CB3σ1
c⊗Id σ2a−1(Id ⊗c)a α B3
(V1⊗V2)⊗V3)∼
→(Vi1⊗Vi2)⊗Vi3(i1, i2, i3)
α(V1⊗V2)⊗V3K3= ker(B3→S3)ϕ∈K3
ψ∈K2
ψ∈K2σ2m
λ= 2m+ 1 ϕ∈K3f(σ2
1, σ2
2)(σ1σ2)3nn∈Z
f(X, Y ) (σ1σ2)3= (σ2σ1)3
B3
(λ, f, n)n= 0
(λ, f)
f(Y, X) = f(X, Y )−1
f(X3, X1)Xm
3f(X2, X3)Xm
2f(X1, X2)Xm
1= 1 X1X2X3= 1, m = (λ−1)/2
f(x12, x23x24)f(x13x23, x34) = f(x23x34)f(x12x13, x24x34)fx12, x23),
16i < j 6n xij = (σj−1· · · σi)σ2
i(σj−1· · · σi)−1Kn
(λ, f)λ∈1 + 2Z
C
C0F:C1→ C2F
C0
1C0
2
(λ1, f1).(λ2, f2) =
(λ, f)
λ=λ1λ2,
f(X, Y ) = f1(f2(X, Y )Xλ2f2(X, Y )−1, , Y λ2).f2(X, Y )
(A, ∆, ε, Φ, R)
A
(λ, f) Φ R
R=R(R21R)m= (RR21)mR m = (λ−1)/2
Φ = Φf(R21R12,Φ−1R32R23Φ)
=f(ΦR21R12Φ−1, R32R23)Φ
(λ, f)