Cocycles of G -Alexander biquandles and G -Alexander multiple conjugation biquandles
Atsushi Ishii, Masahide Iwakiri, Seiichi Kamada, Jieon Kim, Shosaku Matsuzaki, Kanako Oshiro
aa r X i v : . [ m a t h . A T ] F e b Cocycles of G -Alexander biquandles and G -Alexander multiple conjugation biquandles Atsushi Ishii, Masahide Iwakiri, Seiichi KamadaJieon Kim, Shosaku Matsuzaki, and Kanako OshiroFebruary 27, 2020
Abstract
Biquandles and multiple conjugation biquandles are algebras which arerelated to links and handlebody-links in 3-space. Cocycles of them can beused to construct state-sum type invariants of links and handlebody-links.In this paper we discuss cocycles of a certain class of biquandles and mul-tiple conjugation biquandles, which we call G -Alexander biquandles and G -Alexander multiple conjugation biquandles, with a relationship withgroup cocycles. We give a method to obtain a (biquandle or multipleconjugation biquandle) cocycle of them from a group cocycle. A quandle is an algebra related to knots and links (cf. [6, 16, 20]). Usinga (quandle) 2-cocycle or 3-cocycle in the (co)homology theory of a quandle(cf. [3, 7, 8]), we can construct an invariant of knots and links. A multipleconjugation quandle is introduced in [11]. It is an algebra motivated by studyof handlebody-links, which are isotopy classes of embedded handlebodies in 3-space. Another notion called a qualgebra was introduced in [18, 19] which isrelated to trivalent graphs in 3-space.The notions of a quandle and a multiple conjugation quandle are generalizedto biquandles (cf. [1, 4, 5, 17]) and multiple conjugation biquandles (cf. [13, 15]),although the terminologies and definitions of such generalizations vary in theliterature. Homology theory on biquandles is found in [1, 8] and homologytheory on multiple conjugation biquandles is found in [14]. In [14] we discussedthe (co)homology theory of multiple conjugation biquandles and we provided amethod of constructing an invariant of handlebody-links from a 2 or 3-cocycle.Thus, in order to construct such an invariant of links or handlebody-links, it isimportant to find a cocycle of a biquandle or a multiple conjugation biquandle.In this paper, we focus on a certain class of biquandles and multiple conjugationbiquandles which we call G -Alexander biquandles and G -Alexander multipleconjugation biquandles.The quandles associated to G -families of Alexander quandles [12] are animportant class of quandles. In [21], Nosaka studied cocycles of a G -family M of Alexander quandles, for a group G and a G -module M , and gave a methodof obtaining a cocycle of the associated quandle M × G . For a given G -invariantgroup cocycle of the G -family M , we can construct a cocycle of the quandle1 × G by using invariant theory. Our approach in this paper is an analogue ofNosaka’s argument.The paper is organized as follows. In Section 2, we review definitions relatedto the (co)homology of biquandles, multiple conjugation biquandles and groups.We also give the definition of G -Alexander biquandles and G -Alexander multipleconjugation biquandles. In Sections 3 and 4 we focus on G -Alexander biquandles X = M × G and cocycles of them. The case where the biquandle X has the trivial X -set (or has X itself as an X -set, resp.) is discussed in Section 3 (or 4). Weshow how to obtain a (birack) cocycle of the biquandle X = M × G from a groupcocycle: Theorems 3.4 and 3.5 (Theorems 4.4 and 4.5). In Sections 5 and 6 wefocus on G -Alexander multiple conjugation biquandles X = F m ∈ M ( { m } × G ) = M × G and cocycles of them. The case where the biquandle X has the trivial X -set (or has X itself as an X -set, resp.) is discussed in Section 5 (or 6). We showhow to obtain a (multiple conjugation biquandle) cocycle of the G -Alexandermultiple conjugation biquandle from a group cocycle: Theorems 5.7 and 5.8(Theorems 6.8, 6.10 and 6.11). Since any G -invariant multilinear map providesa G -invariant group cocycle, we can obtain many cocycles.This research is supported by JSPS KAKENHI Grant Numbers 16K17600,18K03292 and 19H01788. It is also supported by Young Researchers Programthrough the National Research Foundation of Korea (NRF) funded by the Min-istry of Education, Science and Technology (NRF-2018R1C1B6007021). In this section, we review definitions of biquandles and multiple conjugation bi-quandles, and the (co)homology of those and groups. In Subsection 2.2, we givethe definition of G -Alexander biquandles and G -Alexander multiple conjugationbiquandles. Definition 2.1 ([7, 17]) . A biquandle is a non-empty set X with binary oper-ations ∗ , ∗ : X × X → X satisfying the following axioms.(B1) For any x ∈ X , x ∗ x = x ∗ x .(B2) For any a ∈ X , the map ∗ a : X → X sending x to x ∗ a is bijective.For any a ∈ X , the map ∗ a : X → X sending x to x ∗ a is bijective.The map S : X × X → X × X defined by S ( x, y ) = ( y ∗ x, x ∗ y ) is bijective.(B3) For any x, y, z ∈ X ,( x ∗ y ) ∗ ( z ∗ y ) = ( x ∗ z ) ∗ ( y ∗ z ) , ( x ∗ y ) ∗ ( z ∗ y ) = ( x ∗ z ) ∗ ( y ∗ z ) , ( x ∗ y ) ∗ ( z ∗ y ) = ( x ∗ z ) ∗ ( y ∗ z ) . A birack is a non-empty set X with binary operations ∗ , ∗ : X × X → X satisfying (B2) and (B3). 2 efinition 2.2. For a biquandle (or a birack) X , an X -set is a non-empty set Y with a map ∗ : Y × X → Y satisfying the following axioms. • For any y ∈ Y and a, b ∈ X , ( y ∗ a ) ∗ ( b ∗ a ) = ( y ∗ b ) ∗ ( a ∗ b ). • For any x ∈ X , the map ∗ x : Y → Y ; y y ∗ x is bijective.Let X be the disjoint union of groups G λ ( λ ∈ Λ). We denote by G a thegroup G λ to which a ∈ X belongs. We denote by e λ the identity of G λ . Definition 2.3 ([13]) . A multiple conjugation biquandle is a non-empty set X which is the disjoint union of groups G λ ( λ ∈ Λ) with binary operations ∗ , ∗ : X × X → X satisfying the following axioms. • For any x, y, z ∈ X ,( x ∗ y ) ∗ ( z ∗ y ) = ( x ∗ z ) ∗ ( y ∗ z ) , (1)( x ∗ y ) ∗ ( z ∗ y ) = ( x ∗ z ) ∗ ( y ∗ z ) , (2)( x ∗ y ) ∗ ( z ∗ y ) = ( x ∗ z ) ∗ ( y ∗ z ) . (3) • For any a, x ∈ X , the maps ∗ a : G x → G x ∗ a and ∗ a : G x → G x ∗ a aregroup homomorphisms. • For any a, b ∈ G λ and x ∈ X , x ∗ ab = ( x ∗ a ) ∗ ( b ∗ a ) , x ∗ e λ = x, (4) x ∗ ab = ( x ∗ a ) ∗ ( b ∗ a ) , x ∗ e λ = x, (5) a − b ∗ a = ba − ∗ a. (6)A multiple conjugation biquandle ( X = F λ ∈ Λ G λ , ∗ , ∗ ) is regarded as a bi-quandle ( X, ∗ , ∗ ) by forgetting the decomposition X = F λ ∈ Λ G λ and the groupstructure of each G λ . Definition 2.4.
For a multiple conjugation biquandle X = F λ ∈ Λ G λ , an X -set is a non-empty set Y with a map ∗ : Y × X → Y satisfying the following axioms. • For any y ∈ Y and a, b ∈ G λ , y ∗ e λ = y , y ∗ ( ab ) = ( y ∗ a ) ∗ ( b ∗ a ), where e λ is the identity of G λ . • For any y ∈ Y and a, b ∈ X , ( y ∗ a ) ∗ ( b ∗ a ) = ( y ∗ b ) ∗ ( a ∗ b ).Any singleton set { y } is also an X -set with the map ∗ defined by y ∗ x = y for x ∈ X , which is called a trivial X -set . Any multiple conjugation biquandle X itself is an X -set with the map ∗ or ∗ . G -Alexander biquandles and G -Alexander multiple con-jugation biquandles Definition 2.5.
Let R be a ring, G a group and M a right R [ G ]-module, where R [ G ] is the group ring of G over R . Let ϕ : G → Z ( G ) be a group homomor-phism, where Z ( G ) denotes the center of G . We define binary operations ∗ and ∗ : ( M × G ) → M × G by( m, g ) ∗ ( n, h ) := ( mh + n ( ϕ ( h ) − h ) , h − gh ) , ( m, g ) ∗ ( n, h ) := ( mϕ ( h ) , g ) . M × G, ∗ , ∗ ) is a biquandle (cf.[15]), which we call the G -Alexander bi-quandle of ( M, ϕ ), and ( M × G = F m ∈ M ( { m }× G ) , ∗ , ∗ ) is a multiple conjugationbiquandle by ( m, g )( m, h ) := ( m, gh )for any m ∈ M and any g, h ∈ G (cf.[15]), which we call the G -Alexandermultiple conjugation biquandle of ( M, ϕ ).Unless otherwise stated, we suppose that the ring R is Z . Remark 2.6.
In general, for a given G -family of biquandles, one can associatea biquandle called the associated biquandle of the G -family of biquandles ([15]),and a multiple conjugation biquandle called the associated multiple conjugationbiquandle of the G -family of biquandles or the partially multiplicative biquandleassociated to the G -family of biquandles ([15]). The G -Alexander biquandle (orthe G -Alexander multiple conjugation biquandle) in the definition above is theassociated biquandle (or the associated multiple conjugation biquandle) of a“ G -family of Alexander biquandle” in the sense of [15]. In this subsection, we review the birack chain complex for a biquandle.Let X be a biquandle (or more generally a birack) and Y an X -set. Let C BR n ( X ; Z ) Y denote the free abelian group generated by the elements x =( y, x , . . . , x n ) ∈ Y × X n if n ≥
1, and C BR n ( X ; Z ) Y = 0 otherwise. (Thesuperscripts BR means birack .)For integers n and i such that n ≥ ≤ i ≤ n and for any element x = ( y, x , . . . , x i , . . . , x n ) ∈ Y × X n , we write( x i − , x i +1 ) := ( y, x , . . . , x i − , x i +1 , . . . , x n ) and( x i − ∗ x i , x i +1 ∗ x i ) := ( y ∗ x, x ∗ x i , . . . , x i − ∗ x i , x i +1 ∗ x i , . . . , x n ∗ x i ) , where these elements belong to Y × X n − .Define a boundary map ∂ BR n : C BR n ( X ; Z ) Y → C BR n − ( X ; Z ) Y by ∂ BR n ( x ) := n X i =1 ( − i − n ( x i − , x i +1 ) − ( x i − ∗ x i , x i +1 ∗ x i ) o for n ≥
2, and define ∂ BR n = 0 for n ≤ ∂ BR4 ( y, x , x , x , x ) = + { ( y, x , x , x ) − ( y ∗ x , x ∗ x , x ∗ x , x ∗ x ) }− { ( y, x , x , x ) − ( y ∗ x , x ∗ x , x ∗ x , x ∗ x ) } + { ( y, x , x , x ) − ( y ∗ x , x ∗ x , x ∗ x , x ∗ x ) }− { ( y, x , x , x ) − ( y ∗ x , x ∗ x , x ∗ x , x ∗ x ) } . Proposition 2.7 (cf. [1]) . C BR ∗ ( X ; Z ) Y := ( C BR n ( X ; Z ) Y , ∂ BR n ) n ∈ Z is a chaincomplex. The birack chain complex of X with an X -set Y is the chain complex C BR ∗ ( X ; Z ) Y . It determines the birack homology group H BR n ( X ; Z ) Y and the4ohomology group H n BR ( X ; Z ) Y . For an abelian group A , the chain and cochaincomplexes C BR ∗ ( X ; A ) Y and C ∗ BR ( X ; A ) Y are also defined in the ordinary way.We denote by H BR n ( M ; A ) Y and H n BR ( X ; A ) Y its homology and cohomologygroups. A cocycle of C ∗ BR ( X ; A ) Y is called a birack cocycle with Y of the bi-quandle X with A . We omit Y , that is, we write C BR ∗ ( X ; Z ) := C BR ∗ ( X ; Z ) Y if Y is the trivial X -set. We review chain complexes P ∗ ( X ; Z ) Y , D ∗ ( X ; Z ) Y and C ∗ ( X ; Z ) Y defined in[14] for multiple conjugation biquandles X with X -set Y . Refer to [14] fora geometric interpretation of the chain complexes and an application in knottheory. These chain complexes are an analogy of chain complexes defined in [2]for multiple conjugation quandles.Let X = F λ ∈ Λ G λ be a multiple conjugation biquandle and Y an X -set ofthe MCB X . For n ≥
0, let P n ( X ; Z ) Y denote the free abelian group generatedby the elements h y ih x i · · · h x k i ∈ [ n + ··· + n k = n Y × k Y i =1 [ λ ∈ Λ G n i λ , where h x i i stands for an element h x i , . . . , x in i i ∈ S λ ∈ Λ G n i λ . For n <
0, let P n ( X ; Z ) Y = 0. We write h x i ∗ x i := h x i ∗ x, x i ∗ x, . . . , x in i ∗ x i and h x i ∗ x i := h x i ∗ x, x i ∗ x, . . . , x in i ∗ x i for an element x ∈ X and an element h x i i = h x i , . . . , x in i i ∈ S λ ∈ Λ G n i λ .Define a boundary map ∂ n : P n ( X ; Z ) Y → P n − ( X ; Z ) Y by ∂ n ( h y ih x i · · · h x k i ) := k X i =1 ( − n + ··· + n i − ( h y ∗ x i ih x ∗ x i i · · · h x i − ∗ x i ih e x i ∗ x i ih x i +1 ∗ x i i · · · h x k ∗ x i i + n i X j =1 ( − j h y ih x i · · · h x i − ih x i , . . . , x i ( j − , x i ( j +1) , . . . , x in i ih x i +1 i · · · h x k i ) when n = n + · · · + n k ≥
1, and define ∂ n = 0 if n <
0, where h e x i ∗ x i i := h x − i x i ∗ x i , x − i x i ∗ x i , . . . , x − i x i ( n i − ∗ x i , x − i x in i ∗ x i i . ∂ ( h y ih x ih x ih x i )= h y ∗ x ih x ∗ x ih x ∗ x i − h y ih x ih x i− h y ∗ x ih x ∗ x ih x ∗ x i + h y ih x ih x i + h y ∗ x ih x ∗ x ih x ∗ x i − h y ih x ih x i ,∂ ( h y ih x ih x , x i )= h y ∗ x ih x ∗ x , x ∗ x i − h y ih x , x i − h y ∗ x ih x ∗ x ih x − x ∗ x i + h y ih x ih x i − h y ih x ih x i ,∂ ( h y ih x , x ih x i )= h y ∗ x ih x − x ∗ x ih x ∗ x i − h y ih x ih x i + h y ih x ih x i + h y ∗ x ih x ∗ x , x ∗ x i − h y ih x , x i ,∂ ( h y ih x , x , x i )= h y ∗ x ih x − x ∗ x , x − x ∗ x i − h y ih x , x i + h y ih x , x i − h y ih x , x i . Remark 2.8.
The notations of this paper are different from that of the paper[14], where we used more notations in order to prove propositions clearly.
Proposition 2.9 ([14]) . P ∗ ( X ; Z ) Y := ( P n ( X ; Z ) Y , ∂ n ) n ∈ Z is a chain complex. For positive integers s and t , we put M ( s, t ) := { µ : { , . . . , s } → { , . . . , t } | i < j ⇒ µ ( i ) < µ ( j ) } . For any map µ ∈ M ( s, t ) and any integer j ∈ Z , we define ⌊ j ; µ ⌋ ∈ { , , . . . , s } as ⌊ j ; µ ⌋ := max { ℓ | µ ( ℓ ) ≤ j } .For any λ ∈ Λ and for elements a , . . . , a s , b , . . . , b t ∈ G λ , put h a i := h a , . . . , a s i and h b i := h b , . . . , b t i . We set (cid:10) h a ih b i (cid:11) µ := ( − P sk =1 ( µ ( k ) − k ) (cid:10) a ⌊ µ ⌋ b −⌊ µ ⌋ , . . . , a ⌊ s + t ; µ ⌋ b s + t −⌊ s + t ; µ ⌋ (cid:11) and (cid:10) h a ih b i (cid:11) := X µ ∈ M ( s,s + t ) (cid:10) h a ih b i (cid:11) µ , where a = b = e λ . For example, we have hh a , a ih b , b ii = h a , a , a b , a b i − h a , a b , a b , a b i + h b , a b , a b , a b i + h a , a b , a b , a b i− h b , a b , a b , a b i + h b , b , a b , a b i . for elements a , a , b , b ∈ G λ . Definition 2.10.
Let D n ( X ; Z ) Y be the submodule of P n ( X ; Z ) Y generatedby elements of the form h y ih a i · · · h a ih b i · · · h a k i − h y ih a i · · · hh a ih b ii · · · h a k i .When n ≤
1, we set D n ( X ; Z ) Y = 0.For example, the submodule D ( X ; Z ) Y is generated by the elements of theform h y ih a ih b i − h y ih a, ab i + h y ih b, ab i , for y ∈ Y , a, b ∈ G λ . 6 roposition 2.11 ([14]) . D ∗ ( X ; Z ) Y := ( D n ( X ; Z ) Y , ∂ n ) n ∈ Z is a subcomplexof P ∗ ( X ; Z ) Y . We have a chain complex C ∗ ( X ; Z ) Y := P ∗ ( X ; Z ) Y /D ∗ ( X ; Z ) Y . The ho-mology group H n ( X ; Z ) Y and the cohomology group H n ( X ; Z ) Y of the multi-ple conjugation biquandle X with X -set Y are the homology and cohomologygroups of the chain complex C ∗ ( X ; Z ) Y .For an abelian group A , the chain and cochain complexes C ∗ ( X ; A ) Y and C ∗ ( X ; A ) Y are also defined in the ordinary way. We denote by H n ( X ; A ) Y and H n ( X ; A ) Y its n -th homology and cohomology groups respectively. A cocycleof C ∗ ( X ; A ) Y is called a cocycle with Y of the multiple conjugation biquandle X with A . We omit Y , that is, we write C ∗ ( X ; Z ) := C ∗ ( X ; Z ) Y if Y is thetrivial X -set. In this subsection, we review chain complexes for groups.We review the group (co)homology. Let G be a group and let M be a right Z [ G ]-module, where Z [ G ] is the group ring of G over the ring Z . Let C gp n ( M ; Z )be a free abelian group generated by any element ( m , . . . , m n ) ∈ M n if n ≥ C gp n ( M ; Z ) = 0 otherwise.We define the boundary map ∂ gp n : C gp n ( M ; Z ) → C gp n − ( M ; Z ) by ∂ gp n ( m , . . . , m n ) := ( m , . . . , m n )+ n − X i =1 ( − i ( m , . . . , m i − , m i + m i +1 , m i +2 , . . . , m n )+ ( − n ( m , . . . , m n − )if n ≥
2, and ∂ gp n = 0 otherwise. For example, ∂ gp4 ( m , m , m , m ) = ( m , m , m ) − ( m + m , m , m ) + ( m , m + m , m ) − ( m , m , m + m ) + ( m , m , m ) . Let C gp n ( M ; Z ) G := C gp n ( M ; Z ) ⊗ Z [ G ] Z , where G acts on Z trivially from the left. In other words, under the diagonalaction of G on C gp n ( M ; Z ), C gp n ( M ; Z ) G is the G -coinvariant part. We note that( m g, . . . , m n g ) = ( m , . . . , m n ) in C gp n ( M ; Z ) G for any ( m , . . . , m n ) ∈ M n and any g ∈ G . We have the induced boundary map ∂ gp n : C gp n ( M ; Z ) G → C gp n − ( M ; Z ) G . Proposition 2.12 ([9]) . C gp ∗ ( M ; Z ) G = ( C gp n ( M ; Z ) G , ∂ gp n ) n ∈ Z is a chain com-plex. The chain complex determines the homology group H gp n ( M ; Z ) G and thecohomology group H n gp ( M ; Z ) G . For an abelian group A , the chain and cochaincomplexes C gp ∗ ( M ; A ) G and C ∗ gp ( M ; A ) G are also defined in the ordinary way.We denote by H gp n ( M ; A ) G and H n gp ( M ; A ) G its homology and cohomologygroups respectively. 7et D gp n ( M ; Z ) G be the submodule of C gp n ( M ; Z ) G generated by the elementsof the following set n − [ i =1 n ( m , . . . , m i − , , m i +1 , . . . m n ) (cid:12)(cid:12)(cid:12) m , . . . , m n ∈ M o for n ≥
2. Define D gp n ( M ; Z ) G = 0 for n ≤ Proposition 2.13 ([9]) . D gp ∗ ( M ; Z ) G := ( D gp n ( M ; Z ) G , ∂ gp n ) n ∈ Z is a subcomplexof C gp ∗ ( M ; Z ) G . The chain complex C norgp ∗ ( M ; Z ) G := C gp ∗ ( M ; Z ) G /D gp ∗ ( M ; Z ) G determinesthe homology group H norgp n ( M ; Z ) G called the normalized group homology . Inthe ordinary way, for an abelian group A , the homology group H norgp n ( M ; A ) G and the cohomology group H n norgp ( M ; A ) G are defined. G -Alexander biquandles with triv-ial X -set Throughout this section, let X = M × G be the G -Alexander biquandle of( M, ϕ ), see Definition 2.5. We suppose that X has trivial X -set.We define a chain complex C BR U ∗ ( X ; Z ) and chain maps γ and ψ . It turnsout that the chain complex C BR U ∗ ( X ; Z ) is isomorphic to the birack chain com-plex C BR ∗ ( X ; Z ), and hence the homology group H BR U n ( X ; Z ) is isomorphic to H BR n ( X ; Z ). Through this chain complex, we associate cocycles of the chaincomplex of group G to cocycles of the birack chain complex.Our goal in this section is to give Theorems 3.4 and 3.5. C BR U ∗ ( X ; Z ) and chain maps γ and ψ C BR U ∗ ( X ; Z ) and the homology group H BR U n ( X ; Z )For g = ( g , . . . , g n ) ∈ G n and m = ( m , . . . , m n ) ∈ M n , we use the followingnotations: g { i } :=( g , . . . , g i , g i +2 , . . . , g n ) ∈ G n − , g { ⊳i } :=( g − i +1 g g i +1 , . . . , g − i +1 g i g i +1 , g i +2 , . . . , g n ) ∈ G n − , m { i } :=( m , . . . , m i − , m i + m i +1 , m i +2 , . . . , m n ) ∈ M n − and m { ⊳i } :=( m g i +1 , . . . , m i − g i +1 , m i g i +1 + m i +1 ϕ ( g i +1 ) ,m i +2 ϕ ( g i +1 ) , . . . , m n ϕ ( g i +1 )) ∈ M n − , where n and i are integers such that n ≥ ≤ i ≤ n − C BR U n ( X ; Z ) be the free abelian group generated by the elements ( g ; m ) =( g , . . . , g n ; m , . . . , m n ) ∈ G n × M n if n ≥
1, and C BR U n ( X ; Z ) = 0 otherwise.Define a boundary map ∂ BR U n : C BR U n ( X ; Z ) → C BR U n − ( X ; Z ) by ∂ BR U n ( g ; m ) = n − X i =0 ( − i (cid:8) ( g { i } ; m { i } ) − ( g { ⊳i } ; m { ⊳i } ) (cid:9) n ≥
2, and define ∂ BR U n = 0 for n ≤
1. For example, we have ∂ BR U ( g , g , g ; m , m , m )= (cid:8) ( g , g ; m , m ) − ( g , g ; m ϕ ( g ) , m ϕ ( g )) (cid:9) − (cid:8) ( g , g ; m + m , m ) − ( g − g g , g ; m g + m ϕ ( g ) , m ϕ ( g )) (cid:9) + (cid:8) ( g , g ; m , m + m ) − ( g − g g , g − g g ; m g , m g + m ϕ ( g )) (cid:9) . Lemma 3.1. C BR U ∗ ( X ; Z ) := ( C BR U n ( X ; Z ) , ∂ BR U n ) n ∈ Z is a chain complex.Proof. We fix an integer n with n ≥
2. Define ∂ in : C BR U n ( X ; Z ) → C BR U n − ( X ; Z )by ∂ in ( g ; m ) := ( g { i } ; m { i } ) and δ in : C BR U n ( X ; Z ) → C BR U n − ( X ; Z ) by δ in ( g ; m ) :=( g { ⊳i } ; m { ⊳i } ) for any integer i with 0 ≤ i ≤ n −
1. Then, we have ∂ BR U n = n − X i =0 ( − i ( ∂ in − δ in ) . We assert that δ jn − ◦ δ in = δ in − ◦ δ j +1 n , (7) ∂ jn − ◦ ∂ in = ∂ in − ◦ ∂ j +1 n , (8) ∂ jn − ◦ δ in = δ in − ◦ ∂ j +1 n and (9) δ jn − ◦ ∂ in = ∂ in − ◦ δ j +1 n . (10)We show the equality (7). Since the other equalities are shown in a similar way,we omit the proofs.Let i and j be integers with 0 ≤ i ≤ j ≤ n −
2. We see g { ⊳i }{ ⊳j } = g { ⊳ ( j +1) }{ ⊳i } as follows: g { ⊳i }{ ⊳j } = ( g − j +2 g − i +1 g g i +1 g j +2 , . . . , g − j +2 g − i +1 g i g i +1 g j +2 , g − j +2 g i +2 g j +2 , . . . ,g − j +2 g j +1 g j +2 , g j +3 , . . . , g n )= (( g − j +2 g i +1 g j +2 ) − g − j +2 g g j +2 ( g − j +2 g i +1 g j +2 ) , . . . , ( g − j +2 g i +1 g j +2 ) − g − j +2 g i g j +2 ( g − j +2 g i +1 g j +2 ) , g − j +2 g i +2 g j +2 , . . . ,g − j +2 g j +1 g j +2 , g j +3 , . . . , g n )= g { ⊳ ( j +1) }{ ⊳i } . We see δ jn − ◦ δ in = δ in − ◦ δ j +1 n for 0 < i < j ≤ n − δ jn − ◦ δ in ( g ; m )= ( g { ⊳i }{ ⊳j } ; m g i +1 g j +2 , . . . , m i − g i +1 g j +2 , m i g i +1 g j +2 + m i +1 ϕ ( g i +1 ) g j +2 ,m i +2 ϕ ( g i +1 ) g j +2 , . . . , m j ϕ ( g i +1 ) g j +2 , m j +1 ϕ ( g i +1 ) g j +2 + m j +2 ϕ ( g i +1 ) ϕ ( g j +2 ) ,m j +3 ϕ ( g i +1 ) ϕ ( g j +2 ) , . . . , m n ϕ ( g i +1 ) ϕ ( g j +2 ))= ( g { ⊳ ( j +1) }{ ⊳i } ; m g j +2 ( g − j +2 g i +1 g j +2 ) , . . . , m i − g j +2 ( g − j +2 g i +1 g j +2 ) ,m i g j +2 ( g − j +2 g i +1 g j +2 ) + m i +1 g j +2 ϕ ( g − j +2 g i +1 g j +2 ) , m i +2 g j +2 ϕ ( g − j +2 g i +1 g j +2 ) , . . . ,m j g j +2 ϕ ( g − j +2 g i +1 g j +2 ) , m j +1 g j +2 ϕ ( g − j +2 g i +1 g j +2 ) + m j +2 ϕ ( g j +2 ) ϕ ( g − j +2 g i +1 g j +2 ) ,m j +3 ϕ ( g j +2 ) ϕ ( g − j +2 g i +1 g j +2 ) , . . . , m n ϕ ( g j +2 ) ϕ ( g − j +2 g i +1 g j +2 ))= δ in − ◦ δ j +1 n ( g ; m ) . i = j or i = 0, we can show that the equality (7) holds.Hence, we have the equality (7) for any i, j with 0 ≤ i ≤ j ≤ n − ∂ BR U n − ◦ ∂ BR U n = 0.The n -th homology and cohomology groups of the chain complex C BR U ∗ ( X ; Z )are denoted by H BR U n ( X ; Z ) and H n BR U ( X ; Z ) respectively. γ : C BR ∗ ( X ; Z ) → C BR U ∗ ( X ; Z )For n ≥
1, define a homomorphism γ n : C BR n ( X ; Z ) → C BR U n ( X ; Z ) by γ n (( m , g ) , . . . , ( m n , g n )) := ( g ; m ′ , . . . , m ′ n − , m n ) , where we write m ′ i := m i − m i +1 and g = ( g , . . . , g n ). For n <
1, define γ n = 0. Lemma 3.2. (1) The map γ is a chain map, that is, it holds that γ n − ◦ ∂ BR n = ∂ BR U n ◦ γ n . (2) For any integer n , the map γ n is an isomorphism.Proof. The assertion that γ is a chain map is easily verified by a direct calcu-lation, which is left to the reader. Define a homomorphism r n : C BR U n ( X ; Z ) → C BR n ( X ; Z ) by r n ( g ; m ) := (cid:18)(cid:16) n X i =1 m i , g (cid:17) , (cid:16) n X i =2 m i , g (cid:17) , . . . , (cid:16) n X i = n m i , g n (cid:17)(cid:19) . We can easily see that r n is the inverse map of γ n , and thus, γ n is a bijection. ψ : C BR U ∗ ( X ; Z ) → C gp ∗ ( M ; Z ) G For n ≥
1, define ψ n : C BR U n ( X ; Z ) → C gp n ( M ; Z ) G by ψ n ( g ; m ) := X k =( k ,...,k n ) ∈K n ( − | k | ( m g k , m g k , . . . , m n g k n ) , where K n := { k = ( k , . . . , k n ) ∈ { , } n | k = 0 } and | k | := k + k + · · · + k n and g k i := ϕ ( g k g k · · · g k i i ) g k i +1 i +1 g k i +2 i +2 · · · g k n n ∈ G for an element k = ( k , . . . , k n ) ∈ K n and an integer i with 1 ≤ i ≤ n . For n <
1, define ψ n = 0.For example, we have ψ ( g ; m ) = ( − ( m ϕ ( g ) g g , m ϕ ( g g ) g , m ϕ ( g g g ))+ ( − ( m ϕ ( g ) g g , m ϕ ( g g ) g , m ϕ ( g g g ))+ ( − ( m ϕ ( g ) g g , m ϕ ( g g ) g , m ϕ ( g g g ))+ ( − ( m ϕ ( g ) g g , m ϕ ( g g ) g , m ϕ ( g g g ))= ( m g g , m ϕ ( g ) g , m ϕ ( g g )) − ( m g , m ϕ ( g ) , m ϕ ( g )) − ( m g , m g , m ϕ ( g ))+ ( m , m , m ) . emma 3.3. The map ψ is a chain map, that is, it holds that ψ n − ◦ ∂ BR U n = ∂ gp n ◦ ψ n . Proof.
We fix an integer n ≥
2, and fix g = ( g , . . . , g n ) and m = ( m , . . . , m n ).For any integer i with 2 ≤ i ≤ n , let K i := { k = ( k , . . . , k n ) ∈ { , } n | k = k i = 0 } and K i := { k = ( k , . . . , k n ) ∈ { , } n | k = 0 , k i = 1 } . For any i with 1 ≤ i ≤ n −
1, put A i := ( m g k , . . . , m i − g k i − , m i g k i + m i +1 g k i +1 , m i +2 g k i +2 , . . . , m n g k n (cid:1) ,L := X k ∈K n ( − | k | n − X i =1 ( − i A i . We show that ψ n − ◦ ∂ BR U n ( g ; m ) = L = ∂ gp n ◦ ψ n ( g ; m ).Firstly, we show ψ n − ◦ ∂ BR U n ( g ; m ) = L . We have ψ n − ◦ ∂ BR U n ( g ; m ) = ψ n − (cid:0) ( g { } ; m { } ) − ( g { ⊳ } ; m { ⊳ } ) (cid:1) + n − X i =1 ( − i ψ n − (cid:0) ( g { i } ; m { i } ) − ( g { ⊳i } ; m { ⊳i } ) (cid:1) . (11)Since ( m g, . . . , m n g ) = ( m , . . . , m n ) in C gp n ( M ; Z ) G for any ( m , . . . , m n ) ∈ M n and any g ∈ G , we see that ψ n − ( g { } ; m { } ) − ψ n − ( g { ⊳ } ; m { ⊳ } )= X k ∈K ( − | k | (cid:8) ( m g k , . . . , m n g k n ) − ( m ϕ ( g ) g k , . . . , m n ϕ ( g ) g k n ) (cid:9) = X k ∈K ( − | k | (cid:8) ( m g k , . . . , m n g k n ) − ( m g k ϕ ( g ) , . . . , m n g k n ϕ ( g )) (cid:9) = 0 . (12)For any integer i with 1 ≤ i ≤ n −
1, we have ψ n − ( g { i } ; m { i } )= X k ∈K i +1 ( − | k | (cid:16) m ϕ ( g k ) g k · · · g k i i g i +1 g k i +2 i +2 · · · g k n n , . . . ,m i − ϕ ( g k · · · g k i − i − ) g k i i g i +1 g k i +2 i +2 · · · g k n n ,m i ϕ ( g k · · · g k i i ) g i +1 g k i +2 i +2 · · · g k n n + m i +1 ϕ ( g k · · · g k i i g i +1 ) g k i +2 i +2 · · · g k n n ,m i +2 ϕ ( g k · · · g k i i g i +1 g k i +2 i +2 ) g k i +3 i +3 · · · g k n n , . . . ,m n ϕ ( g k g k · · · g k i i g i +1 g k i +2 i +2 · · · g k n n ) (cid:17) = X k ∈K i +1 ( − | k | A i . (13)11or any integer i with 1 ≤ i ≤ n −
1, we also have ψ n − ( g { ⊳i } ; m { ⊳i } )= X k ∈K i +1 ( − | k | (cid:16) m g i +1 ϕ (cid:0) ( g − i +1 g g i +1 ) k (cid:1) ( g − i +1 g g i +1 ) k · · · ( g − i +1 g i g i +1 ) k i g i +1 g k i +2 i +2 · · · g k n n , . . . ,m i − g i +1 ϕ (cid:0) ( g − i +1 g g i +1 ) k · · · ( g − i +1 g i − g i +1 ) k i − (cid:1) ( g − i +1 g i g i +1 ) k i g i +1 g k i +2 i +2 · · · g k n n ,m i g i +1 ϕ (cid:0) ( g − i +1 g g i +1 ) k · · · ( g − i +1 g i g i +1 ) k i (cid:1) g i +1 g k i +2 i +2 · · · g k n n + m i +1 ϕ ( g i +1 ) ϕ (cid:0) ( g − i +1 g g i +1 ) k · · · ( g − i +1 g i g i +1 ) k i g i +1 (cid:1) g k i +2 i +2 · · · g k n n ,m i +2 ϕ ( g i +1 ) ϕ (cid:0) ( g − i +1 g g i +1 ) k · · · ( g − i +1 g i g i +1 ) k i g i +1 g k i +2 i +2 (cid:1) g k i +3 i +3 · · · g k n n , . . . ,m n ϕ ( g i +1 ) ϕ (cid:0) ( g − i +1 g g i +1 ) k · · · ( g − i +1 g i g i +1 ) k i g i +1 g k i +2 i +2 · · · g k n n (cid:1)(cid:17) = X k ∈K i +1 ( − | k | (cid:16) m g i +1 ϕ (cid:0) g k (cid:1) g − i +1 g k · · · g k i i g i +1 g i +1 g k i +2 i +2 · · · g k n n , . . . ,m i − g i +1 ϕ (cid:0) g k g k · · · g k i − i − (cid:1) g − i +1 g k i i g i +1 g i +1 g k i +2 i +2 · · · g k n n ,m i g i +1 ϕ (cid:0) g k g k · · · g k i i (cid:1) g i +1 g k i +2 i +2 · · · g k n n + m i +1 ϕ ( g i +1 ) ϕ (cid:0) g k g k · · · g k i i g i +1 (cid:1) g k i +2 i +2 · · · g k n n ,m i +2 ϕ ( g i +1 ) ϕ (cid:0) g k g k · · · g k i i g i +1 g k i +2 i +2 (cid:1) g k i +3 i +3 · · · g k n n , . . . ,m n ϕ ( g i +1 ) ϕ (cid:0) g k g k · · · g k i i g i +1 g k i +2 i +2 · · · g k n n (cid:1)(cid:17) = X k ∈K i +1 ( − | k | (cid:16) m ϕ (cid:0) g k (cid:1) g k · · · g k i i g i +1 g k i +2 i +2 · · · g k n n , . . . ,m i − ϕ (cid:0) g k g k · · · g k i − i − (cid:1) g k i i g i +1 g k i +2 i +2 · · · g k n n ,m i ϕ (cid:0) g k g k · · · g k i i (cid:1) g i +1 g k i +2 i +2 · · · g k n n + m i +1 ϕ (cid:0) g k g k · · · g k i i g i +1 (cid:1) g k i +2 i +2 · · · g k n n ,m i +2 ϕ (cid:0) g k g k · · · g k i i g i +1 g k i +2 i +2 (cid:1) g k i +3 i +3 · · · g k n n , . . . ,m n ϕ (cid:0) g k g k · · · g k i i g i +1 g k i +2 i +2 · · · g k n n (cid:1)(cid:17) = X k ∈K i +1 ( − | k |− A i = − X k ∈K i +1 ( − | k | A i . (14)Using (13) and (14), we have ψ n − (cid:0) ( g { i } ; m { i } ) − ( g { ⊳i } ; m { ⊳i } ) (cid:1) = X k ∈K i +1 ( − | k | A i + X k ∈K i +1 ( − | k | A i = X k ∈K n ( − | k | A i . (15)Using (11), (12) and (15), we have ψ n − ◦ ∂ BR U n ( g ; m ) = n − X i =1 ( − i X k ∈K n ( − | k | A i = L. ∂ gp n ◦ ψ n ( g ; m ) = L . We have ∂ gp n ◦ ψ n ( g ; m ) = X k ∈K n ( − | k | ∂ gp n ( m g k , . . . , m n g k n )= X k ∈K n ( − | k | ( m g k , . . . , m n g k n ) + L + X k ∈K n ( − | k | ( − n ( m g k , . . . , m n − g k n − ) . (16)We have X k ∈K n ( − | k | ( m g k , . . . , m n g k n )= X k ∈K ( − | k | ( m g k , . . . , m n g k n ) + X k ∈K ( − | k | ( m g k , . . . , m n g k n )= 0 . (17)This is because it holds that X k ∈K ( − | k | ( m g k , . . . , m n g k n )= X k ∈K ( − | k | (cid:16) m ϕ ( g k g ) g k · · · g k n n , . . . , m n ϕ ( g k g g k · · · g k n n ) (cid:17) = X k ∈K ( − | k | (cid:16) m ϕ ( g k ) g k · · · g k n n , . . . , m n ϕ ( g k g k · · · g k n n ) (cid:17) = X k ∈K ( − | k | +1 (cid:16) m ϕ ( g k ) g g k · · · g k n n , . . . , m n ϕ ( g k g g k · · · g k n n ) (cid:17) = − X k ∈K ( − | k | ( m g k , . . . , m n g k n ) . Similarly, we have X k ∈K n ( − | k | ( − n ( m g k , . . . , m n − g k n − )= ( − n (cid:16) X k ∈K n ( − | k | ( m g k , . . . , m n − g k n − ) + X k ∈K n ( − | k | ( m g k , . . . , m n − g k n − ) (cid:17) = 0 . (18)By (16), (17) and (18), we have ∂ gp n ◦ ψ n ( g ; m ) = L . G -Alexander biquandles Let γ = ( γ n ) and ψ = ( ψ n ) be the chain maps defined in Subsection 3.1. Wehave a sequence C BR n ( X ; Z ) γ n −→ C BR U n ( X ; Z ) ψ n −→ C gp n ( M ; Z ) G of chain groups C BR n ( X ; Z ) , C BR U n ( X ; Z ) , C gp n ( M ; Z ) G and chain maps γ, ψ for n ≥
1. 13 heorem 3.4.
For any n -cocycle f : C gp n ( M ; Z ) G → A of M , the map Φ f := f ◦ ψ n ◦ γ n : C BR n ( X ; Z ) → A is a birack n -cocycle of the G -Alexander biquandle X .Proof. Since ψ n ◦ γ n is a chain map, we see the result.An A -multilinear map f : M n → A is G -invariant if f ( m g, . . . , m n g ) = f ( m , . . . , m n ) for any g ∈ G and ( m , . . . , m n ) ∈ M n . Note that any G -invariant A -multilinear map f : M n → A induces a cocycle f : C gp n ( M ; Z ) G → A . The following theorem follows from a direct calculation. Theorem 3.5. (1) Let f : M → A be a G -invariant A -multilinear map.The birack -cocycle Φ f = f ◦ ψ ◦ γ : C BR2 ( X ; Z ) → A is formulated as Φ f (( m , g ) , ( m , g )) = f (cid:0) m − m , m (1 − ϕ ( g ) g − ) (cid:1) for (( m , g ) , ( m , g )) ∈ X ⊂ C BR2 ( X ; Z ) .(2) Let f : M → A be a G -invariant A -multilinear map. The birack -cocycle Φ f = f ◦ ψ ◦ γ : C BR3 ( X ; Z ) → A is formulated as Φ f (( m , g ) , ( m , g ) , ( m , g ))= f (cid:0) ( m − m )(1 − ϕ ( g ) − g ) , m − m , m (1 − ϕ ( g ) g − ) (cid:1) for (( m , g ) , ( m , g ) , ( m , g )) ∈ X ⊂ C BR3 ( X ; Z ) . G -Alexander biquandles with the X -set X Throughout this section, let X = M × G be the G -Alexander biquandle of( M, ϕ ) and we assume that X is also an X -set with the action ∗ := ∗ , that is,it holds that ( x ∗ x ) ∗ ( x ∗ x ) = ( x ∗ x ) ∗ ( x ∗ x ) for any x , x , x ∈ X .We discuss birack cocycles of X with the X -set X .Since we apply a similar argument as shown in Section 3 to this case, wesummarize all the properties without proof.We define a chain complex of X = M × G with the X -set X , denoted by C BR U ∗ ( X ; Z ) X , and define chain maps γ and ψ . It turns out that the chaincomplex C BR U ∗ ( X ; Z ) X is isomorphic to the birack chain complex C BR ∗ ( X ; Z ) X and hence the homology group H BR U n ( X ; Z ) X is isomorphic to H BR n ( X ; Z ) X .Our goal in this section is to give Theorems 4.4 and 4.5.14 .1 The chain complex C BR U ∗ ( X ; Z ) X and chain maps γ and ψ C BR U ∗ ( X ; Z ) X and the homology group H BR U n ( X ; Z ) X For g = ( g , g , . . . , g n ) ∈ G × G n and m = ( m , m , . . . , m n ) ∈ M × M n , weuse the following notations: g { i } := ( g , g , . . . , g i , g i +2 , . . . , g n ) ∈ G × G n − , g { ⊳i } := ( g − i +1 g g i +1 , g − i +1 g g i +1 , . . . , g − i +1 g i g i +1 , g i +2 , . . . , g n ) ∈ G × G n − , m { i } := ( m , m , . . . , m i − , m i + m i +1 , m i +2 , . . . , m n ) ∈ M × M n − and m { ⊳i } := (cid:0) m g i +1 , m g i +1 , . . . , m i − g i +1 , m i g i +1 + m i +1 ϕ ( g i +1 ) ,m i +2 ϕ ( g i +1 ) , . . . , m n ϕ ( g i +1 ) (cid:1) ∈ M × M n − , where n and i are integers with n ≥ ≤ i ≤ n − C BR U n ( X ; Z ) X be the free abelian group generated by the elements( g , m ) = ( g , g , . . . , g n ; m , m , . . . , m n ) ∈ ( G × G n ) × ( M × M n )if n ≥
1, and C BR U n ( X ; Z ) X := 0 otherwise.We define a boundary map ∂ BR U n : C BR U n ( X ; Z ) X → C BR U n − ( X ; Z ) X by ∂ BR U n ( g ; m ) = n − X i =0 ( − i (cid:8) ( g { i } ; m { i } ) − ( g { ⊳i } ; m { ⊳i } ) (cid:9) if n ≥
2, and ∂ BR U n = 0 otherwise. For example, we have ∂ BR U ( g , g , g , g ; m , m , m , m )= ( − (cid:8) ( g , g , g ; m + m , m , m ) − ( g − g g , g , g ; m g + m ϕ ( g ) , m ϕ ( g ) , m ϕ ( g )) (cid:9) + ( − (cid:8) ( g , g , g ; m , m + m , m ) − ( g − g g , g − g g , g ; m g , m g + m ϕ ( g ) , m ϕ ( g )) (cid:9) + ( − (cid:8) ( g , g , g ; m , m , m + m ) − ( g − g g , g − g g , g − g g ; m g , m g , m g + m ϕ ( g )) (cid:9) . Lemma 4.1. C BR U ∗ ( X ; Z ) X = ( C BR U n ( X ; Z ) X , ∂ BR U n ) n ∈ Z is a chain complex. γ : C BR ∗ ( X ; Z ) X → C BR U ∗ ( X ; Z ) X For n ≥
1, define a homomorphism γ n : C BR n ( X ; Z ) X → C BR U n ( X ; Z ) X by γ n (( m , g ) , ( m , g ) , . . . , ( m n , g n )) := ( g ; m ′ , m ′ , . . . , m ′ n − , m n ) , where we write m ′ i := m i − m i +1 and g = ( g , . . . , g n ). Define γ n = 0 for i < Lemma 4.2. (1) The map γ is a chain map, that is, it holds that γ n − ◦ ∂ BR n = ∂ BR U n ◦ γ n . (2) For each integer n , the map γ n is an isomorphism. .1.3 The chain map ψ : C BR U ∗ ( X ; Z ) X → C gp ∗ +1 ( M ; Z ) G For n ≥
1, define ψ n : C BR U n ( X ; Z ) X → C gp n +1 ( M ; Z ) G by ψ n ( g , g , . . . , g n ; m , m , . . . , m n ) := X k ∈K n ( − | k | ( m g k , m g k , . . . , m n g k n ) , where K n := { , } n and | k | := k + k + · · · + k n , g k := g k g k · · · g k n n ∈ G and g k i := ϕ ( g k g k · · · g k i i ) g k i +1 i +1 g k i +2 i +2 · · · g k n n ∈ G for an element k = ( k , . . . , k n ) ∈ K n and an integer i with 1 ≤ i ≤ n . Define ψ n = 0 for n <
1. We note that the codomain of ψ n is C gp n +1 ( M ; Z ) G .For example, we have ψ ( g ; m ) = ( − ( m g g g , m ϕ ( g ) g g , m ϕ ( g g ) g , m ϕ ( g g g ))+ ( − ( m g g g , m ϕ ( g ) g g , m ϕ ( g g ) g , m ϕ ( g g g ))+ ( − ( m g g g , m ϕ ( g ) g g , m ϕ ( g g ) g , m ϕ ( g g g ))+ ( − ( m g g g , m ϕ ( g ) g g , m ϕ ( g g ) g , m ϕ ( g g g ))+ ( − ( m g g g , m ϕ ( g ) g g , m ϕ ( g g ) g , m ϕ ( g g g ))+ ( − ( m g g g , m ϕ ( g ) g g , m ϕ ( g g ) g , m ϕ ( g g g ))+ ( − ( m g g g , m ϕ ( g ) g g , m ϕ ( g g ) g , m ϕ ( g g g ))+ ( − ( m g g g , m ϕ ( g ) g g , m ϕ ( g g ) g , m ϕ ( g g g )) . Lemma 4.3.
The map ψ = ( ψ n ) is a chain map, that is, it holds that ψ n − ◦ ∂ BR U n = ∂ gp n +1 ◦ ψ n . G -Alexander biquandles with the X -set X As a consequence of Subsection 4.1, we have a sequence C BR n ( X ; Z ) X γ n −→ C BR U n ( X ; Z ) X ψ n −→ C gp n +1 ( M ; Z ) G of chain groups C BR n ( X ; Z ) X , C BR U n ( X ; Z ) X , C gp n ( M ; Z ) G and chain maps γ, ψ for n ≥
1. Therefore we have the following theorem.
Theorem 4.4.
For any ( n + 1) -cocycle f : C gp n +1 ( M ; Z ) G → A , the map Φ f := f ◦ ψ n ◦ γ n : C BR n ( X ; Z ) X → A is a birack n -cocycle of the G -Alexander biquandle X = M × G . Theorem 4.5. (1) Let f : M → A be a G -invariant A -multilinear map. Thebirack -cocycle Φ f = f ◦ ψ ◦ γ : C BR2 ( X ; Z ) X → A of the G -Alexanderbiquandle X = M × G is formulated as Φ f (( m , g ) , ( m , g ) , ( m , g ))= f (cid:0) m ′ (1 − ϕ ( g ) − g ) , m ′ , m (1 − ϕ ( g ) g − ) (cid:1) for (( m , g ) , ( m , g ) , ( m , g )) ∈ X × X ⊂ C BR2 ( X ; Z ) X , where m ′ i := m i − m i +1 .
2) Let f : M → A be a G -invariant A -multilinear map. The birack -cocycle Φ f = f ◦ ψ ◦ γ : C BR3 ( X ; Z ) X → A of the G -Alexander biquandle X = M × G is formulated as Φ f (cid:0) ( m , g ) , ( m , g ) , ( m , g ) , ( m , g ) (cid:1) = f (cid:0) m ′ (1 − ϕ ( g ) − g ) , m ′ , m ′ , m (1 − ϕ ( g ) g − ) (cid:1) − f (cid:0) m ′ (1 − ϕ ( g ) − g ) g , m ′ g , m ′ ϕ ( g ) , m (1 − ϕ ( g ) g − ) ϕ ( g ) (cid:1) for (( m , g ) , ( m , g ) , ( m , g ) , ( m , g )) ∈ X × X ⊂ C BR3 ( X ; Z ) X , where m ′ i := m i − m i +1 . G -Alexander multiple conjugationbiquandles Throughout this section, let X = F m ∈ M ( { m }× G ) = M × G be the G -Alexandermultiple conjugation biquandle of ( M, ϕ ), see Definition 2.5. Our goal in thissection is to give Theorem 5.7. D BR ∗ ( X ; Z ) , D BR U ∗ ( X ; Z ) andthe induced homomorphisms γ n , ψ n,λ D BR ∗ ( X ; Z ) of C BR ∗ ( X ; Z )Let D BR n ( X ; Z ) be the subgroup of C BR n ( X ; Z ) generated by the elements of thefollowing two sets n − [ i =1 n ( x i − , ( m, g ) , ( m, h ) , x i +2 ) (cid:12)(cid:12)(cid:12) x ∈ X n , m ∈ M, g, h ∈ G o and n [ i =1 ( ( x i − , ( m, gh ) , x i +1 ) − ( x i − , ( m, g ) , x i +1 ) − (cid:0) x i − ∗ ( m, g ) , (cid:0) ( m, h ) , x i +1 (cid:1) ∗ ( m, g ) (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x ∈ X n ,m ∈ M, g, h ∈ G ) for n ≥
2. We define D BR n ( X ; Z ) = 0 for n ≤
1. We note that (cid:0) x i − ∗ ( m, g ) , (cid:0) ( m, h ) , x i +1 (cid:1) ∗ ( m, g ) (cid:1) = ( x ∗ ( m, g ) , . . . , x i − ∗ ( m, g ) , ( m, h ) ∗ ( m, g ) , x i +1 ∗ ( m, g ) , . . . , x n ∗ ( m, g )) . Lemma 5.1. D BR ∗ ( X ; Z ) := ( D BR n ( X ; Z ) , ∂ BR n ) n ∈ Z is a subcomplex of C BR ∗ ( X ; Z ) .Proof. We fix an integer n ≥ ∂ BR n ( D BR n ( X ; Z )) ⊂ D BR n − ( X ; Z ). Itsuffices to show ∂ BR n ( x i − , ( m, g ) , ( m, h ) , x i +2 ) ≡ i with 1 ≤ i ≤ n − ∂ BR n ( x i − , ( m, gh ) , x i +1 ) ≡ ∂ BR n ( x i − , ( m, g ) , x i +1 )+ ∂ BR n (cid:0) x i − ∗ ( m, g ) , (cid:0) ( m, h ) , x i +1 (cid:1) ∗ ( m, g ) (cid:1) for any i with 1 ≤ i ≤ n in C BR n − ( X ; Z ) /D BR n − ( X ; Z ).17e verify the first equality in the quotient group. Put x i := ( m, g ), x i +1 :=( m, h ) and x := ( x , . . . , x n ). ∂ BR n ( x )= n X j =1 ( − j − (cid:8) ( x j − , x j +1 ) − ( x j − ∗ x j , x j +1 ∗ x j ) (cid:9) = i − X j =1 ( − j − { ( x j − , x j +1 ) − ( x j − ∗ x j , x j +1 ∗ x j ) } ( A ) + ( − i − (cid:8) ( x i − , ( m, h ) , x i +2 ) − (cid:0) x i − ∗ ( m, g ) , (( m, h ) , x i +2 ) ∗ ( m, g ) (cid:1)(cid:9) ( B ) + ( − i (cid:8) ( x i − , ( m, g ) , x i +2 ) − (cid:0) ( x i − , ( m, g )) ∗ ( m, h ) , x i +2 ∗ ( m, h ) (cid:1)(cid:9) ( C ) + n X j = i +2 ( − j − { ( x j − , x j +1 ) − ( x j − ∗ x j , x j +1 ∗ x j ) } ( D ) . (19)Since ( m, gh ) = ( m, h ( h − gh )) and ( m, g ) ∗ ( m, h ) = ( mh + m ( ϕ ( h ) − h ) , h − gh ) = ( m, h − gh ) ∗ ( m, h ), we see that for ( B ) and ( C ) of (19),( B ) − ( C )= ( x i − , ( m, h ) , x i +2 ) − ( x i − ∗ ( m, g ) , (( m, h ) , x i +2 ) ∗ ( m, g )) − ( x i − , ( m, g ) , x i +2 )+ (cid:0) ( x i − , ( m, g )) ∗ ( m, h ) , x i +2 ∗ ( m, h ) (cid:1) ≡ ( x i − , ( m, h ) , x i +2 ) − ( x i − , ( m, gh ) , x i +2 )+ (cid:0) ( x i − , ( m, g )) ∗ ( m, h ) , x i +2 ∗ ( m, h ) (cid:1) = ( x i − , ( m, h ) , x i +2 ) − ( x i − , ( m, h ( h − gh )) , x i +2 )+ ( x i − ∗ ( m, h ) , (( m, h − gh ) , x i +2 ) ∗ ( m, h )) ≡ . When j = i and j = i + 1, we have ( x j − , x j +1 ) ≡ m, g ) ∗ x j is equal to that of ( m, h ) ∗ x j and the first elementof ( m, g ) ∗ x j is equal to that of ( m, h ) ∗ x j , we have (cid:0) x j − ∗ x j , (cid:0) x i − j +1 , ( m, g ) , ( m, h ) , x i +2 (cid:1) ∗ x j (cid:1) ∈ D BR n − ( X ; Z ) and (cid:16)(cid:0) x i − , ( m, g ) , ( m, h ) , x j − i +2 (cid:1) ∗ x j , x j +1 ∗ x j (cid:17) ∈ D BR n − ( X ; Z ) , where x ba means the sequence x a , x a +1 , . . . , x b − , x b . Then, ( x j − ∗ x j , x j +1 ∗ x i ) ≡ A ) ≡ ( D ) ≡
0. Hence, we have ∂ BR n ( x ) ≡ x i := ( m, gh ) and18 := ( x , . . . , x n ) = ( x , . . . , x i − , ( m, gh ) , x i +1 , . . . , x n ). We have ∂ BR n ( x ) = n X j =1 ( − j − (cid:8) ( x j − , x j +1 ) − ( x j − ∗ x j , x j +1 ∗ x j ) (cid:9) = i − X j =1 ( − j − (cid:8) ( x j − , x j +1 ) − ( x j − ∗ x j , x j +1 ∗ x j ) (cid:9) + i X j = i ( − j − (cid:8) ( x j − , x j +1 ) − ( x j − ∗ x j , x j +1 ∗ x j ) (cid:9) + n X j = i +1 ( − j − (cid:8) ( x j − , x j +1 ) − ( x j − ∗ x j , x j +1 ∗ x j ) (cid:9) . When 1 ≤ j ≤ i −
1, we have( x j − , x j +1 ) = ( x j − , x i − j +1 , ( m, gh ) , x i +1 ) ≡ (cid:0) x j − , x i − j +1 , ( m, g ) , x i +1 (cid:1) ( A ) + (cid:0)(cid:0) x j − , x i − j +1 (cid:1) ∗ ( m, g ) , (( m, h ) , x i +1 ) ∗ ( m, g ) (cid:1) ( B ) . Put x j := ( m j , g j ). Since ( m, g ′ ) ∗ x j = ( mϕ ( g j ) , g ′ ) for any g ′ ∈ G , we have( x j − ∗ x j , x j +1 ∗ x j )= ( x j − ∗ x j , (cid:0) x i − j +1 , ( m, gh ) , x i +1 (cid:1) ∗ x j )= ( x j − ∗ x j , x i − j +1 ∗ x j , ( mϕ ( g j ) , gh ) , x i +1 ∗ x j ) ≡ (cid:0) x j − ∗ x j , x i − j +1 ∗ x j , ( mϕ ( g j ) , g ) , x i +1 ∗ x j (cid:1) + (cid:0)(cid:0) x j − ∗ x j , x i − j +1 ∗ x j (cid:1) ∗ ( mϕ ( g j ) , g ) , (( mϕ ( g j ) , h ) , x i +1 ∗ x j ) ∗ ( mϕ ( g j ) , g ) (cid:1) = ( x j − ∗ x j , (cid:0) x i − j +1 , ( m, g ) , x i +1 (cid:1) ∗ x j ) ( C ) + (cid:0)(cid:0) x j − ∗ x j , x i − j +1 ∗ x j (cid:1) ∗ (( m, g ) ∗ x j ) , (( m, h ) ∗ x j , x i +1 ∗ x j ) ∗ (( m, g ) ∗ x j ) (cid:1) ( D ) . Then, we have( D ) = (cid:16)(cid:0) x j − ∗ ( m, g ) (cid:1) ∗ ( x j ∗ ( m, g )) , (cid:0) x i − j +1 ∗ ( m, g ) , ( m, h ) ∗ ( m, g ) , x i +1 ∗ ( m, g ) (cid:1) ∗ ( x j ∗ ( m, g )) (cid:17) ( D ′ ) . When j = i , we have( x j − , x j +1 ) = ( x i − , x i +1 ) ( E ) , ( x j − ∗ x j , x j +1 ∗ x j )= ( x i − ∗ ( m, gh ) , x i +1 ∗ ( m, gh ))= (cid:0)(cid:0) x i − ∗ ( m, g ) (cid:1) ∗ (( m, h ) ∗ ( m, g )) , ( x i +1 ∗ ( m, g )) ∗ (( m, h ) ∗ ( m, g )) (cid:1) ( F ) . When i + 1 ≤ j ≤ n , we have 19 x j − , x j +1 ) = ( x i − , ( m, gh ) , x j − i +1 , x j +1 ) ≡ ( x i − , ( m, g ) , x j − i +1 , x j +1 ) ( G ) + ( x i − ∗ ( m, g ) , { ( m, h ) , x j − i +1 , x j +1 } ∗ ( m, g )) ( H ) . (20)Put x j := ( m j , g j ). We have ( m ′ , g ′ ) ∗ x j = ( m ′ g j + m j ( ϕ ( g j ) − g j ) , g − j g ′ g j ) and( m ′ , g ′ ) ∗ m j = ( m ′ ϕ ( g j ) , g ′ ) for any ( m ′ , g ′ ) ∈ X . Put A := mg j + m j ( ϕ ( g j ) − g j ).We have( x j − ∗ x j , x j +1 ∗ x j )= ( x i − ∗ x j , ( A, g − j gg j g − j hg j ) , x j − i +1 ∗ x j , x j +1 ∗ x j ) ≡ ( x i − ∗ x j , ( A, g − j gg j ) , x j − i +1 ∗ x j , x j +1 ∗ x j )+ (cid:0)(cid:0) x i − ∗ x j (cid:1) ∗ ( A, g − j gg j ) , (cid:16) ( A, g − j hg j ) , x j − i +1 ∗ x j , x j +1 ∗ x j (cid:17) ∗ ( A, g − j gg j ) (cid:1) = (cid:0)(cid:0) x i − , ( m, g ) , x j − i +1 (cid:1) ∗ x j , x j +1 ∗ x j (cid:1) ( I ) + (cid:0) ( x i − ∗ x j ) ∗ (cid:0) ( m, g ) ∗ x j (cid:1) , (cid:16) ( m, h ) ∗ x j , x j − i +1 ∗ x j , x j +1 ∗ x j (cid:17) ∗ (cid:0) ( m, g ) ∗ x j (cid:1)(cid:1) ( J ) . (21)Then, we have J = (cid:16)(cid:0) x i − ∗ ( m, g ) , ( m, h ) ∗ ( m, g ) , x j − i +1 ∗ ( m, g ) (cid:1) ∗ ( x j ∗ ( m, g )) , (cid:0) x j +1 ∗ ( m, g ) (cid:1) ∗ ( x j ∗ ( m, g )) (cid:17) ( J ′ ) . Hence, we have ∂ BR n ( x ) ≡ i − X j =1 ( − j − ( A + B − ( C + D ′ ))+ ( − i − ( E − F ) + n X j = i +1 ( − j − ( G + H − ( I + J ′ )) . By the definition of the map ∂ BR n , we have ∂ BR n ( x i − , ( m, g ) , x i +1 ) = i − X j =1 ( − j − ( A − C )+ ( − i − ( E − ( x i − ∗ ( m, g ) , x i +1 ∗ ( m, g ))) + n X j = i +1 ( − j − ( G − I )and ∂ BR n ( x i − ∗ ( m, g ) , (( m, h ) , x i +1 ) ∗ ( m, g )) = i − X j =1 ( − j − ( B − D ′ )+ ( − i − (( x i − ∗ ( m, g ) , x i +1 ∗ ( m, g )) − F ) + n X j = i +1 ( − j − ( H − J ′ ) . ∂ BR n ( x ) ≡ ∂ BR n ( x i − , ( m, g ) , x i +1 ) + ∂ BR n ( x i − ∗ ( m, g ) , (( m, h ) , x i +1 ) ∗ ( m, g )) . This completes the proof.The normalized birack chain complex is C norBR ∗ ( X ; Z ) := C BR ∗ ( X ; Z ) /D BR ∗ ( X ; Z ).It determines the normalized birack homology group H norBR n ( X ; Z ). In the or-dinary way, for an abelian group A , we have the (co)homology theory with thecoefficient group A and the homology group H norBR n ( X ; A ) and the cohomologygroup H n norBR ( X ; A ) are defined. D BR U ∗ ( X ; Z ) of C BR U ∗ ( X ; Z )We introduce the degenerate subcomplex D BR U ∗ ( X ; Z ) of C BR U ∗ ( X ; Z ), which isa counterpart of the degenerate subcomplex D BR ∗ ( X ; Z ) of C BR ∗ ( X ; Z ).Let D BR U n ( X ; Z ) be the subgroup of C BR U n ( X ; Z ) generated by the elementsof the following sets n − [ i =1 n ( g ; m i − , , m i +1 ) (cid:12)(cid:12)(cid:12) g ∈ G n , m ∈ M n o and n [ i =1 n ( g i − , g i h, g i +1 ; m ) − ( g ; m ) − ( g − i g i − g i , h, g i +1 ; m i − g i , m i ϕ ( g i )) (cid:12)(cid:12)(cid:12) h ∈ G , g ∈ G n , m ∈ M n o for n ≥
2, where we write( g ; m i − , , m i +1 ) := ( g , . . . , g n ; m , . . . , m i − , , m i +1 , . . . , m n ) , ( g i − , g i h, g i +1 ; m ) := ( g , . . . , g i − , g i h, g i +1 , . . . , g n ; m , . . . , m n ) and( g − i g i − g i , h, g i +1 ; m i − g i , m i ϕ ( g i )) := (( g − i g g i ) , . . . , ( g − i g i − g i ) ,h, g i +1 , . . . , g n ; m g i , . . . , m i − g i , m i ϕ ( g i ) , . . . , m n ϕ ( g i )) . We define D BR U n ( X ; Z ) = 0 for n ≤ Lemma 5.2. D BR U ∗ ( X ; Z ) := ( D BR U n ( X ; Z ) , ∂ BR U n ) n ∈ Z is a subcomplex of C BR U ∗ ( X ; Z ) .Proof. In Lemma 5.3, we will show that the isomorphism γ n defined in Subsec-tion 3.1 gives an isomorphism D BR n ( X ; Z ) ∼ = D BR U n ( X ; Z ).The normalized U-birack chain complex is C norBR U ∗ ( X ; Z ) := C BR U ∗ ( X ; Z ) /D BR U ∗ ( X ; Z ).It determines the homology group H norBR U n ( X ; Z ). In the ordinary way, for anabelian group A , H norBR U n ( X ; A ) and H n norBR U ( X ; A ) are defined. γ n Next lemma shows that the isomorphism γ n : C BR n ( X ; Z ) → C BR U n ( X ; Z ), whichis defined in Subsection 3.1, induces the isomorphism γ n : C norBR n ( X ; Z ) → C norBR U n ( X ; Z ), where we denote it by the same symbol γ n for simplicity. Lemma 5.3.
It holds that γ n ( D BR n ( X ; Z )) = D BR U n ( X ; Z ) . Therefore, γ n in-duces the isomorphism γ n : C norBR n ( X ; Z ) → C norBR U n ( X ; Z ) . roof. Let x ∈ D BR n ( X ; Z ). If x = ( x i − , ( m, g i ) , ( m, g i +1 ) , x i +2 ), then we put γ n ( x ) = ( g , . . . , g n ; m . . . , m n ). By the definition of γ n , we have m i = m − m =0. Then, γ n ( x ) ∈ D BR U n ( X ; Z ).Suppose that x = ( x i − , ( m i , g i h ) , x i +1 ) − ( x i − , ( m i , g i ) , x i +1 ) − (cid:0) x i − ∗ ( m i , g i ) , (( m i , h ) , x i +1 ) ∗ ( m i , g i ) (cid:1) . We have (cid:0) x i − ∗ ( m i , g i ) , (cid:0) ( m i , h ) , x i +1 (cid:1) ∗ ( m i , g i ) (cid:1) = (cid:0) ( A , g − i g g i ) , . . . , ( A i − , g − i g i − g i ) , ( m i ϕ ( g i ) , h ) , ( m i +1 ϕ ( g i ) , g i +1 ) , . . . , ( m n ϕ ( g i ) , g n ) (cid:1) , where A j = m j g i + m i ( ϕ ( g i ) − g i ). We note that A j − A j +1 = ( m j − m j +1 ) g i for j with 1 ≤ j < i − A i − − m i ϕ ( g i ) = ( m i − − m i ) g i . Then, γ n (cid:0) ( x i − ∗ ( m i , g i ) , (cid:0) ( m i , h ) , x i +1 (cid:1) ∗ ( m i , g i )) (cid:1) = ( g − i g i − g i , h, g i +1 ; m ′ g i , . . . , m ′ i − g i , m ′ i ϕ ( g i ) , . . . , m ′ n − ϕ ( g i ) , m n ϕ ( g i )) , where m ′ i := m i − m i +1 . Then, we have γ n ( x ) = ( g i − , g i h, g i +1 ; m ′ , . . . , m ′ n − , m n ) − ( g ; m ′ , . . . , m ′ n − , m n ) − ( g − i g i − g i , h, g i +1 ; m ′ g i , . . . , m ′ i − g i ,m ′ i ϕ ( g i ) , . . . , m ′ n − ϕ ( g i ) , m n ϕ ( g i )) ∈ D BR n ( X ; Z ) . Hence γ n ( D BR n ( X ; Z )) ⊂ D BR U n ( X ; Z ).Let z = ( g ; x i − , , x i +1 ) be an element of D BR U n ( X ; Z ). Put a k := P nj = k m j .Then we have a n = m n and a k − a k +1 = m k for all 1 ≤ k ≤ n −
1. Moreover, a i = a i +1 . Then, we have γ n (cid:0) ( a , g ) , . . . , ( a n , g n ) (cid:1) = z ∈ D BR U n ( X ; Z ) . Let z = ( g i − , g i h, g i +1 ; x ) − ( g ; x ) − ( g − i g i − g i , h, g i +1 ; x i − g i , x i ϕ ( g i )) ∈ D BR U n ( X ; Z ) . Put a k := P nj = k m j . Then we have a n = m n and a k − a k +1 = m k for all 1 ≤ k ≤ n −
1. Define x k := ( a k , g k ) and x := ( x , . . . , x n ). Then, wehave γ n (cid:16) ( x i − , ( a i , g i h ) , x i +1 ) − ( x i − , ( a i , g i ) , x i +1 ) − ( x i − ∗ ( a i , g i ) , (( a i , h ) , x i +1 ) ∗ ( a i , g i )) (cid:17) = z . Hence γ − n ( D BR U n ( X ; Z )) ⊂ D BR n ( X ; Z ). This completes the proof. ψ n,λ We fix a group homomorphism λ : G → A . We also define a map ˜ λ : C BR U n ( X ; Z ) → A as ˜ λ ( g , . . . , g n ; m ) := λ ( g ). In addition, we define a map ψ n,λ : C BR U n ( X ; Z ) → C gp n ( M ; Z ) G ⊗ A by ψ n,λ ( g ; m ) := ψ n ( g ; m ) ⊗ ˜ λ ( g ; m ) for any integer n ≥ Lemma 5.4.
It holds that ψ n,λ ( D BR U n ( X ; Z )) ⊂ D gp n ( M ; Z ) G ⊗ A . Therefore, ψ n,λ induces the homomorphism ψ n,λ : C norBR U n ( X ; Z ) → C norgp n ( M ; Z ) G ⊗ A. roof. It suffices to show that ψ n,λ ( g ; m i − , , m i +1 ) ∈ D gp n ( M ; Z ) G ⊗ A for i with 1 ≤ i ≤ n − ψ n,λ ( z i ) ∈ D gp n ( M ; Z ) G ⊗ A for i with 1 ≤ i ≤ n , where z i := ( g i − , g i h, g i +1 ; m ) − ( g ; m ) − ( g − i g i − g i , h, g i +1 ; m i − g i , m i ϕ ( g i )) . We have ψ n ( g ; m i − , , m i +1 )= X k ∈K n ( − | k | ( m g k , . . . , m i − g k i − , , m i +1 g k i +1 , . . . , m n g k n ) ∈ D gp n ( M ; Z ) G . This inclusion ψ n ( g ; m i − , , m i +1 ) ∈ D gp n ( M ; Z ) G implies ψ n,λ ( g ; m i − , , m i +1 ) ∈ D gp n ( M ; Z ) G ⊗ A .Next, we show ψ n,λ ( z i ) ∈ D gp n ( M ; Z ) G ⊗ A .For i = 1, we can prove that ψ n ( z i ) = 0 ∈ D gp n ( M ; Z ) G as follows. Let s be aninteger with 1 < s < i −
1. By the direct calculation, we have ψ n ( g − i g g i , . . . , g − i g i − g i , h, g i +1 , . . . , g n ; m g i , . . . , m i − g i , m i ϕ ( g i ) , . . . , m n ϕ ( g i ))= X k ∈K i ( − | k | (cid:16) m ϕ ( g k ) g k · · · g k i − i − g i h g k i +1 i +1 · · · g k n n , . . . ,m s ϕ ( g k · · · g k s s ) g k s +1 s +1 · · · g k i − i − g i h g k i +1 i +1 · · · g k n n , . . . ,m i − ϕ ( g k · · · g k i − i − ) g i h g k i +1 i +1 · · · g k n n ,m i ϕ ( g k · · · g k i − i − g i h ) g k i +1 i +1 · · · g k n n , . . . ,m n ϕ ( g k · · · g k i − i − g i h g k i +1 i +1 · · · g k n n ) (cid:17) (22)+ X k ∈K i ( − | k | (cid:16) m ϕ ( g k ) g k · · · g k i − i − g i h g k i +1 i +1 · · · g k n n , . . . ,m s ϕ ( g k · · · g k s s ) g k s +1 s +1 · · · g k i − i − g i h g k i +1 i +1 · · · g k n n , . . . ,m i − ϕ ( g k · · · g k i − i − ) g i h g k i +1 i +1 · · · g k n n ,m i ϕ ( g k · · · g k i − i − g i h ) g k i +1 i +1 · · · g k n n , . . . ,m n ϕ ( g k · · · g k i − i − g i h g k i +1 i +1 · · · g k n n ) (cid:17) . (23)We also have ψ n ( g , . . . , g i − , g i , g i +1 , . . . , g n ; m , . . . , m n )= X k ∈K i ( − | k | (cid:16) m ϕ ( g k ) g k · · · g k i − i − g i g k i +1 i +1 · · · g k n n , . . . ,m s ϕ ( g k · · · g k s s ) g k s +1 s +1 · · · g k i − i − g i g k i +1 i +1 · · · g k n n , . . . ,m i − ϕ ( g k · · · g k i − i − ) g i g k i +1 i +1 · · · g k n n ,m i ϕ ( g k · · · g k i − i − g i ) g k i +1 i +1 · · · g k n n , . . . ,m n ϕ ( g k · · · g k i − i − g i g k i +1 i +1 · · · g k n n ) (cid:17) (24)+ X k ∈K i ( − | k | (cid:16) m ϕ ( g k ) g k · · · g k i − i − g i g k i +1 i +1 · · · g k n n , . . . ,m s ϕ ( g k · · · g k s s ) g k s +1 s +1 · · · g k i − i − g i g k i +1 i +1 · · · g k n n , . . . ,m i − ϕ ( g k · · · g k i − i − ) g i g k i +1 i +1 · · · g k n n ,m i ϕ ( g k · · · g k i − i − g i ) g k i +1 i +1 · · · g k n n , . . . ,m n ϕ ( g k · · · g k i − i − g i g k i +1 i +1 · · · g k n n ) (cid:17) . (25)23ince (25) = − (22), we have ψ n ( g , . . . , g i − , g i h, g i +1 , . . . , g n ; m , . . . , m n )= (23) + (24)= (22) + (23) + (24) + (25) . Then, we have ψ n ( z i ) = 0 ∈ D gp n ( M ; Z ) G .Therefore, we have ψ n,λ ( z i ) = 0 ∈ D gp n ( M ; Z ) G ⊗ A .For the case that i = 1, we have ψ n ( h, g , . . . , g n ; m ϕ ( g ) , . . . , m n ϕ ( g ))= X k ∈K n ( − | k | (cid:0) m ϕ ( g ) ϕ ( h k ) g k · · · g k n n , . . . , m n ϕ ( g ) ϕ ( h k g k · · · g k n n ) (cid:1) = X k ∈K n ( − | k | (cid:0) m ϕ ( h k ) g k · · · g k n n , . . . , m n ϕ ( h k g k · · · g k n n ) (cid:1) = X k ∈K n ( − | k | (cid:0) m ϕ (( g h ) k ) g k · · · g k n n , . . . , m n ϕ (( g h ) k g k · · · g k n n ) (cid:1) = ψ n ( g h, g , . . . , g n ; m , . . . , m n )= ψ n ( g , g , . . . , g n ; m , . . . , m n )since k = 0. Then, we have ψ n,λ ( g h, g , . . . , g n ; m , . . . , m n )= ψ n ( g h, g , . . . , g n ; m , . . . , m n ) ⊗ λ ( g h )= ψ n ( g h, g , . . . , g n ; m , . . . , m n ) ⊗ ( λ ( g ) + λ ( h ))= ψ n ( g , g , . . . , g n ; m , . . . , m n ) ⊗ λ ( g )+ ψ n ( h, g , . . . , g n ; m ϕ ( g ) , . . . , m n ϕ ( g )) ⊗ λ ( h )= ψ n,λ ( g , g , . . . , g n ; m , . . . , m n ) + ψ n,λ ( h, g , . . . , g n ; m ϕ ( g ) , . . . , m n ϕ ( g )) . Then, we have ψ n,λ ( z i ) = 0 ∈ D gp n ( M ; Z ) G ⊗ A . G -Alexander MCB with the trivial X -set Let X = ⊔ m ∈ M ( { m } × G ) be the G -Alexander multiple conjugation biquandleof ( M, ϕ ). We consider the chain complex C ∗ ( X ; Z ) defined in Subsection 2.4,where the X -set Y is the trivial X -set. Thus the first element of C n ( X ; Z ) isomitted. For example, an element h y ih a ih b i − h y ih a, ab i + h y ih b, ab i ∈ C ( X ; Z )is written by h a ih b i − h a, ab i + h b, ab i for simplicity.Define a homomorphism proj n : P n ( X ; Z ) → C BR n ( X ; Z ) byproj n ( h x i · · · h x k i ) = (cid:26) ( x , x , . . . , x k ) ( k = n )0 (otherwise)for n ≥
2, where h x j i means h x j , x j , . . . , x jn j i for each j ∈ { , . . . , k } . Wedefine proj n = 0 for n ≤ Lemma 5.5.
It holds that proj n ( D n ( X ; Z )) ⊂ D BR n ( X ; Z ) . roof. Let ( m , g ) , . . . , ( m n , g n ) be elements of X such that m i = m i +1 forsome i . Put x i := ( m i , g j ). We show thatproj n (cid:16) h x i · · · h x i − ih x i ih x i +1 ih x i +2 i · · · h x n i−h x i · · · h x i − ihh x i ih x i +1 iih x i +2 i · · · h x n i (cid:17) ∈ D BR n ( X ; Z ) . We haveproj n ( h x i · · · h x i − ih x i ih x i +1 ih x i +2 i · · · h x n i− h x i · · · h x i − ihh x i ih x i +1 iih x i +2 i · · · h x n i )= proj n ( h x i · · · h x i − ih x i ih x i +1 ih x i +2 i · · · h x n i− h x i · · · h x i − i ( h x i , x i x i +1 i − h x i +1 , x i x i +1 i ) h x i +2 i · · · h x n i )= proj n ( h x i · · · h x i − ih x i ih x i +1 ih x i +2 i · · · h x n i ) − proj n ( h x i · · · h x i − ih x i , x i x i +1 ih x i +2 i · · · h x n i )+ proj n ( h x i · · · h x i − ih x i +1 , x i x i +1 ih x i +2 i · · · h x n i )= proj n ( h x i · · · h x i − ih x i ih x i +1 ih x i +2 i · · · h x n i )= proj n ( h ( m , g ) i · · · h ( m i , g i ) ih ( m i , g i +1 ) i · · · h ( m n , g n ) i )= (( m , g ) , . . . , ( m i , g i ) , ( m i , g i +1 ) , . . . , ( m n , g n )) ∈ D BR n ( X ; Z ) . We have proj n ( D n ( X ; Z )) ⊂ D BR n ( X ; Z ).Therefore, the homomorphism proj n : P n ( X ; Z ) → C BR n ( X ; Z ) induces thehomomorphism proj n : C n ( X ; Z ) → C norBR n ( X ; Z ). Lemma 5.6.
The map proj : C ∗ ( X ; Z ) → C norBR ∗ ( X ; Z ) is a chain map, thatis, it holds that for any integer n proj n − ◦ ∂ n = ∂ norBR n ◦ proj n . Proof.
It is sufficient to consider h x i · · · h x n i in C n ( X ; Z ). For h x i · · · h x n i , wehave ∂ norBR n ◦ proj n ( h x i · · · h x n i )= ∂ norBR n ( x , . . . , x n )= n X i =1 ( − i − { ( x i − , x i +1 ) − ( x i − ∗ x i , x i +1 ∗ x i ) } = n X i =1 ( − i − (cid:8) proj n − ( h x i · · · h x i − ih x i +1 i · · · h x n i ) − proj n − ( h x ∗ x i i · · · h x i − ∗ x i ih x i +1 ∗ x i i · · · h x n ∗ x i i ) (cid:9) = proj n − ◦ ∂ n ( h x i · · · h x n i ) . We have proj n − ◦ ∂ n = ∂ norBR n ◦ proj n .Fix a group homomorphism λ : G → A . As a consequence of Lemma 5.6and Subsection 5.1, we have the sequence C n ( X ; Z ) proj n −→ C norBR n ( X ; Z ) γ n −→ C norBR U n ( X ; Z ) ψ n,λ −→ C norgp n ( M ; Z ) G ⊗ A of chain groups. Therefore, we have the following theorem.25 heorem 5.7. For any n -cocycle f : C norgp n ( M ; Z ) G → A , the map Φ f,λ := ( f ⊗ id A ) ◦ ψ n,λ ◦ γ n ◦ proj n : C n ( X ; Z ) → A is an n -cocycle of the G -Alexander MCB X = ⊔ m ∈ M ( { x } × G ) . The following theorem follows from the direct calculation.
Theorem 5.8. (1) Let f : M → A be a G -invariant A -multilinear map.The -cocycle Φ f,λ = ( f ⊗ id A ) ◦ ψ ,λ ◦ γ ◦ proj : C ( X ; Z ) → A of the G -Alexander MCB X = F m ∈ M ( { m } × G ) is formulated as Φ f,λ ( h ( m , g ) ih ( m , g ) i ) = f (cid:16) m − m , m (1 − ϕ ( g ) g − ) (cid:17) ⊗ λ ( g ) for h ( m , g ) ih ( m , g ) i ∈ X ⊂ C ( X ; Z ) .(2) Let f : M → A be a G -invariant A -multilinear map. The -cocycle Φ f,λ = ( f ⊗ id A ) ◦ ψ ,λ ◦ γ ◦ proj : C ( X ; Z ) → A of the G -AlexanderMCB X = F m ∈ M ( { m } × G ) is formulated as Φ f,λ ( h ( m , g ) ih ( m , g ) ih ( m , g ) i )= f (cid:16) ( m − m )(1 − ϕ ( g ) − g ) , m − m , m (1 − ϕ ( g ) g − ) (cid:17) ⊗ λ ( g ) for h ( m , g ) ih ( m , g ) ih ( m , g ) i ∈ X ⊂ C ( X ; Z ) . G -Alexander multiple conjugationbiquandles with the X -set X Throughout this section, let X = F m ∈ M ( { m } × G ) = M × G the G -Alexandermultiple conjugation biquandle of ( M, ϕ ). We assume that the X -set Y is X itself. Our goal in this section is to give Theorem 6.8. D BR ∗ ( X ; Z ) X , D BR U ∗ ( X ; Z ) X andthe induced homomorphisms γ n , ψ n and ψ n,λ D BR ∗ ( X ; Z ) X of C BR ∗ ( X ; Z ) X Let D BR n ( X ; Z ) X be the subgroup of C BR n ( X ; Z ) X generated by the elements ofthe following sets n − [ i =1 n ( x i − , ( m, g ) , ( m, h ) , x i +2 ) (cid:12)(cid:12)(cid:12) x ∈ X × X n , m ∈ M, g, h ∈ G o and n [ i =1 ( ( x i − , ( m, gh ) , x i +1 ) − ( x i − , ( m, g ) , x i +1 ) − ( x i − ∗ ( m, g ) , (cid:0) ( m, h ) , x i +1 (cid:1) ∗ ( m, g )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x ∈ X × X n ,m ∈ M, g, h ∈ G ) for n ≥
2, where we write (cid:0) x i − ∗ ( m, g ) , (cid:0) ( m, h ) , x i +1 (cid:1) ∗ ( m, g ) (cid:1) = ( x ∗ ( m, g ) , . . . , x i − ∗ ( m, g ) , ( m, h ) ∗ ( m, g ) , x i +1 ∗ ( m, g ) , . . . , x n ∗ ( m, g )) . Define D BR n ( X ; Z ) X := 0 for n ≤
1. 26 emma 6.1. D BR ∗ ( X ; Z ) X := ( D BR n ( X ; Z ) X , ∂ BR n ) n ∈ Z is a subcomplex of C BR ∗ ( X ; Z ) X . The chain complex C norBR ∗ ( X ; Z ) X := C BR ∗ ( X ; Z ) X /D BR ∗ ( X ; Z ) X determines the homology group H norBR n ( X ; Z ) X . In the ordinary way, for anabelian group A , we have the (co)homology theory with the coefficient group A and the homology group H norBR n ( X ; A ) X and the cohomology group H n norBR ( X ; A ) X are defined. D BR U ∗ ( X ; Z ) X of C BR U ∗ ( X ; Z ) X Let D BR U n ( X ; Z ) X be the subgroup of C BR U n ( X ; Z ) X generated by the elementsof the following sets n − [ i =1 n ( g ; m i − , , m i +1 ) (cid:12)(cid:12)(cid:12) g ∈ G × G n , m ∈ M × M n o and n [ i =1 n ( g i − , g i h, g i +1 ; m ) − ( g ; m ) − ( g − i g i − g i , h, g i +1 ; m i − g i , m i ϕ ( g i )) (cid:12)(cid:12)(cid:12) h ∈ G , g ∈ G × G n , m ∈ M × M n o for n ≥
2, where we write( g ; m i − , , m i +1 ) := ( g , g , . . . , g n ; m , m , . . . , m i − , , m i +1 , . . . , m n ) , ( g i − , g i h, g i +1 ; m ) := ( g , g , . . . , g i − , g i h, g i +1 , . . . , g n ; m , m , . . . , m n ) and( g − i g i − g i , h, g i +1 ; m i − g i , m i ϕ ( g i )) := (cid:0) ( g − i g g i ) , ( g − i g g i ) , . . . , ( g − i g i − g i ) ,h, g i +1 , . . . , g n ; m g i , m g i , . . . , m i − g i , m i ϕ ( g i ) , . . . , m n ϕ ( g i ) (cid:1) . Define D BR U n ( X ; Z ) X := 0 for n ≤ Lemma 6.2. D BR U ∗ ( X ; Z ) X := ( D BR U n ( X ; Z ) X , ∂ BR U n ) n ∈ Z is a subcomplex of C BR U ∗ ( X ; Z ) X . The chain complex C norBR U ∗ ( X ; Z ) X := C BR U ∗ ( X ; Z ) X /D BR U ∗ ( X ; Z ) X determines the homology group H norBR U n ( X ; Z ) X . In the ordinary way, foran abelian group A , we have the (co)homology theory with the coefficientgroup A and the homology group H norBR U n ( X ; A ) X and the cohomology group H n norBR U ( X ; A ) X are defined. γ n Next lemma shows that the isomorphism γ n : C BR n ( X ; Z ) X → C BR U n ( X ; Z ) X defined in Subsection 4.1 induces the isomorphism γ n : C norBR n ( X ; Z ) X → C norBR U n ( X ; Z ) X , where we denote it by the same symbol γ n for simplicity. Lemma 6.3.
It holds that γ n ( D BR n ( X ; Z ) X ) = D BR U n ( X ; Z ) X . Therefore γ n induces the isomorphism γ n : C norBR n ( X ; Z ) X → C norBR U n ( X ; Z ) X . .1.4 The induced homomorphisms ψ n and ψ n,λ Next lemma shows that the map ψ n : C BR U n ( X ; Z ) X → C gp n +1 ( M ; Z ) G de-fined in Subsection 4.1 induces the homomorphism ψ n : C norBR U n ( X ; Z ) X → C norgp n +1 ( M ; Z ) G , where we denote it by the same symbol ψ n for simplicity. Lemma 6.4.
It holds that ψ n ( D BR U n ( X ; Z ) X ) ⊂ D gp n +1 ( M ; Z ) G . Therefore ψ n induces the homomorphism ψ n : C norBR U n ( X ; Z ) X → C norgp n +1 ( M ; Z ) G . In addition, a homomorphism ψ n,λ : C norBR U n ( X ; Z ) X → C norgp n +1 ( M ; Z ) G ⊗ A is also defined as follows: We fix a group homomorphism λ : G → A . We define amap ˜ λ : C BR U n ( X ; Z ) X → A as ˜ λ ( g , g , . . . , g n ; m , m , . . . , m n ) = λ ( g ). Definea map ψ n,λ : C BR U n ( X ; Z ) X → C gp n +1 ( M ; Z ) G ⊗ A by ψ n,λ = ψ n ⊗ ˜ λ for n ≥ ψ n,λ := 0 for n < Lemma 6.5.
It holds that ψ n,λ ( D BR U n ( X ; Z ) X ) ⊂ D gp n +1 ( M ; Z ) G ⊗ A . Therefore ψ n,λ induces the homomorphism ψ n,λ : C norBR U n ( X ; Z ) X → C norgp n +1 ( M ; Z ) G ⊗ A. G -Alexander MCB with the X -set X Define a homomorphism proj n : P n ( X ; Z ) X → C BR n ( X ; Z ) X byproj n ( h x ih x i · · · h x k i ) = (cid:26) ( x , x , . . . , x k ) ( k = n )0 (otherwise)for n ≥
2, where h x j i means h x j , x j , . . . , x jn j i for each j ∈ { , . . . , k } . Defineproj n = 0 for n ≤ Lemma 6.6.
It holds that proj n ( D n ( X ; Z ) X ) ⊂ D BR n ( X ; Z ) X . Therefore, the homomorphism proj n : P n ( X ; Z ) X → C BR n ( X ; Z ) X inducesthe homomorphism proj n : C n ( X ; Z ) X → C norBR n ( X ; Z ) X . Lemma 6.7.
The map proj is a chain map, that is, it holds that proj n − ◦ ∂ n = ∂ norBR n ◦ proj n . As a consequence of Lemma 6.7 and Subsection 6.1, we have two sequences C n ( X ; Z ) X proj n −→ C norBR n ( X ; Z ) X γ n −→ C norBR U n ( X ; Z ) X ψ n −→ C norgp n +1 ( M ; Z ) G ⊗ A,C n ( X ; Z ) X proj n −→ C norBR n ( X ; Z ) X γ n −→ C norBR U n ( X ; Z ) X ψ n,λ −→ C norgp n +1 ( M ; Z ) G ⊗ A. Therefore, we have the following theorem.
Theorem 6.8.
For any ( n + 1) -cocycle f : C norgp n +1 ( M ; Z ) G → A , the maps Φ f := f ◦ ψ n ◦ γ n ◦ proj n : C n ( X ; Z ) X → A and Φ f,λ := ( f ⊗ id A ) ◦ ψ n,λ ◦ γ n ◦ proj n : C n ( X ; Z ) X → A are n -cocycles of the G -Alexander MCB X = F m ∈ M ( { m } × G ) . emark 6.9. The cocycle Φ f does not use the information g of ( m , g ) ∈ X = M × G . Therefore, we may replace the X -set X = M × G with M , thatis, Φ f can be also defined as the map from C n ( X ; Z ) X to A . Theorem 6.10. (1) Let f : M → A be a G -invariant A -multilinear map.The -cocycle Φ f = f ◦ ψ ◦ γ : C ( X ; Z ) X → A of the G -Alexander MCB X = M × G = F m ∈ M ( { m } × G ) is formulated as Φ f (cid:0) h ( m , g ) ih ( m , g ) ih ( m , g ) i (cid:1) = f (cid:0) m ′ (1 − ϕ ( g ) − g ) , m ′ , m (1 − ϕ ( g ) g − ) (cid:1) for any h ( m , g ) ih ( m , g ) ih ( m , g ) i ∈ X × X , where m ′ i := m i − m i +1 .(2) Let f : M → A be a G -invariant A -multilinear map. The -cocycle Φ f = f ◦ ψ ◦ γ : C ( X ; Z ) X → A of the G -Alexander MCB X = M × G = F m ∈ M ( { m } × G ) is formulated as Φ f (cid:0) h ( m , g ) ih ( m , g ) ih ( m , g ) ih ( m , g ) i (cid:1) = f (cid:0) m ′ (1 − ϕ ( g ) − g ) , m ′ , m ′ , m (1 − ϕ ( g ) g − ) (cid:1) − f (cid:0) m ′ (1 − ϕ ( g ) − g ) g , m ′ g , m ′ ϕ ( g ) , m (1 − ϕ ( g ) g − ) ϕ ( g ) (cid:1) for any h ( m , g ) ih ( m , g ) ih ( m , g ) ih ( m , g ) i ∈ X × X , where m ′ i := m i − m i +1 . Theorem 6.11. (1) Let f : M → A be a G -invariant A -multilinear map.The -cocycle Φ f,λ = ( f ⊗ id A ) ◦ ψ ,λ ◦ γ : C ( X ; Z ) X → A of the G -Alexander MCB X = ⊔ m ∈ M ( { x } × G ) is formulated as Φ f,λ ( h ( m , g ) ih ( m , g ) ih ( m , g ) i )= f (cid:16) m ′ (1 − ϕ ( g ) − g ) , m ′ , m (1 − ϕ ( g ) g − ) (cid:17) ⊗ λ ( g ) for any h ( m , g ) ih ( m , g ) ih ( m , g ) i ∈ X × X , where m ′ i := m i − m i +1 .(2) Let f : M → A be a G -invariant A -multilinear map. The -cocycle Φ f,λ = ( f ⊗ id A ) ◦ ψ ,λ ◦ γ : C ( X ; Z ) X → A of the G -Alexander MCB X = ⊔ m ∈ M ( { x } × G ) is formulated as Φ f,λ ( h ( m , g ) ih ( m , g ) ih ( m , g ) ih ( m , g ) i )= f (cid:16) m ′ (1 − ϕ ( g ) − g ) , m ′ , m ′ , m (1 − ϕ ( g ) g − ) (cid:17) ⊗ λ ( g ) − f (cid:16) m ′ (1 − ϕ ( g ) − g ) g , m ′ g , m ′ ϕ ( g ) , m (1 − ϕ ( g ) g − ) ϕ ( g ) (cid:17) ⊗ λ ( g ) for any h ( m , g ) ih ( m , g ) ih ( m , g ) ih ( m , g ) i ∈ X × X , where m ′ i := m i − m i +1 . References [1] S. Carter, M. Elhamdadi and M. Saito,
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