∂Y ] and cup products;thus, this computation is not so hard. In fact, we give some examples; see §
4. In addition,as an application (Theorem 3.1), when Y is a 3-fold covering space of S branched along ahyperbolic link L and M is a trivial coefficient, we give a diagrammatic computation of thetrilinear pairing (1).This paper is organized as follows. Section 2 formulates the trilinear forms in terms of thequandle cocycle invariants, and states the main theorems. Section 3 discusses a relation to3-fold branched coverings. Section 4 describes some computations. Section 5 gives the proofsof the theorems. Notation.
Every link L is smoothly embedded in the 3-sphere S with orientation. We write E L for the 3-manifold which is obtained from S by removing an open neighborhood of L . Our purpose in this section is to give a link invariant of trilinear form (Theorem 2.3), andto state the main results in § § G and a right G -module M over a ring A . We need some notation from [IIJO, N2] before proceeding. Denote M × G by X . Further,define a binary operation on X by ⊳ : ( M × G ) × ( M × G ) −→ M × G, ( a, g, b, h ) ( ( a − b ) · h + b, h − gh ) , (2)which was first introduced in [IIJO, Lemma 2.2], and satisfies “the quandle axiom”. Further-more, we choose a link L ⊂ S with a group homomorphism f : π ( S \ L ) → G . ext, we review colorings. Choose an oriented diagram D of L. Then, it follows from theWirtinger presentation of D that the homomorphism f is regarded as a map { arcs of D } → G .Furthermore, a map C : { arcs of D } → X is an X - coloring if it satisfies C ( α τ ) ⊳ C ( β τ ) = C ( γ τ )at each crossings of D illustrated as Figure 1. It is worth noticing that the set of all coloringsis regarded as a subset of the direct product X α D , where α D is the number of arcs of D . LetCol X ( D f ) denote the set of all X -colorings over f , that is,Col X ( D f ) := { C ∈ ( M × G ) α D | C is an X -coloring , p G ◦ C = f } , (3)where p G is the projection X = M × G → G . Then, we can easily verify from the linearoperation (2) that Col X ( D f ) is made into an abelian subgroup of M α ( D ) , and that the diagonalsubset M diag ⊂ M α D is a direct summand in Col X ( D f ). Denoting another summand byCol red X ( D f ), we have a decomposition Col X ( D f ) ∼ = Col red X ( D f ) ⊕ M diag . The previous paper [N2] gave a topological meaning of the coloring sets as follows:
Theorem 2.1 ([N2]) . Let E L be a link complement in S as in §
1. Regard the G -module M as a local system of E L via f : π ( E L ) → G . Then, there are isomorphisms Col X ( D f ) ∼ = H ( E L , ∂E L ; M ) ⊕ M, Col red X ( D f ) ∼ = H ( E L , ∂E L ; M ) . (4)Furthermore, let us review shadow colorings [CKS, IIJO]. A shadow coloring is a pair ofa coloring C over f and a map λ from the complementary regions of D to M , satisfying thecondition depicted in the right side of Figure 1 for every arcs. Let SCol X ( D f ) denote the set ofshadow colorings of D such that the unbounded exterior region is assigned by 0 ∈ M . Noticethat, by the coloring rules, assignments of the other regions are uniquely determined from theunbounded region, and admit, therefore, a shadow coloring; we thus obtain a bijectionCol X ( D f ) ≃ SCol X ( D f ) . (5) α τ β τ γ τ C ( α τ ) ⊳ C ( β τ ) = C ( γ τ ) δ RR ′ λ ( R ′ ) = ( λ ( R ) − b ) · h + b ,where C ( δ ) = ( b, h ) . Figure 1: The coloring conditions at each crossing τ and around each arcs. In addition, we will explain Definition 2.2 below, and show Theorem 2.3.For this, we need two things: first, we take three G -modules M , M , M and the associated X i = M i × G. Let A be an abelian group. On the other hand, we prepare a trilinear map ψ : M × M × M → A over Z satisfying the G -invariance, that is, ψ ( a · g, a · g, a · g ) = ψ ( a , a , a ) , (6)holds for any a i ∈ M i and g ∈ G . ext, let us consider the map X × X × X → A by the formula (cid:0) ( b , g ) , ( b , g ) , ( b , g ) (cid:1) ψ (cid:0) ( b − b ) · (1 − g ) , b − b , b − b · g − (cid:1) , (7)for a i ∈ M i and g , g , g ∈ G . This map was first defined in [N1, Corollary 4.6]. Furthermore,given three shadow colorings S i ∈ SCol X i ( D f ) with i ≤ τ of D , we canfind assignments as illustrated in Figure 2. Inspired by the formula (7), we define a weight of τ to be W ψ,τ ( S , S , S ) := ψ (cid:0) ( a − b )(1 − g ǫ τ ) , b − c , c − c · h − (cid:1) ∈ A, where ǫ τ ∈ {± } is the sign of τ.a a a ( b , g ) ( c , h ) ( b , g ) ( c , h ) ( b , g ) ( c , h ) ∈ M × G Figure 2: Colors around a crossing with respect to three shadow colorings.
Definition 2.2.
Given a G -invariant trilinear map ψ : M × M × M → A , we define atrilinear map T ψ : Y i =1 SCol X i ( D f ) −→ A ; ( S , S , S ) X τ W ψ,τ ( S , S , S ) , where τ runs over all the crossings of D .The point is that, given a diagram D , we can diagrammatically deal with the trilinear T ψ by definitions; see § § T ψ up to trilinear equivalence: Theorem 2.3.
Let two diagrams D and D ′ differ by a Reidemeister move. There is a canonicalisomorphism B i : SCol X i ( D f ) ≃ SCol X i ( D ′ f ) , for which the equality T ψ = T ′ ψ ◦ ( B ⊗ B ⊗ B ) holds as a map.In particular, the equivalence class of the trilinear map T ψ depends on only the homomor-phism f : π ( S \ L ) → G and the input data ( M , M , M , ψ ) .Proof. We first focus on Reidemeister move of type III; see Figure 3. Then, considering thecorrespondence in Figure 3 with x i , y i , z i ∈ X i , we have the bijection B i . Moreover, we supposethat the left region is colored by r i ∈ M . Thus, it is enough to show the desired equality. Forthis, take a i , b i , c i ∈ M i and g, h, k ∈ G such that x i = ( a i , g ) , y i = ( b i , h ) , z i = ( c i , k ) ∈ X i .Then, the sum from the left side is, by definition and examining the figure, computed as ψ (cid:0) ( r − a )(1 − g ) , a − c , c (1 − k − ) (cid:1) + ψ (cid:0) ( r g − a g + a − b )(1 − h ) , b − c , c (1 − k − ) (cid:1) + ψ (cid:0) ( r − a ) k (1 − k − gk ) , ( a − b ) k, ( b k − c k + c )(1 − k − h − k ) (cid:1) . On the other hand, the sum from the right side is formulated as ψ (cid:0) ( r − a )(1 − g ) , a − b , b (1 − h − ) (cid:1) + ψ (cid:0) ( r − b )(1 − h ) , b − c , c (1 − k − ) (cid:1) ψ (cid:0) ( r − a ) h (1 − h − gh ) , ( b − a ) h + a − c , c (1 − k − ) (cid:1) . Then, an elementary calculation from (6) can show the two sums are equal. However, sincethe calculation is a little tedious, we omit the detail.Finally, the required equality concerning Reidemeister moves of type I immediately followsfrom ψ (0 , y, z ) = 0, and the invariance of type II is clear by a similar discussion. r i r i x i y i z i x i y i z i z i y i ⊳ z i ( x i ⊳ z i ) ⊳ ( y i ⊳ z i ) z i y i ⊳ z i ( x i ⊳ y i ) ⊳ z i ←→ Figure 3: The 1:1-correspondence associated with a Reidemeister move of type III.
Remark 2.4.
In this way, the construction for trilinear forms is applicable to not only tamelinks in S , but also handlebody-knots H g in S . In fact, as a similar discussion to [IIJO], wecan easily check that the trilinear form is invariant with respect to the diagrammatic movesof handlebody-knots; see [IIJO, Figures 1 and 2] for the moves. As mentioned in the introduction, we will show (Theorems 2.5 and 2.6) that the trilinear formsof some links are equal to the trilinear pairings (The proofs of the theorems appear in § Theorem 2.5.
Let M , M , M be G -modules as in Definition 2.2. Furthermore, choose afundamental class [ E L , ∂E L ] in H ( E L , ∂E L ; Z ) ∼ = Z .We assume that L is either a hyperbolic link or a prime knot which is neither a cable knotnor a torus knot. Then, via the identification (4) , the trilinear form T ψ is equal to the followingcomposite map: O i : 1 ≤ i ≤ H ( E L , ∂E L ; M i ) ⌣ −−−→ H ( E L , ∂E L ; M ⊗ M ⊗ M ) ψ ◦h• , [ E L ,∂E L ] i −−−−−−−−−−→ A. (8)In addition, we mention the torus knot, although we need a condition. More precisely, Theorem 2.6.
Let M , M , M , ψ , and [ E L , ∂E L ] be as above.Assume that L is the ( m, n ) -torus knot. Then, the trilinear form T ψ is equal to the composite (8) modulo the integer nm ∈ Z . As a concluding remark, while the triple cup product of a link often is considered tobe speculative and uncomputable, it become computable from only a link diagram withoutdescribing [ E L , ∂E L ] and any triangulation in S \ L . emark 2.7. Finally, we compare the trilinear forms in Definition 2.2 with the existing resultson “the quandle cocycle invariants”, in detail. Briefly speaking, the link invariant in [CKS]is constructed from a quandle X and a map Φ : X → A which satisfy “the quandle cocyclecondition”, and is defined to be a certain map J Φ : SCol X ( D ) → A . Then, we note that ourtrilinear form is a trilinearization of the quandle cocycle invariants with respect to quandles ofthe form X = M × G . To be precise, if M = M = M = M , we can see that the associatedinvariant J Φ : SCol X ( D ) → A is equal to the composite T ψ ◦ ( △ × id) ◦ △ by definitions. Inconclusion, the theorems also suggest topological meanings of the quandle cocycle invariantswith X = M × G . Although we considered relative cohomology, for n ∈ Z ≥ and a closed 3-manifold N , let usconsider the triple cup product H ( N ; Z /n Z ) ⊗ ⌣ −−−→ H ( N ; Z /n Z ) h• , [ N,∂N ] i −−−−−−−→ Z /n, (9)where the coefficient module Z /n is trivial. Although there are studies of this map (see, e.g.,[S, CGO, Tur1]), there are few examples of the computation. As an application of the theoremsabove, this section gives a recover of the triple cup products of N , when N is a 3-fold cycliccovering of S branched over a link.To state Theorem 3.1, we need some terminology. Let G be Z / h t | t = 1 i . Consider theepimorphism f : π ( S \ L ) → G which sends every meridian to t , and the associated 3-foldcyclic branched covering e C L → S . Theorem 3.1.
Let M , M , and M be Z [ t ± ] / ( n, t + t +1) . Let p : Z [ t ± ] / ( n, t + t +1) → Z /n be the map which sends a + tb to a . Set up the map ψ : M → Z /n which takes ( x, y, z ) to xyz . As in Theorem 2.5, assume that L is either a hyperbolic link or a prime knot which isneither a cable knot nor a torus knot.Then, there is an isomorphism Col red X i ( D f ) ∼ = H ( e C L ; Z /n Z ) such that the trilinear map T p ◦ ψ is equivalent to (9) with N = e C L .Proof of Theorem 3.1. We first show the isomorphism Col red X i ( D f ) ∼ = H ( e C L ; Z /n Z ). Let R bethe ring Z [ t ] / ( n, t + t + 1). By Theorem 2.1, we have Col red X i ( D f ) ∼ = H ( E L ; ∂E L ; M ). Noticethat H i ( ∂E L ; M ) is annihilated by 1 − t . Since 1 − t and 1 + t + t are coprime, we have H ( E L ; ∂E L ; M ) ∼ = H ( E L ; M ) ∼ = Hom R -mod ( H ( E L , M ) , R ) . (10)Let e E L → E L = S \ L be the 3-fold covering. Then, by Shapiro’s Lemma (see, e.g. [Bro]),the canonical inclusion ι : Z /n → R yields the isomorphisms: H ∗ ( e E L : Z /n ) ∼ = H ∗ ( E L ; Z [ t ] / ( n, t − ∼ = H ∗ ( E L ; R ) ⊕ H ∗ ( E L ; Z [ t ] / ( n, t − . (11)Here, the second isomorphism is obtained from the ring isomorphism Z [ t ] / ( n, t − ∼ = R ⊕ Z [ t ] / ( n, t − i : e E L ֒ → e C L be the inclusion. According to [Kaw, Theorem 5.5.1], the ho-mology H ( e C L ; Z ) is annihilated by 1+ t + t , and the induced map i ∗ : H ( e E L ; Z ) → H ( e C L ; Z ) s a splitting surjection. Thus, dually, the induced map i ∗ : H ( e C L ; Z /n ) → H ( e E L ; Z /n ) isinjective and the image is isomorphic to H ( E L ; R ). In summary, we obtained the desiredisomorphism.We will complete the proof. By (11), we have a splitting injection S : H ∗ ( e E L , ∂ e E L ; R ) → H ∗ ( E L , ∂E L ; Z /n ). Take the canonical maps j : ( e E L , ∂ e E L ) → ( e C L , e C L \ e E L ) and k : ( e C L , ∅ ) → ( e C L , e C L \ e E L ). Then, we have the commutative diagrams on the cup products: H ( E L , ∂E L ; M ) ⊗ S (cid:15) (cid:15) ψ ◦ ⌣ / / H ( E L , ∂E L ; M ) S (cid:15) (cid:15) h• , [ E L ,∂E L ] i / / RH ( e E L , ∂ e E L ; Z /n ) ⊗ ⌣ / / H ( e E L , ∂ e E L ; Z /n ) h• , [ e E L ,∂ e E L ] i / / Z /n ι O O H ( e C L , e C L \ e E L ; Z /n ) ⊗ ⌣ / / ∼ = j ∗ O O k ∗ (cid:15) (cid:15) H ( e C L , e C L \ e E L ; Z /n ) ∼ = j ∗ O O k ∗ (cid:15) (cid:15) h• , [ e C L , e C L \ e E L ] i / / Z /nH ( e C L ; Z /n ) ⊗ ⌣ / / H ( e C L ; Z /n ) h• , [ e C L ] i / / Z /n. Here, the vertical maps j ∗ are isomorphisms by the excision axiom. Moreover, by the discussionin the above paragraph, the composite k ∗ ◦ ( j ∗ ) − ◦ S is an isomorphism from H ( E L , ∂E L ; M ).Hence, since p ◦ ι : Z /n → Z /n is an isomorphism, the following two composites are equivalent: p ◦ ψ ◦ h• , [ E L , ∂E L ] i◦ ⌣, h• , [ e C L ] i◦ ⌣ . By Theorem 2.5, the left hand side is equal to the trilinear map T p ◦ ψ . Hence, T p ◦ ψ is equivalentto (9) with N = e C L as desired. α β γ α α α α α α m α i ......Figure 4: The trefoil knot and the figure eight knot We will compute the trilinear forms T ψ associated with some homomorphisms f : π L → G ,where L is either the trefoil knot or the figure eight knot.As a simple example, we will focus on the the trefoil knot 3 . Let D be the diagram of K as illustrated in Figure 4. Note the Wirtinger presentation π L ∼ = h α, β | αβα = βαβ i . Then,we can easily see that a correspondence C : { α, β, γ } → X with C ( α ) = ( a i , g ) , C ( β ) = ( b i , g ) , C ( α ) = ( c i , g ) ∈ M i × G s an X -coloring C over f : π L → G , if and only if it satisfies the four equations g = f ( α ) , h = f ( β ) , ghg = hgh, (12) c i = a i · h + b i · (1 − h ) , (13)( a i − b i ) · (1 − g + hg ) = ( a i − b i ) · (1 − h + gh ) = 0 . (14)Furthermore, given a G -invariant linear form ψ , the sum T ψ is equal to ψ (cid:0) − a · (1 − g ) , a − b , a · (1 − h − ) (cid:1) + ψ (cid:0) − b · (1 − h ) , b − c , c · (1 − h − g − h ) (cid:1) + ψ (cid:0) − c · (1 − h − gh ) , c − a , a · (1 − g − ) (cid:1) , by definition. Then, by canceling out c i by using (13) and (12), we can easily obtain theresulting computation: for (( a i , g ) , ( b i , h )) ∈ SCol X i ( D f ) ⊂ M i , T ψ (cid:0) ( a , b ) , ( a , b ) , ( a , b ) (cid:1) = ψ (cid:0) ( a − b ) g − , ( a − b ) · h, a − b (cid:1) ∈ A. (15)Next, we will compute T ψ of the figure eight knot. However, the computation can be donein a similar way to the trefoil case. So we only describe the outline.Let D be the diagram with arcs as illustrated in Figure 4. Similarly, we can see that acorrespondence C : { α , α , α , α } → X with C ( α i ) = ( x i , z i ) ∈ M i × G is an X -coloring C over f : π L → G , if and only if it satisfies the following equations: z i = f ( α i ) , z − z z = z − z − z z z − z z ∈ G, (16) x = ( x − x ) · z + x , x = ( x − x ) · z + x , (17)( x − x ) · ( z + z −
1) = ( x − x ) · (1 − z − ) z z = ( x − x ) · (1 − z − ) z z ∈ M. (18)Accordingly, it follows from (17) that the set Col X ( D f ) is generated by x , x .Given a G -invariant trilinear form ψ , it can be seen that the trilinear form T ψ (cid:0) ( x , x ) , ( x ′ , x ′ ) , ( x ′′ , x ′′ ) (cid:1) is expressed as ψ (( x − x ) · z z − , x ′ − x ′ , ( x ′′ − x ′′ ) · (1 − z − ))+ ψ (( x − x ) · z − z , ( x ′ − x ′ ) · (1 − z ) , ( x ′′ − x ′′ ) · (1 − z − ) z ) . Remark 4.1.
Here, we should give some examples from concrete M and ψ. In particular,the author attempted to get non-trivial trilinear form T ψ when G is a Lie group and M is arepresentation of G . However, even if G = SL ( R ) or G = SL ( C ) and M = C or C , theauthor computed the resulting T ψ equal to zero. Indeed, the author could not find non-trivialexamples of T ψ except those in § T ψ from representations with respectto Lie groups. .2 The ( m, m ) -torus link T m,m We also calculate the trilinear form T ψ concerning the ( m, m )-torus link, following from Def-inition 2.2. These calculations will be useful in the paper [N4], which suggests invariants of“Hurewitz equivalence classes”.Let L be the ( m, m )-torus link T m,m with m ≥
2, and let α , . . . , α m be the arcs depicted inFigure 4. Furthermore, let us identity α i + m with α i of period m . By Wirtinger presentation,we have a presentation of π L as h a , . . . , a m | a · · · a m = a m a a · · · a m − = a m − a m a · · · a m − = · · · = a · · · a m a i . Given a homomorphism f : π L → G with f ( α i ) ∈ G , let us discuss X -colorings C over f .Then, concerning the coloring condition on the ℓ -th link component, it satisfies the equation (cid:0) · · · ( C ( α ℓ ) ⊳ C ( α ℓ +1 )) ⊳ · · · (cid:1) ⊳ C ( α ℓ + m − ) = C ( α ℓ ) , for any 1 ≤ ℓ ≤ m. (19)With notation C ( α i ) := ( x i , z i ) ∈ X , this equation (19) reduces to a system of linear equations( x ℓ − − x ℓ ) + X ℓ ≤ j ≤ ℓ + m − ( x j − x j +1 ) · z j +1 z j +2 · · · z m + ℓ = 0 ∈ M, for any 1 ≤ ℓ ≤ m. (20)Conversely, we can easily verify that, if a map C : { arcs of D } → X satisfies the equation (20),then C is an X -coloring. Denoting the left side in (20) by Γ f,k ( ~x ), consider a homomorphismΓ f : M m −→ M m ; ( x , . . . , x m ) (Γ f, ( ~x ) , . . . , Γ f,m ( ~x )) . To conclude, the set Col X ( D f ) coincides with the kernel of Γ f .Next, we precisely formulate the resulting trilinear form. Proposition 4.2.
Let f : π ( S \ T m,m ) → G be as above. Let ψ : M → A be a G -invariant linear functions. Then, the trilinear form T ψ : Ker(Γ f ) ⊗ → A sends ( w , . . . , w m ) ⊗ ( x , . . . , x m ) ⊗ ( y , . . . , y m ) to m X ℓ =1 m − X k =1 ψ (cid:0) w ℓ · (1 − z ℓ )ˆ z ℓ +1; ℓ + k − , k X j =1 ( x j + ℓ − − x j + ℓ ) · ˆ z j + ℓ ; k + ℓ − , y k + ℓ · (1 − z − k + ℓ ) (cid:1) . (21) Here, for s ≤ t , we use notation ˆ z s ; t := z s z s +1 · · · z t and ˆ z s +1; s := 1 ∈ G. The formula is obtained by direct calculation and definitions.
We will give some examples in Theorem 3.1. Thus, we should suppose the situation of Theorem3.1 as follows. Let G = Z / h t | t = 1 i , and f : π ( S \ L ) → Z / t . Furthermore, take M i = A = Z [ t ] / ( n, t + t + 1) for some n ∈ Z ≥ , let ψ : M × M × M → A send ( x, y, z ) to xyz. In this paragraph, we focus on only knots, K , such that H ( E K , ∂E K ; A ) ∼ = H ( e B K ; Z /n )is isomorphic to either A or 0. We will write the trilinear map T ψ as a cubic polynomial withrespect to a, b, c ∈ (cid:0) H ( E K , ∂E K ; Z /n ) (cid:1) . Then, we give the resulting computation of T ψ ,when K is a prime knot with crossing number <
7. The list of the computation is as follows: n T ψ abc abc t ) abc any 06 any 06 any 0 We will complete the proofs of Theorems 2.5–2.6 in § § We will spell out the relative group (co)homology in the non-homogeneous terms. Throughoutthis subsection, we fix a group Γ and a homomorphism f : Γ → G . Then, we have the actionof Γ on the right G -module M via f .Let C n gr (Γ; M ) be Map(Γ n , M ). For φ ∈ C n gr (Γ; M ), define the coboundary ∂ n ( φ ) ∈ C n +1gr (Γ; M )by the formula ∂ n ( φ )( g , . . . , g n +1 ) = φ ( g , . . . , g n +1 ) + X ≤ i ≤ n ( − i φ ( g , . . . , g i − , g i g i +1 , g i +2 , . . . , g n +1 ) + ( − n φ ( g , . . . , g n ) g n +1 . Furthermore, we set subgroups K j and the inclusions ι j : K j ֒ → Γ, where the index j runsover 1 ≤ j ≤ m . Then, we can define the mapping cone of ι j : More precisely, C n (Γ , K J ; M ) := Map(Γ n , M ) ⊕ (cid:0)M j Map(( K j ) n − , M ) (cid:1) . For ( h, k , . . . , k m ) ∈ C n (Γ , K J ; M ), let us define ∂ n ( h, k , . . . , k m ) in C n +1 (Γ , K J ; M ) by ∂ n (cid:0) h, k , . . . , k m (cid:1) ( a, b , . . . , b m ) = (cid:0) ∂ n h ( a ) , h ( b ) − ∂ n − k ( b ) , . . . , h ( b m ) − ∂ n − k m ( b m ) (cid:1) , where ( a, b , . . . , b m ) ∈ Γ n +1 × K n × · · · × K nm . Then we have a complex ( C ∗ (Γ , K J ; M ) , ∂ ∗ ),and can define the cohomology.We now observe the submodule consisting of 1-cocycles Z (Γ , K J ; M ). Let us define thesemi-direct product M ⋊ G by( a, g ) ⋆ ( a ′ , g ′ ) := ( a · g ′ + a ′ , gg ′ ) , for a, a ′ ∈ M, g, g ′ ∈ G. Let Hom f (Γ , M ⋊ G ) be the set of group homomorphisms Γ → M ⋊ G over the homomorphism f . Consider a map Z (Γ , K J ; M ) → Hom f (Γ , M ⋊ G ) ⊕ M m ; ( h, y , . . . , y m ) ( γ ( h ( γ ) , f ( γ )) , y , . . . , y m ) . emma 5.1 ([N2, Lemma 5.2]) . This map gives an isomorphism between Z (Γ , K J ; M ) andthe following set: (cid:8) ( e f , y , . . . , y m ) ∈ Hom f (Γ , M ⋊ G ) ⊕ M m (cid:12)(cid:12) e f ( h j ) = ( y j − y j · h j , f j ( h j )) , for any h j ∈ K j (cid:9) . Moreover, the image of ∂ , i.e., B (Γ , K J ; M ) , is equal to the subset { ( e f a , a, . . . , a ) } a ∈ M . Here, for a ∈ M, the map e f a : Γ → M ⋊ G is defined as a homomorphism which sends γ to ( a − a · γ, f ( γ )) . In particular, if K J is non-empty, B (Γ , K J ; M ) is a direct summand of Z (Γ , K J ; M ) . Furthermore, we review the cup product. When K J is the empty set, the product of u ∈ C p (Γ; M ) and v ∈ C q (Γ; M ′ ) is defined to be u ⌣ v ∈ C p + q (Γ; M ⊗ M ′ ) given by( u ⌣ v )( g , . . . , g p + q ) := ( − pq (cid:0) u ( g , . . . , g p ) g p +1 · · · g p + q (cid:1) ⊗ v ( g p +1 , . . . , g p + q ) . (22)Furthermore, if K J is not empty, for two elements ( f, k , . . . , k m ) ∈ C p (Γ , K J ; M ) and( f ′ , k ′ , . . . , k ′ m ) ∈ C q (Γ , K J ; M ′ ), let us define the cup product to be the formula( f ⌣ f ′ , k ⌣ f ′ , . . . , k m ⌣ f ′ ) ∈ C p + q (Γ , K J ; M ⊗ M ′ ) . This formula yields a bilinear map, by passage to cohomology.Finally, we observe another complex. Consider the module of the form C n red (Γ) := { ( c , . . . , c m ) ∈ Map( Z [Γ n ] , M ) m | c + c + · · · + c m = 0 ∈ Map( Z [Γ n ] , M ) } . Then, this complex canonically has an inclusion into the direct sum of C n (Γ , K j ): P n : C n red (Γ) −→ M j : 1 ≤ j ≤ m C n (Γ , K j ) . Then, we define a quotient complex, D n (Γ , K J ; M ), to be the cokernel of P n . Then, C n (Γ , K J ; M )is isomorphic to D n (Γ , K J ; M ), since the kernel of the inclusions ⊕ mj =1 C n (Γ , K j ) → C n (Γ , K )is the image of P n . Remark 5.2.
We give a natural relation to usual cohomology. Take the Eilenberg-MacLanespaces of Γ and of K j , and consider the map ( ι j ) ∗ : K ( K j , → K (Γ ,
1) induced by theinclusions. Then the relative homology H n (Γ , K J ; M ) is isomorphic to the homology of themapping cone of ⊔ j K ( K j , → K (Γ ,
1) with local coefficients. Further, the cup product ⌣ above coincides with that on the singular cohomology groups.We mention the case where L is either a knot or a hyperbolic link. Then, the complement S \ L is known to be an Eilenberg-MacLane space. Since we only use Γ as π ( S \ L ) in thispaper, we may discuss only the relative group cohomology. Throughout this section, we denote π ( S \ L ) by π L , and the union of the fundamental groupsof the boundaries of S \ L by ∂π L , for brevity. Let m = L , and choose a diagram D. heorem 5.3 ([N2, Theorem 2.2]) . Let X be M × G , as mentioned in (2) . Let κ : X → M ⋊ G be a map which sends ( m, g ) to ( m − mg, g ) . Given an X -coloring C over f , consider a map { arcs of D } → M ⋊ G which assigns α to κ (cid:0) C ( α ) (cid:1) . This assignment yields isomorphisms Col X ( D f ) ∼ = Z ( π L , ∂π L ; M ) , Col red X ( D f ) ∼ = H ( π L , ∂π L ; M ) . Next, we explain Theorem 5.4. Choose a relative 1-cocycle ˜ f : π L → M ⋊ π L with y , . . . , y m .We define the subgroup K ℓ to be { ( y ℓ − y ℓ m aℓ l bℓ , m aℓ l bℓ ) ∈ M ⋊ π L | a, b ∈ Z } . Furthermore, given a G -invariant trilinear map ψ : M → A , consider the map θ ℓ : ( M ⋊ π ( S \ L )) −→ A ;(( a, g ) , ( b, h ) , ( c, k )) ψ (( a + y ℓ − y ℓ g ) · hk, ( b + y ℓ − y ℓ h ) · k, c + y ℓ − y ℓ k ) . (23)Then, we can easily check that each θ ℓ is a 3-cocycle in C ( M ⋊ π L ; A ). Then, the collectionΨ := ( θ , . . . , θ L ) represents a relative 3-cocycle in D ( M ⋊ π L , K ; A ). Proposition 5.4 ([N5, Proposition 6.7]) . Under the notation above, fix a shadow coloring S ˜ f corresponding the relative 1-cocycle ( ˜ f , y , . . . , y L ) . If L is either a hyperbolic link or a prime knot which is neither a cable knot nor a torusknot, as in Theorem 2.5, then the diagonal restriction of T ψ is equal to the pairing of the3-class [ E L , ∂E L ] and the above 3-cocycle Ψ . To be precise, T ψ ( S ˜ f , S ˜ f , S ˜ f ) = ψ h Ψ , ˜ f ∗ [ E L , ∂E L ] i . (24) Furthermore, if L is the ( m, n ) -torus knot, the same equality (24) holds modulo mn. Proof of Theorem 2.5.
First, we observe (25) below. Consider a 0-cochain ~y := ( y , . . . , y L ) ∈ D ( M ⋊ π L , M ). Then, e f − ∂ ~y is represented by another 1-cocycle C ′ := (( e f − ¯ y , . . . , e f − ¯ y L ) , (0 , . . . , ∈ D ( M ⋊ π L , M ) , where ¯ y ℓ means a map π L → M which takes g to y − y g . Then, the 3-cocycle Ψ explained in(23) is equal to the cup product C ′ ⌣ C ′ ⌣ C ′ , by definition. Hence, Proposition 5.4 implies T ψ ( S ˜ f , S ˜ f , S ˜ f ) = ψ hC ′ ⌣ C ′ ⌣ C ′ , [ E L , ∂E L ] i = ψ hC ⌣ C ⌣ C , [ E L , ∂E L ] i . (25)Finally, we will deal with non-diagonal parts, and complete the proof. Here, we define M to be the direct product M × M × M , and consider the j -th inclusion ι j : M j −→ M = M × M × M ; x ( δ j x, δ j x, δ j x ) . Thus, we can decompose S ˜ f as ( S , S , S ) ∈ Col X ( D f ) × Col X ( D f ) × Col X ( D f ) componen-twise. In addition, we define a G -invariant trilinear form ψ : M × M × M −→ A ; (cid:0) ( a, b, c ) , ( d, e, f ) , ( g, h, i ) (cid:1) ψ ( a, e, f ) . hen, the transformation of the coefficients ι × ι × ι yields a diagram Q i =1 H ( E L , ∂E L ; M i ) ⌣ / / ( ι × ι × ι ) ∗ (cid:15) (cid:15) H ( E L , ∂E L ; M × M × M ) ψ ◦h• , [ E L ,∂E L ] i / / ( ι × ι × ι ) ∗ (cid:15) (cid:15) AH ( E L , ∂E L ; M ) ⌣ ∆ / / H ( E L , ∂E L ; M × M × M ) ψ ◦h• , [ E L ,∂E L ] i / / A. Here, the left bottom ⌣ ∆ is defined by a a ⌣ a ⌣ a . Then, we can verify the commutativityby definitions. By Proposition 5.4, the bottom arrow is equal to the left hand side in (24).Hence, the pullback to Q i =1 H ( E L , ∂E L ; M i ) is equal to the trilinear T ψ as desired. Proof of Theorem 2.6.
Let L be the ( m, n )-torus knot. According to the latter part in Theorem5.4, we need discussions modulo mn . However, the proof runs well in the same manner. Acknowledgments
The work is partially supported by JSPS KAKENHI Grant Number 17K05257.
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