Computing Adem Cohomology Operations

arXiv:1112.2494v1 [math.AT] 12 Dec 2011 Computing Adem Cohomology Operations Rocio Gonzalez–Diaz, Pedro Real Depto. de Matematica Aplicada I, E.T.S.I. Informatica, Universidad de Sevilla (Spain) e-mail: {rogodi,real}@us.es July, 2006 Abstract We deal with the problem of obtaining explicit simplicial formulae defining the classical Adem cohomology operations at the cochain level. Having these formulae at hand, we design an algorithm for computing these operations for any finite simplicial set. Keywords: Cohomology operations, computational topology, simplicial topology 1 Introduction Cohomology operations are algebraic operations on the cohomology groups of spaces. This machinery is useful when the (co)homology and the cup product on cohomology fail to distinguish two spaces. In the literature, there exists classical methods for computing the (co)homology of spaces (see [Mun84]). Our aim is the computation of cohomology operations on any finite simplicial set. Steenrod squares [Ste47], Steenrod reduced powers [Ste52] and Adem secondary cohomology operations [Ade52] constitute important classes of cohomology operations. In [GR98, GR99a, GR02b, GR03], the classical definition of Steenrod cohomology operations are reinterpreted in terms of permutations and contractions (special type of homotopy equivalences) of explicit chains for obtaining 1 explicit formulae for these operations at the cochain level. The underlying combinatorial structures of these formulae are also studied in detail. In [GRb], this point of view is extended to a general theory for obtaining explicit formulae for any cohomology operation that can be expressed at the cochain level. [GRa] is devoted to give general techniques for simplifying the underlying simplicial structures of the formulae obtained using the process explained in [GRb]. In this paper, we deal with the problem of the computation of Adem cohomology operations. For this task, we apply the machinery developed in [GRb] to this special case in order to obtain explicit formulae at the cochain level of these operations. Some properties are also given in order to obtain simplicial “economical” formulae in the sense that they are expressed only in terms of face operators. Finally, since an explicit contraction of the cochain complex to the cohomology can be constructed [GR03], an algorithm for computing Adem cohomology operations at the cohomology level for nay finite simplicial set is designed in detail. The organization of the paper follows. In Section 2, we introduce the theoretical background from Algebraic Topology on the concepts we use here. In Section 3, we give a procedure for obtaining explicit formulae for the morphisms E3i+3 that are involved in the definition of Adem cohomology operations at the cochain level. In Section 4, we design an algorithm for computing Adem cohomology operations using the explicit formulae obtained before. 2 Background We introduce the basic notation and terminology we use throughout this paper. References for the material in this section are [May67, McL75]. Asimplicial set Kis a graded set indexed by the non–negative integers together with face and degeneracy operators ∂i:Kq→Kq−1and si:Kq→ Kq+1, 0 ≤i≤q, satisfying the following identities: ∂i∂j=∂j−1∂iif i < j, sisj=sj+1siif i≤j, ∂isj=sj−1∂iif i < j,∂jsj= 1K=∂j+1sj,∂isj=sj∂i−1if i > j+1. 2 The elements of Kqare called q–simplices. A simplex xis degenerate if x=siyfor some simplex yand degeneracy operator si; otherwise, xis non– degenerate. The cartesian product K×Lis a simplicial set whose simplices and face and degeneracy operators are given by (K×L)q=Kq×Lq,∂i(x, y) = (∂ix, ∂iy) and si(x, y) = (six, siy). The semi–direct product G ×χG′is the group G×G′with the group–operation (g1, g′ 1)·(g2, g′ 2) = (g1·χ(g′ 1, g2), g′ 1·g′ 2), where χ:G′⊗G → G satisfies that χ(¯ 0, g1) = g1, χ(g′ 1, χ(g′ 2, g1)) = χ(g′ 1·g′ 2, g1) and χ(g′ 1, g1·g2) = χ(g′ 1, g1)·χ(g′ 1, g2). Let Rbe a commutative ring with identity 1 6= 0. A chain complex is a graded R–module C=⊕n∈ZCntogether with an R–module endomorphism of degree −1, d=Pn∈Zdn:Cn→Cn−1, such that dd =d2is zero. The map dis called the differential of C. Ker dnis the module of n–cycles in C; Im dn+1 is the module of n–boundaries in C; the quotient Hn(C) = Ker dn/Im dn+1 is the nth homology module of C. The homology class of a cycle a∈Ker dnis denoted by [a]. The nth homology of Cwith coefficients in a ring Gis defined by Hn(C;G) = Hn(C⊗G). Whenever two graded objects x and yof degree pand qare interchanged we apply Koszul convention and introduce the sign (−1)pq .The tensor product of chain complexes Cand Dis the chain complex C⊗Dwith differential dC⊗D=dC⊗1D+ 1C⊗dD. Thus if x∈Cqand y∈Dr,an application of Koszul convention gives dC⊗D(x⊗y) = dC(x)⊗y+ (−1)qx⊗dD(y). A module homomorphism f:C→Dof degree zero such that df =fd is a chain map. If f:C→Dand g:C′→D′are chain maps, so is f⊗g:C⊗C′→D⊗D′. Let Cbe a chain complex and Gan R–module. Form the abelian group Cn= HomR(Cn, G), for all n; its elements are the module homomorphisms c:Cn→G, called n–cochains of C∗. The differential d:C→Cinduces a map of degree +1, δ:C∗→C∗, defined by δnc= (−1)n+1cdn+1 :Cn+1 →G, for all c∈Cnand for all n. The cohomology of Cis the family of abelian groups Hn(C, G) =Ker δn/Im δn−1. An element of Im δn−1is called an n–coboundary and an element of Ker δnan n–cocycle. The chain complex of a simplicial set Kwith coefficients in R, denoted by C(K) is constructed as follows. Let Cn(K) denote the free R–module on the set Kn. The face operators ∂iyield module maps Cn(K)→Cn−1(K), which we also call ∂i; their alternating sum d=Pi≥0(−1)i∂iis the differential of C(K). The normalized chain complex CN(K) is the chain complex defined as the quotient CN n(K) = Cn(K)/s(Cn−1(K)), where s(Cn−1(K)) denotes the free R–module on the set of all degenerate n–simplices of K. 3 Since we will always work with normalized chain complexes, we simplify notation and write C(K) instead of CN(K). The (co)homology of Kis, by definition, the (co)homology of C(K). The cohomology of Kis an algebra with the cup product ⌣:Hi(K;G)⊗Hj(K;G)→Hi+j(K;G) defined by [c]⌣[c′] = [c ⌣ c′], where Gis a group, c∈Ker δi,c′∈Ker δjand c ⌣ c′(x) = µ(c(∂i+1 ···∂i+jx)⊗c′(∂0···∂i−1x)) ,with x∈Ci+j(K) and µ being the operation on G. The following chain maps are needed to be defined: •The diagonal map ∆ : C(K)→C(K×n) is defined by ∆ x= (x, ntimes ... , x). •The cyclic permutations t:C(Kn)→C(K×n) such that t(x1, x2,...,xn) = (x2,...,xn, x1); and T:C(K)⊗n→C(K)⊗ndefined by T(x1⊗x2⊗ · · · ⊗ xn) = (−1)|x1|(|x2|+···+|xn|)x2⊗ · · · ⊗ xn⊗x1. Adifferential graded module (DG–module, for short) Mis a chain complex such that Mn= 0 for all n < 0. A DGA–module (M, ξ, η) (we will write it simply Mwhen no confusion can arise) is a DG–module Mendowed with two morphisms called the augmentation ξ:M0→Rand the coaugmentation, η:R→M0. It is required that ξ η = 1Rand ξ d = 0. ADGA–algebra (A, µ) (resp. DGA–coalgebra (B, ∇)) is a DGA–module endowed with a morphism µ:A⊗A→A, called product on A, such that µ(µ⊗1A) = µ(1A⊗µ) and µ(ηA⊗1A) = 1A=µ(1A⊗ηA). Resp. ∇: B→B⊗B, called coproduct on B, where (∇ ⊗ 1B)∇= (1B⊗ ∇)∇and (ηB⊗1B)∇= 1B= (1B⊗ηB)∇. The free R–algebra generated by a group Gis a DGA–algebra denoted by (R[G], ξG, ηG, µG) such that it is zero in each degree except for degree zero, where R0[G] = {Pa∈Aλaa:λa∈Rand Ais a finite subset of G}. The product µG, the augmentation ξGand the coaugmentation ηGare given by µG((Pλaa)⊗(Pλa′a′)) = Pλaλa′(a+a′), ξG(Pλaa) = Pλaand ηG(λ) = λ¯ 0, where a, a′∈ G and λa, λa′, λ ∈R. The reduced bar construction ¯ B(G) of the DGA–algebra R[G] is defined (as a graded module) in degree n > 0 by ¯ Bn(G) = R[G]⊗n/s(R[G]⊗n) where s(R[G]⊗n) is the R–module generated by all the elements of R[G]⊗nof the form a1⊗ · · · ⊗ ¯ 0⊗ · · · ⊗ an; and ¯ B0(G) = R. The element of ¯ B0(G) corresponding to the identity in Ris denoted by [ ] and an element a1⊗ · · · ⊗ an of ¯ Bn(G) is denoted by [a1|···|an]. The differential of ¯ B(G) is given by 4 d([a1|···|an]) = ξG(a1)[a2|···|an] + (−1)n[a1|···|an−1]ξG(an) +X 1≤i≤n−1 (−1)i[a1|···|ai−1|µG(ai⊗ai+1)|ai+2|···|an]. Observe that d[a1] = 0 for all [a1]∈¯ B1(G). The augmentation and the coaugmentation on ¯ B(G) coincide with the identity on R. Moreover, ¯ B(G) is a DGA–coalgebra with the coproduct: ∇([a1|···|an]) = X 0≤i≤n [a1|···|ai]⊗[ai+1|···|an]. Let (B, ∇) be a DGA–coalgebra and (A, µ) a DGA–algebra. A twisting cochain κ, is a graded module morphism of degree −1, κ:B→A, satisfying that dAκ+κ dB+µ(κ⊗κ)∇= 0, ξAκ= 0 and κ ηB= 0. Let Mbe an A–DG–module (where ν:M⊗A→Mis the (right) A–module structure on A). Define the morphism dκ:M⊗B→M⊗Bwhere dκ(m⊗b) = (dM⊗B+κ∩)(m⊗b) and κ∩= (ν⊗1B)(1M⊗κ⊗1B)(1M⊗ ∇). The graded module M⊗Bendowed with dκis a DG–module denoted by M⊗κBand called twisted tensor product by the twisting cochain κ. An example of twisted tensor product is R[G]⊗θ¯ B(G), where the twisting cochain θcalled the universal twisting cochain is given by θ([a1]) = a1−¯ 0 and θ([a1|···|an]) = 0 for n > 1. We deal with an special type of homotopy equivalences. A contraction rof a DG–module Nto a DG–module M, consists in three morphisms (f, g, φ) where f:N→M(projection) and g:M→N(inclusion) are DG- module morphisms of degree zero, and φ:N→N(homotopy operator) is a morphism of degree −1 satisfying that fg = 1M,φd +dφ =gf −1N. Moreover, it is required that φg = 0, fφ = 0, φφ = 0. A contraction will be denoted by r= (f, g, φ) : N⇒Mor briefly Nr ⇒M. Note that the importance of having this structure of Mto Nis that Nis “smaller” than Malthough both have the same homology. Let r= (f, g, φ) : N⇒M and r′= (f′, g′, φ′) : N′⇒M′be two contractions, then the following contractions can be constructed: •The tensor product contraction: r⊗r′= (f⊗f′, g⊗g′, φ⊗g′f′+1N⊗φ′) : N⊗N′⇒M⊗M′. •If M=N′, the composition contraction: r′r= (f′f, gg′, φ +gφ′f) : N⇒M′. 5 Let pand qbe non–negative integers, a (p, q)–shuffle (α, β) is a partition of the set {0,1, . . . , p +q−1}in two disjoint subsets, α1<··· < αpand β1<···< βq, of pand qintegers, respectively. The signature of the shuffle (α, β) is defined by sig(α, β) = P1≤i≤pαi−(i−1). An Eilenberg–Zilber contraction [EZ59] of C(K×L) to C(K)⊗C(L), where Kand Lare simplicial sets, is a triple rEZ = (Aw, Em, Sh) where: •Alexander–Whitney operator Aw :C(K×L)→C(K)⊗C(L) is defined by: Aw(xm×ym) = X 0≤i≤m ∂i+1 ···∂mxm⊗∂0···∂i−1ym. •Eilenberg–Mac Lane operator Em :C(K)⊗C(L)→C(K×L) by: Em(xp⊗yq) = X (α,β)∈{(p,q)−shuffles} (−1)sig(α,β)(sβxp, sαyq) where sβ=sβq···sβ1and sα=sαp···sα1. •Shih operator Sh :C(K×L)→C(K×L) by: Sh((xm, ym)) = X T(m) (−1)sg(s¯ β+ ¯m∂m−q+1 ···∂mxm, sα+ ¯m∂¯m···∂m−q−1ym) where sg = ¯m+sig(α, β) + 1, ¯m=m−p−q,¯ β+ ¯m={βq+ ¯m, ···β1+ ¯m, ¯m−1},α+ ¯m={αp+1 + ¯m, ···α1+ ¯m}and T(m) ={0≤p≤m−q−1≤m−1,(α, β)∈ {(p+ 1, q)–shuffles}}. It is evident that Alexander–Whitney operator has a polynomial nature. However, Eilenberg–MacLane and Shih operator have an essential “exponential” character because shuffles of degeneracy operators are involved in their respective formulation. A recursive formula for the Shih operator is given in [EM54]. An explicit formula for this operator was stated by J. Rubio [Rub91] and proved by F. Morace [Rea00]. It is possible to construct a contraction rEZ(p)= (Awp, Emp, Shp) of C(K×p) to C(K)⊗p(p≥0), appropriately composing Eilenberg–Zilber contractions. Note that the cup product defined on page 4 can be written as c ⌣ c′= µ(c⊗c′)Aw∆ where µis the operation defined on G. Generalizations of the cup product are the cochain mappings called cup–iproduct [Ste47], ⌣i: Cn(K;Z)⊗Cm(K;Z)→Cn+m−i(K;Z) given by c ⌣ic′=µ(c⊗c′)Di∆ where Di=Aw(t Sh)i. The following relation holds (up to sign): δ(c ⌣ic′) = ±c ⌣i−1c′±c′⌣i−1c+δc ⌣ic′+c ⌣iδc′.(1) 6 Taking [c]∈Hj(K;Z) and defining Sqi[c] = [c ⌣j−ic], then Sqi(c)∈ Hj−i(K;Z2). These cohomology operations are called Steenrod squares [Ste47]. Explicit formulae for Diand therefore for computing cup–iproducts, Steenrod squares and Steenrod reduced powers [Ste52] (a generalization of Steenrod squares), can be found in [GR98, GR99a, GR99b, GR02b]. In [Ade52], J. Adem constructed secondary cohomology operations using relations on iterated Steenrod squares. He proved the relation Sq2Sq2+ Sq3Sq1= 0 by means of the existence of cochain mappings Ej:Cp(K×4;Z)→ Cp−j(K;Z2) such that mod 2 (c ⌣ic)⌣i+2 (c ⌣ic) + (c ⌣i+1 c)⌣i(c ⌣i+1 c) = δE3i+3(c4) (2) where ⌣kis the cup–kproduct, cis a q–cocycle and i=q−2. If cis a q–cocycle such that Sq2(c) is a coboundary (that is, there exists a cochain b such that c ⌣ic=δb), then w=b ⌣i+1 b+b ⌣i+2 δb +E3i+3(c4) + η(c)⌣i−1η(c) + η(c)⌣iδη(c) is a mod 2 cocycle, where η(c) = 1 2(c ⌣i+2 c) + c. Adem secondary cohomology operations are defined as Ψq[c] = [w] + Sq2Hq+1(K;Z)∈Hq+3(K;Z2). In Section 4 we give a procedure for computing the Adem secondary cohomology operation Ψq(α) for any integer qand any cohomology class α. For doing this, the obtention of explicit formuale for E3i+3 is essential. We will obtain these formulae in the following section. 3 Adem Cocyclic Operations In order to obtain “economical” formulae for E3i+3 in the sense that they are written only in terms of face operators, we first write E3i+3 in terms of the component morphisms of Eilenberg–Zilber contractions. After this, we make a simplification of the given formulae based on the fact that any composition of face and degeneracy operators can be put in a unique form. For the first aim, we will use the following result whose demonstration is given in [GRb]. Lemma 1 Let Gbe a group. Let Mand Nbe two R[G]–DG–modules, where ν:M⊗R[G]→Mand ν′:N⊗R[G]→Nare the (right) R[G]–module structures on R[G]. Let r= (f, g, φ) : M⇒Nbe a contraction such that (g⊗1¯ B[G])ν′=νg. 7 Then (r⊗1¯ B[G])θ∩= ((f⊗1¯ B[G])θ∩, g ⊗1¯ B[G],(φ⊗1¯ B[G])θ∩)of M⊗θ¯ B(G)to N⊗θ¯ B(G)is a new contraction where (f⊗1¯ B[G])θ∩=Pi≥0(f⊗1¯ B[G])(θ∩(φ⊗ 1¯ B[G]))iand (φ⊗1¯ B[G])θ∩=−Pi≥0((φ⊗1¯ B[G])θ∩)i(φ⊗1¯ B[G]). Now, let Gbe the semi–direct product Z×2 2×χZ2where χ((a, b),¯ 1) = (b, a). Let M=C(K×4) and N=C(K)⊗4be two Z2[G]–modules: ν(x, a1) = z(x) = (x1, x3, x2, x4), ν′(y, a1) = z′(y) = y1⊗y3⊗y2⊗y4, ν(x, a2) = t×2(x) = (x2, x1, x4, x3), ν′(y, a2) = T⊗2(y) = y2⊗y1⊗y4⊗ y3, ν(x, a3) = t(x) = (x3, x4, x1, x2), ν′(y, a3) = T(y) = y3⊗y4⊗y1⊗y2; where a1= ((¯ 0,¯ 0),¯ 1), a2= ((¯ 1,¯ 0),¯ 0), a3= ((¯ 0,¯ 1),¯ 0), x= (x1, x2, x3, x4)∈ C(K×4) and y=y1⊗y2⊗y3⊗y4∈C(K)⊗4. Let rbe the Eilenberg–Zilber contraction rEZ(4) = (r⊗2 EZ )rEZ = (Aw4, Em4, Sh4) : C(K×4)⇒C(K)⊗4which commutes with the structures given above. Observe that Aw4= (Aw⊗2)Aw, Em4=Em(Em⊗2) and Sh4=Sh+Em(Sh⊗EmAw+1⊗Sh)Aw). Lemma 1 produces the new contraction: (rEZ(4) ⊗1¯ B(Z×2 2×χZ2))θ∩:C(K×4)⊗θ¯ B(Z×2 2×χZ2)⇒C(K)⊗4⊗θ¯ B(Z×2 2×χZ2). As we will see now, Adem relation (2) can be obtained from the fact that the projection (Aw4⊗1¯ B(G))θ∩is a DG–module morphism. Theorem 2 Adem relation (2) is obtained from the identity: µc⊗4(1N⊗ξ¯ B(G))(Aw4⊗1¯ B(G))θ∩(dM⊗¯ B(G)+θ∩)(∆(x)⊗e3i+3) =µc⊗4(1N⊗ξ¯ B(G))(dN⊗¯ B(G)+θ∩)(Aw4⊗1¯ B(G))θ∩(∆(x)⊗e3i+3) where µis the product on Z2,c∈Ker δq,x∈Cq+3(K)and e3i+3 ∈¯ B(G)is: e3i+3 =e(3,0) for i= 0, e3i+3 =e(3i+3,i)+e(3i+3,i−1) for i= 2,4, e3i+3 =e(3i+3,i)for i= 1,3,5and for i≥6,ibeing odd and i−7 2being even, e3i+3 =e(3i+3,i)+e(3i+3,i−2) for i≥6,i, i−7 2being odd, e3i+3 =e(3i+3,i)+e(3i+3,i−1) +e(3i+3,i−2) for i≥6,ibeing even, 8 such that e(3i+3,ℓ)=X S(3i+2,ℓ) [(cπ1,¯ 0)|...|(cπj,¯ 0)|((¯ 0,¯ 0),¯ 1)|(¯cπj+1 ,¯ 0)|...|(¯cπ3i+2 ,¯ 0)] where S(3i+ 2, ℓ) = {π∈ {(3i+ 2 −ℓ, ℓ)-sh.}} ∪ {j: 0 ≤j≤3i+ 2}, [c1,...,c3i+2] = [(¯ 1,¯ 0),3i+2−ℓ . . . , (¯ 1,¯ 0),(¯ 0,¯ 1),ℓ . . ., (¯ 0,¯ 1)] and if c= (a, b)then ¯c= (b, a). The proof of this theorem is given in an appendix at the end of this paper. Corollary 3 The explicit formulae for E3i+3 are: E3=˜ E(3,0) +c ⌣0c ⌣1c ⌣2cfor i= 0. E6=˜ E(6,1) +c ⌣1c ⌣2c ⌣3cfor i= 1. E9=˜ E(9,2) +˜ E(9,1) +c ⌣2c ⌣3c ⌣4c+c ⌣0c ⌣7c ⌣2for i= 2. E12 =˜ E(12,3) +c ⌣3c ⌣4c ⌣5c+c ⌣2c ⌣7c ⌣3c+c ⌣3c ⌣8c ⌣1c for i= 3. E15 =˜ E(15,4) +˜ E(15,3) +c ⌣4c ⌣5c ⌣6cfor i= 4. E18 =˜ E(18,5)+c ⌣5c ⌣6c ⌣7c+c ⌣4c ⌣9c ⌣5c+c ⌣4c ⌣11 c ⌣3c for i= 5. E3i+3 =˜ E(3i+3,i)+˜ E(3i+3,i−1) +˜ E(3i+3,i−2) +c ⌣ic ⌣i+1 c ⌣i+2 for i≥6,ibeing even, i−6 2being odd. E3i+3 =˜ E(3i+3,i)+˜ E(3i+3,i−1) +˜ E(3i+3,i−2) +c ⌣ic ⌣i+1 c ⌣i+2 +c ⌣ic ⌣i+5 c ⌣i−2for i≥6,i, i−6 2being even. E3i+3 =˜ E(3i+3,i)+˜ E(3i+3,i−2) +c ⌣ic ⌣i+1 c ⌣i+2 +c ⌣ic ⌣i+4 c ⌣i−1 +c ⌣ic ⌣i+5 c ⌣i−2+c ⌣i−1c ⌣i+6 c ⌣i−2 for i≥7,i, i−7 2being odd. E3i+3 =˜ E(3i+3,i)+c ⌣ic ⌣i+1 c ⌣i+2 +c ⌣ic ⌣i+4 c ⌣i−1 +c ⌣ic ⌣i+5 c ⌣i−2+c ⌣i−1c ⌣i+6 c ⌣i−2 for i≥7,ibeing odd, i−7 2being even. 9 =                        X 0≤j≤n0 j+n0is odd (Dj+k+Pn2j−1⊗TjD−j+k−1+Pn2jt)TDmif n2k−1is even X 0≤j≤n0 j+n0is odd (Dj+k+Pn2j−1⊗TjD−j+k−1+Pn2jt)TDm +X 0≤j≤n0 j+n0is odd (Dj+k+Pn2j−1⊗TjD−j+k+Pn2j)Dm−1 otherwise for k, n0, n1, . . . , n2k−1>0, m≥0 and nibeing odd for 1 ≤i≤2k−2. where Di=Aw(t Sh)n. Let us observe that all the summands we will obtain when (7) is simplified, has the form: µc⊗4(Da⊗Db)D2i+2−ℓ∆(x) and µc⊗4(Dd⊗De)Di−k∆(x) where cis a (i+ 2)–cocycle, a+b=i+ℓ,d+e= 2i+ 2 + kand 0 ≤ℓ, k ≤i. Let us recall that if cis an m–cocycle and c′is an n–cocycle, then c ⌣rc′= µ(c⊗c′)Dris null if m < r or n < r. Taking in mind this last fact, we have that ℓ+ 2 ≥aand ℓ+ 2 ≥b. Since a+b=i+ℓ, then i−4≤ℓ≤i. On the other hand, d≤i+ 2 and e≤i+ 2. Since d+e= 2i+ 2 + kthen 0 ≤k≤2. Therefore, the possible non–null summands of (7) have the form: µc⊗4(Di+2 ⊗Di+2)Di−2∆(x), µc⊗4(Di+1 ⊗Di+2)Di−1∆(x), µc⊗4(Di+2 ⊗Di+1)Di−1∆(x), µc⊗4(Di⊗Di+2)Di∆(x), µc⊗4(Di+1 ⊗Di+1)Di∆(x), µc⊗4(Di+2 ⊗Di)Di∆(x), µc⊗4(Di+2 ⊗Di−1)Di+1∆(x), µc⊗4(Di+1 ⊗Di)Di+1∆(x), µc⊗4(Di⊗Di+1)Di+1∆(x), µc⊗4(Di−1⊗Di+2)Di+1∆(x), µc⊗4(Di−2⊗Di+2)Di+2∆(x), µc⊗4(Di−1⊗Di+1)Di+2∆(x), µc⊗4(Di⊗Di)Di+2∆(x), µc⊗4(Di+1 ⊗Di−1)Di+2∆(x), µc⊗4(Di+2 ⊗Di−2)Di+2∆(x), µc⊗4(Di−2⊗Di+1)Di+3∆(x), µc⊗4(Di−1⊗Di)Di+3∆(x), µc⊗4(Di⊗Di−1)Di+3∆(x), µc⊗4(Di+1 ⊗Di−2)Di+3∆(x), µc⊗4(Di−2⊗Di)Di+4∆(x), µc⊗4(Di−1⊗Di−1)Di+4∆(x), µc⊗4(Di⊗Di−2)Di+4∆(x), µc⊗4(Di−2⊗Di−1)Di+5∆(x), µc⊗4(Di−1⊗Di−2)Di+5∆(x), µc⊗4(Di−2⊗Di−2)Di+6∆(x). Using the properties mentioned above, the possible non–null summands of (7) for ℓ=i−2 are: µc⊗4Aw4(t×2Sh4)2i+4(t Sh4)i−2∆(x) = µc⊗4(Di+2 ⊗Di+2)Di−2∆(x), µc⊗4Aw4t×2Sh4(t Sh4t×2Sh4)i−3(t Sh4)i+7∆(x) = µc⊗4(Di−2⊗Di−2)Di+2∆(x). The possible non–null summands of (7) for ℓ=i−1 are: µc⊗4Aw4(t×2Sh4)2i+3(t Sh4)i−1∆(x) =µc⊗4(Di+1 ⊗Di+2 +Di+2 ⊗Di+1)Di−1∆(x), µc⊗4Aw4(t×2Sh4)2i+3−jt Sh4(t×2Sh4)j(t Sh4)i−2∆(x) = 0 for 1 ≤j≤2i+ 2, 16 µc⊗4Aw4t×2Sh4(t Sh4t×2Sh4)i−2(t Sh4)i+5∆(x) =µc⊗4((Di−2⊗Di−1)Di+5 + (Di−1⊗Di−1)Di+4)∆(x), µc⊗4Aw4(t×2Sh4)2(t Sh4t×2Sh4)i−3(t Sh4)i+6∆(x) =µc⊗4(Di−1⊗Di−2)Di+5∆(x), µc⊗4Aw4t×2Sh4(t Sh4t×2Sh4)i−4t Sh4(t×2Sh4)2(t Sh4)i+6∆(x) = 0, µc⊗4Aw4(t×2Sh4)3(t Sh4t×2Sh4)i−4(t Sh4)i+7∆(x) = 0, µc⊗4Aw4t×2Sh4(t Sh4t×2Sh4)i−5t Sh4(t×2Sh4)3(t Sh4)i+7∆(x) = 0, µc⊗4Aw4t×2Sh4t Sh4(t×2Sh4)3(t Sh4t×2Sh4)i−5(t Sh4)i+7∆(x) = 0, µc⊗4Aw4t×2Sh4(t Sh4t×2Sh4)i−5−jt Sh4(t×2Sh4)3(t Sh4t×2Sh4)j(t Sh4)i+7∆(x) = 0 for 1 ≤j≤i−6. Finally, the possible non–null summands of (7) for ℓ=iare: µc⊗4Aw4(t×2Sh4)2i+2(t Sh4)i∆(x) =(µc⊗4((Di⊗Di+2 +Di+1 ⊗Di+1 +Di+2 ⊗Di)Di+ (Di+1 ⊗Di+2)Di−1)∆(x) if iis odd µc⊗4((Di⊗Di+2 +Di+1 ⊗Di+1 +Di+2 ⊗Di)Di+ (Di+2 ⊗Di+1)Di−1)∆(x) if iis even, µc⊗4Aw4(t×2Sh4)2i+2−jt Sh(t×2Sh4)j(t Sh4)i−1∆(x) =     µc⊗4(Di+1 ⊗Di+2)Di−1∆(x) if iis odd µc⊗4(Di+2 ⊗Di+1)Di−1∆(x) if j, i are even µc⊗4((Di+2 ⊗Di+1)Di−1+Di+2 ⊗Di+2)Di−2)∆(x) if jis odd, iis even for 1 ≤j≤2i+ 1, µc⊗4Aw4(t×2Sh4)2i+2−j−kt Sh(t×2Sh4)jt Sh(t×2Sh4)k(t Sh4)i−2∆(x) =     µc⊗4(Di+2 ⊗Di+2)Di−2∆(x) if k, i are odd µc⊗4(Di+2 ⊗Di+2)Di−2∆(x) if k, i are even 0 otherwise for 1 ≤j, k,j+k≤2i+ 1 and jbeing odd, µc⊗4Aw4t×2Sh4(t Sh t×2Sh4)i−1(t Sh4)i+3∆(x) =(µc⊗4((Di−1⊗Di)Di+3 + (Di⊗Di)Di+2)∆(x) if iis odd µc⊗4((Di⊗Di−1)Di+3 + (Di⊗Di)Di+2)∆(x) if iis even, µc⊗4Aw4(t×2Sh4)2(t Sh t×2Sh4)i−2(t Sh4)i+4∆(x) =(µc⊗4((Di⊗Di−2)Di+4 + (Di⊗Di−1)Di+3)∆(x) if iis odd µc⊗4((Di−1⊗Di−1)Di+4 + (Di⊗Di−1)Di+3)∆(x) if iis even µc⊗4Aw4t×2Sh4(t Sh t×2Sh4)i−3t Sh(t×2Sh4)2(t Sh4)i+4∆(x) =(µc⊗4(Di⊗Di−2)Di+4∆(x) if iis odd µc⊗4(Di−2⊗Di)Di+4∆(x) if iis even, µc⊗4Aw4(t×2Sh4)3(t Sh t×2Sh4)i−3(t Sh4)i+5∆(x) =(µc⊗4((Di−1⊗Di−2)Di+5 + (Di−2⊗Di+Di⊗Di−2)Di+4)∆(x) if iis odd µc⊗4((Di−2⊗Di−1)Di+5 + (Di−2⊗Di+Di⊗Di−2)Di+4)∆(x) if iis even, µc⊗4Aw4(t×2Sh4)2(t Sh t×2Sh4)i−4t Sh(t×2Sh4)2(t Sh4)i+5∆(x) 17 =(µc⊗4(Di−2⊗Di−1)Di+5∆(x) if iis odd 0 if iis even, µc⊗4Aw4t×2Sh4(t Sh t×2Sh4)i−4t Sh(t×2Sh4)3(t Sh4)i+5∆(x) =(µc⊗4(Di−2⊗Di)Di+4∆(x) if iis odd µc⊗4(Di⊗Di−2)Di+4∆(x) if iis even, µc⊗4Aw4t×2Sh4t Sh(t×2Sh4)3(t Sh t×2Sh4)i−4(t Sh4)i+5∆(x) =(µc⊗4((Di−1⊗Di−2)Di+5 + (Di⊗Di−2)Di+4)∆(x) if iis odd µc⊗4(Di⊗Di−2)Di+4∆(x) if iis even, µc⊗4Aw4t×2Sh4(t Sh t×2Sh4)i−4−jt Sh(t×2Sh4)3(t Sh t×2Sh4)j(t Sh4)i+5∆(x) =         µc⊗4((Di−1⊗Di−2)Di+5 + (Di⊗Di−2)Di+4)∆(x) if j, i are odd µc⊗4(Di−2⊗Di)Di+4∆(x) if jis even, iis odd µc⊗4((Di−2⊗Di−1)Di+5 + (Di−2⊗Di)Di+4)∆(x) if jis odd, iis even µc⊗4(Di⊗Di−2)Di+4∆(x) if j, i is even for 1 ≤j≤i−5, µc⊗4Aw4(t×2Sh4)4(t Sh t×2Sh4)i−4(t Sh4)i+6∆(x) =(µc⊗4((Di−2⊗Di−2)Di+6 + (Di−2⊗Di−1)Di+5)∆(x) if iis odd µc⊗4(Di−2⊗Di−1)Di+5∆(x) if iis even, µc⊗4Aw4(t×2Sh4)3(t Sh t×2Sh4)i−5t Sh(t×2Sh4)2(t Sh4)i+6∆(x) =µc⊗4(Di−2⊗Di−2)Di+6∆(x), µc⊗4Aw4(t×2Sh4)2(t Sh t×2Sh4)i−5t Sh(t×2Sh4)3(t Sh4)i+6∆(x) =(0 if iis odd µc⊗4(Di−2⊗Di−1)Di+5∆(x) if iis even, µc⊗4Aw4(t×2Sh4)2t Sh(t×2Sh4)3(t Sh t×2Sh4)i−5(t Sh4)i+6∆(x) = 0, µc⊗4Aw4(t×2Sh4)2(t Sh t×2Sh4)i−5−jt Sh(t×2Sh4)3(t Sh t×2Sh4)j(t Sh4)i+6∆(x) =     µc⊗4((Di−2⊗Di−2)Di+6 + (Di−2⊗Di−1)Di+5)∆(x) if j, i are odd µc⊗4(Di−2⊗Di−1)Di+5∆(x) if j, i are even 0 otherwise for 1 ≤j≤i−6, µc⊗4Aw4t×2Sh4(t Sh t×2Sh4)i−5t Sh(t×2Sh4)4(t Sh4)i+6∆(x) = 0, µc⊗4Aw4t×2Sh4t Sh(t×2Sh4)3(t Sh t×2Sh4)i−6t Sh(t×2Sh4)2(t Sh4)i+6∆(x) =(µc⊗4(Di−2⊗Di−2)Di+6∆(x) if iis odd 0 if iis even, µc⊗4Aw4t×2Sh4(t Sh t×2Sh4)i−6−jt Sh(t×2Sh4)3(t Sh t×2Sh4)jt Sh(t×2Sh4)2(t Sh4)i+6∆(x) =(µc⊗4(Di−2⊗Di−2)Di+6∆(x) if jis even 0 otherwise for 1 ≤j≤i−7, µc⊗4Aw4t×2Sh4(t Sh t×2Sh4)i−6t Sh(t×2Sh4)3t Sh(t×2Sh4)2(t Sh4)i+6∆(x) 18 =µc⊗4(Di−2⊗Di−2)Di+6∆(x), µc⊗4Aw4(t×2Sh4)5(t Sh t×2Sh4)i−5(t Sh4)i+7∆(x) =µc⊗4(Di−2⊗Di−2)Di+6∆(x), µc⊗4Aw4t×2Sh4(t Sh t×2Sh4)i−6t Sh(t×2Sh4)5(t Sh4)i+7∆(x) = 0, µc⊗4Aw4t×2Sh4(t Sh t×2Sh4)i−6−jt Sh(t×2Sh4)5(t Sh t×2Sh4)j(t Sh4)i+7∆(x) = 0 for 1 ≤j≤i−7, µc⊗4Aw4(t×2Sh4)3(t Sh t×2Sh4)i−6t Sh(t×2Sh4)3(t Sh4)i+7∆(x) =µc⊗4(Di−2⊗Di−2)Di+6∆(x), µc⊗4Aw4(t×2Sh4)3t Sh(t×2Sh4)3(t Sh t×2Sh4)i−6(t Sh4)i+7∆(x) =µc⊗4(Di−2⊗Di−2)Di+6∆(x), µc⊗4Aw4(t×2Sh4)3(t Sh t×2Sh4)i−6−jt Sh(t×2Sh4)3(t Sh t×2Sh4)j(t Sh4)i+7∆(x) =µc⊗4(Di−2⊗Di−2)Di+6∆(x) for 1 ≤j≤i−7, µc⊗4Aw4t×2Sh4t Sh(t×2Sh4)3(t Sh t×2Sh4)i−7t Sh(t×2Sh4)3(t Sh4)i+7∆(x) =(µc⊗4(Di−2⊗Di−2)Di+6∆(x) if iis odd 0 if iis even, µc⊗4Aw4t×2Sh4(t Sh t×2Sh4)i−7t Sh(t×2Sh4)3t Sh(t×2Sh4)3(t Sh4)i+7∆(x) =µc⊗4(Di−2⊗Di−2)Di+6∆(x), µc⊗4Aw4t×2Sh4(t Sh t×2Sh4)i−7−jt Sh(t×2Sh4)3(t Sh t×2Sh4)jt Sh(t×2Sh4)3(t Sh4)i+7∆(x), =(µc⊗4(Di−2⊗Di−2)Di+6∆(x) if jis even 0 otherwise for 1 ≤j≤i−8. Now, using property (1), the simplified expression for (7) mod 2 is: µc⊗4((D0⊗D2+D2⊗D0+D1⊗D1)D0+ (D0⊗D0)D2)∆(x) =δ(c ⌣0c ⌣1c ⌣2c) + c ⌣1c ⌣0c ⌣1c+c ⌣0c ⌣2c ⌣0cfor i= 0. µc⊗4((D1⊗D3+D3⊗D1+D2⊗D2)D1+ (D1⊗D1)D3)∆(x) =δ(c ⌣1c ⌣2c ⌣3c) + c ⌣2c ⌣1c ⌣2c+c ⌣1c ⌣3c ⌣1cfor i= 1. µc⊗4((D2⊗D4+D4⊗D2+D3⊗D3)D2+ (D2⊗D2)D4 +(D0⊗D2+D2⊗D0)D6)∆(x) =δ(c ⌣2c ⌣3c ⌣4c+c ⌣0c ⌣7c ⌣2c) + c ⌣3c ⌣2c ⌣3c +c ⌣2c ⌣4c ⌣2cfor i= 2. µc⊗4((D3⊗D5+D5⊗D3+D4⊗D4)D3+ (D3⊗D3)D5 +(D2⊗D3+D3⊗D2)D6+ (D3⊗D1+D1⊗D3)D7)∆(x) =δ(c ⌣3c ⌣4c ⌣5c+c ⌣2c ⌣7c ⌣3c+c ⌣3c ⌣8c ⌣1c) +c ⌣3c ⌣5c ⌣3c+c ⌣4c ⌣3c ⌣4cfor i= 3. µc⊗4((D4⊗D6+D6⊗D4+D5⊗D5)D4+ (D4⊗D4)D6)∆(x) =δ(c ⌣4c ⌣5c ⌣6c) + c ⌣5c ⌣4c ⌣5c+c ⌣4c ⌣6c ⌣4cfor i= 4. µc⊗4((D5⊗D7+D7⊗D5+D6⊗D6)D5+ (D5⊗D5)D7 19 +(D4⊗D5+D5⊗D4)D8+ (D4⊗D3+D3⊗D4)D10)∆(x) =δ(c ⌣5c ⌣6c ⌣7c+c ⌣4c ⌣9c ⌣5c+c ⌣4c ⌣11 c ⌣3c) +c ⌣5c ⌣7c ⌣5c+c ⌣6c ⌣5c ⌣6cfor i= 5. µc⊗4((Di⊗Di+2 +Di+2 ⊗Di+Di+1 ⊗Di+1)Di+ (Di⊗Di)Di+2)∆(x) =δ(c ⌣ic ⌣i+1 c ⌣i+2 c) + c ⌣i+1 c ⌣ic ⌣i+1 c+c ⌣ic ⌣i+2 c ⌣ic for i≥6, ibeing even, i−6 2being odd. µc⊗4((Di⊗Di+2 +Di+2 ⊗Di+Di+1 ⊗Di+1)Di+ (Di⊗Di)Di+2 +(Di⊗Di−2+Di−2⊗Di)Di+4 =δ(c ⌣ic ⌣i+1 c ⌣i+2 c+c ⌣ic ⌣i+5 c ⌣i−2c) +c ⌣i+1 c ⌣ic ⌣i+1 c+c ⌣ic ⌣i+2 c ⌣ic for i≥6 and i, i−6 2being even. µc⊗4((Di⊗Di+2 +Di+2 ⊗Di+Di+1 ⊗Di+1)Di+ (Di⊗Di)Di+2 +(Di−1⊗Di+Di⊗Di−1)Di+3 + (Di⊗Di−2+Di−2⊗Di)Di+4 +(Di−1⊗Di−2+Di−2⊗Di−1)Di+5)∆(x) =δ(c ⌣ic ⌣i+1 c ⌣i+2 c+c ⌣ic ⌣i+4 c ⌣i−1c+c ⌣ic ⌣i+5 c ⌣i−2c +c ⌣i−1c ⌣i+6 c ⌣i−2c) + c ⌣i+1 c ⌣ic ⌣i+1 c+c ⌣ic ⌣i+2 c ⌣ic for i≥7 and i, i−7 2being odd. µc⊗4((Di⊗Di+2 +Di+2 ⊗Di+Di+1 ⊗Di+1)Di+ (Di⊗Di)Di+2 +(Di−1⊗Di+Di⊗Di−1)Di+3 + (Di⊗Di−2+Di−2⊗Di)Di+4 +(Di−1⊗Di−2+Di−2⊗Di−1)Di+5)∆(x) =δ(c ⌣ic ⌣i+1 c ⌣i+2 c+c ⌣ic ⌣i+4 c ⌣i−1c+c ⌣ic ⌣i+5 c ⌣i−2c +c ⌣i−1c ⌣i+6 c ⌣i−2c) + c ⌣i+1 c ⌣ic ⌣i+1 c+c ⌣ic ⌣i+2 c ⌣ic for i≥7, ibeing odd, i−7 2being even. This completes the proof. 20