A combinatorial method for computing Steenrod squares

A COMBINATORIAL METHOD FOR COMPUTING STEENROD SQUARES Roc´ıo Gonz´alez-D´ıaz and Ped o Real Dp o. de Ma em´a ica Aplicada I Facul ad de In o m´a ica y Es ad´ıs ica Uni e sidad de Se illa ogo[email p o ec ed], [email p o ec ed] Abs ac We p esen he e a combina o ial me hod o compu ing cup-ip oduc s and S een od squa es o a simplicial se X. This me hod is essen ially based on he de e mina ion o explici o mulae o he componen mo phisms o a highe diagonal app oxima- ion (i.e., a amily o mo phisms measu ing he lack o commu a i i y o he cup p oduc on he cochain le el) in e ms o ace ope a o s o X. A gene aliza ion o his me hod o S een od educed powe s is ske ched. Bo h au ho s a e pa ially suppo ed by he PAICYT esea ch p ojec FQM-0143 om Jun a de Andaluc´ıa and he DGESIC esea ch p ojec PB97-1025-C02-02 om Educa ion and Science Minis y (Spain). 1 1 In oduc ion Cohomology ope a ions a e algeb aic ope a ions on he cohomology g oups o spa- ces commu ing wi h he homomo phisms induced by con inuous mappings. This machine y is use ul when he g aded ec o space s uc u e and he cup p oduc on cohomology ail o dis inguish wo spaces by hei cohomology. S een od squa- es [20] cons i u e an ex emely impo an class o cohomology ope a ions no only in Algeb aic Topology bu also in he a ea o simplicial me hods in Homological Algeb a (cohomology o g oups, Hochschild cohomology o algeb as, ...). We he e s udy in de ail he unde lying combina o ial s uc u es o he de ini ion o hese cohomology ope a ions. Mo e conc e ely, we will mo e in he amewo k o Simplicial Topology [12] in which he basic objec s a e simplicial se s, ha is, g aded se s endowed wi h ace (∂i) and degene acy (si) ope a o s, sa is ying se e al commu a i i y ela ions. Roughly speaking, a simplicial se can be conside ed as an algeb aic gene aliza ion o he s uc u e o a iangula ed polyhed on al hough he o me ea u es a mo e igid combina o ial s uc u e han he la e . In his con ex , we de elop a sui able se ing in which all S een od cohomology ope a ions can be s udied simul aneously. I is well-known ha he e a e se e al me hods o cons uc ing S een od squa- es. One o hem consis s o making hese ope a ions using he cohomology o Eilenbe g-Mac Lane spaces (see, o ins ance, [[12]; p. 107]). Ano he me hod is de e mined by he cons uc ion o a amily o mo phisms {Di}measu ing he lack o commu a i i y o he cup p oduc on he cochain le el [20]. This sequence o mo p- hisms is called highe diagonal app oxima ion. I s exis ence is always gua an eed by he acyclic models me hod [5]. In his way, i is possible o de i e a ecu si e p ocedu e o ob ain he explici o mula o any Di(see [4], [[13]; Sec . 7]). In his pape , we p esen an al e na i e me hod o ob aining he explici o mula o a highe diagonal app oxima ion. In [15], he o mula o Diis es ablished in e ms o he componen mo phisms o a gi en Eilenbe g-Zilbe con ac ion (a special homo opy equi alence) om CN ∗(X×X) on o CN ∗(X)⊗CN ∗(X), whe e CN ∗(X) deno es he no malized chain complex o a gi en simplicial se X. Now, we ha e he ollowing p oblems. On one hand, he componen mo phisms o he con ac ion abo e a e de ined in e ms o ace and degene acy ope a o s o he simplicial se X. On he o he hand, he o mula o a mo phism Dialways in ol es he use o he homo opy ope a o o he Eilenbe g-Zilbe con ac ion and he explici o mula o his las mo phism is de e mined by shu les (a special ype o pe mu a ion) o degene acy ope a o s. In consequence, i we y o exp ess in 2 his way he mo phisms Diin e ms o ace and degene acy ope a o s o X, he numbe o summands appea ing in he o mula o a mo phism Die alua ed o e an elemen o deg ee nis, in gene al, a leas 2n. The e o e, an algo i hm ha would be designed s a ing om hese o mulae would be oo slow o p ac ical implemen a ion. Because o his, he idea o simpli ying hese o mulae a ises in a na u al way. This simpli ica ion o no maliza ion is based on he ac ha any composi ion o ace and degene acy ope a o s o he simplicial se Xcan be pu in a “canonical” way. Tha is, a composi ion o his ype can be exp essed in he unique o m: sj · · · sj1∂i1· · · ∂is, whe e j >· · · > j1≥0 and is>· · · > i1≥0. Mo eo e , aking in o accoun ha he image o a mo phism Dilies in CN ∗(X)⊗CN ∗(X), hose summands o he simpli ied o mula o Diwi h a ac o ha ing a degene acy ope a o in i s exp ession, can be elimina ed. The eason is ha his ac o applied o an elemen o he simplicial se Xis ze o in he no - malized chain complex associa ed o X. In his way, we he e ob ain mo e simple o mula o he mo phism Di. In his way, he classical de ini ion o S een od squa es: Sqi(c)(x) = (µ(< c ⊗c, Dj−i(x)>), i ≤j 0, i > j (1) whe e c∈Hom(CN j(X),Z2), x∈CN i+j(X) and µis he homomo phism induced by he mul iplica ion in Z2, is complemen ed by a manageable combina o ial o mula- ion o he highe diagonal app oxima ion. As we will ema k la e , his desc ip ion can be conside ed as a di ec ansla ion o he mos ancien de ini ion o S een od squa es (see [18]) o he gene al se ing o he Simplicial Topology. We gi e a mo e de ailed explana ion in he hi d sec ion. S ill, we hink ha his combina o ial machine y could be subs an ially imp o ed in he u u e by exploi ing he well-known p ope ies o S een od squa es [1, 20] and ad anced echniques o calcula ing cocycles (see, o example, [9, 10]). In his way, a “ easonably e icien ” algo i hm compu ing he cohomology algeb a o se e al impo an simplicial se s could be de i ed. Finally, we s a an analogous s udy o ha gi en in [15] o S een od edu- ced powe s. Mo e p ecisely, we p o ide a simplicial desc ip ion o hese ope a- ions in e ms o he componen mo phisms o an Eilenbe g-Zilbe con ac ion om 3 CN ∗(X×p imes · · · ×X) o CN ?(X)⊗p imes · · · ⊗CN ∗(X), and a 0-sequence {γi}i≥0in he symme ic g oup Gp. He e is a summa y o he p esen pape . Sec ion 2 is dedica ed o no a ion, e minology and a p esen a ion o he p oblem. In Sec ion 3, we show an explici combina o ial de ini ion o cup-ip oduc s and, consequen ly, o S een od squa es. In Sec ion 4, we s udy S een od educed powe s om his poin o iew. Finally, a p oo o he main heo em enuncia ed in Sec ion 3 is gi en in Sec ion 5. We a e g a e ul o P o . Tom´as Recio o his help ul sugges ions o making he exposi ion mo e eadable and, hope ully, mo e in o ma i e. We also wish o acknowledge ou deb o he e e ees o hei many aluable indica ions. 2 The highe homo opy commu a i i y o he Alexande -Whi ney ope a o The aim o his sec ion is o gi e some p elimina ies and a b ie accoun o he wo k done in [15] in o de o acili a e he unde s anding o he es o he pape . Mos o he ma e ial gi en in his sec ion can be ound in [12], [11], [17] and [21]. Asimplicial se Xis a sequence o se s X0, X1, . . ., oge he wi h ace ope a o s ∂i:Xn→Xn−1and degene acy ope a o s si:Xn→Xn+1 (i= 0,1, . . . , n), which sa is y he ollowing “simplicial” iden i ies: (s1) ∂i∂j=∂j−1∂i,i i < j; (s2) sisj=sj+1si,i i≤j; (s3) ∂isj=sj−1∂i,i i < j, (s4) ∂isj=sj∂i−1,i i > j + 1, (s5) ∂jsj= 1X=∂j+1sj. The elemen s o Xna e called n-simplices. A simplex xis degene a ed i x=siy o some simplex yand degene acy ope a o si; o he wise, xis non degene a ed. The ollowing elemen a y lemma will be essen ial in he p oo o he main heo em o his pape . 4 Lemma 2.1 [12] Any composi ion µ:Xm→Xno ace and degene acy ope a o s o a simplicial se Xcan be pu in a unique “canonical” o m: sj · · · sj1∂i1· · · ∂is, whe e n>j >· · · > j1≥0,m≥is>· · · > i1≥0and n− +s=m. Le Rbe a ing which is commu a i e wi h uni . A chain ( esp. cochain)complex M={Mn, dn}( esp. C={Cn, δn}) is a g aded (o e he in ege s) R-module oge he wi h a R-module map do deg ee −1 ( esp. o deg ee +1), called he di e en ial, such ha dndn+1 = 0 ( esp. δn+1 δn= 0). An elemen co Cnis called n-cochain. The homology H∗(M) ( esp. he cohomology H∗(C) is he amily o modules Hn(M) = Ke dn/Im dn+1 ( esp. Hn(C) = Ke δn/Im δn−1). I Mis a chain complex o e Rand Gis an R-module, he e is a cochain complex Hom(M, G) = {Hom(Mn, G), δn}, whe e, i c∈Hom(Mn, G), hen δnc∈Hom(Mn+1, G) is de ined by (δnc)(x) = c(dn+1(x)), x ∈Mn+1. We also w i e < c, x > ins ead o c(x) and se < c, x >= 0 i he deg ee o he cochain cis no equal o he deg ee o he elemen x. In his no a ion, < δnc, x > =< c, dn+1(x)> . Now, gi en a simplicial se X, le C∗(X) deno es he chain complex {Cn(X), dn}, in which Cn(X) is he ee R-module gene a ed by Xnand dn:Cn(X)→Cn−1(X) is de ined by dn= n X i=0 (−1)i∂i. Le us deno e by s(C∗(X)) he g aded R-module gene a ed by all he degene a ed simplices. In C∗(X), we ha e ha dn(s(Cn−1(X))) ⊂s(Cn−2(X)) and, hen CN ∗(X) = {Cn(X)/s(Cn−1(X)), dn}is a chain complex called he no malized chain complex associa ed o X. Gi en a R-module G, le us de ine by C∗(X;G) he cochain complex associa ed o CN ∗(X). In his way, we de ine he homology and cohomology o Xwi h coe icien s in a 5 R-module Gby H∗(X;G) = H∗(CN ∗(X)⊗G) and H∗(X;G) = H∗(C∗(X;G)), es- pec i ely. Eilenbe g and Mac Lane de ined in [6] a con ac ion o chain complexes om N on o M, as a iple ( , g, φ) in which :N→M(p ojec ion) and g:M→N (inclusion) a e chain maps, and φ:N→N(homo opy ope a o ) is a map o R-module aising deg ee by 1. Mo eo e , i is equi ed ha (c1) g = 1M,(c2) φd +dφ +g = 1N, (c3) φg = 0,(c4) φ = 0,(c5) φφ = 0. Hence, his de ini ion implies ha he “big” complex Nis homology equi alen o he “small” complex Min a s ong way. In ac , a con ac ion is a special homo opy equi alence be ween chain complexes. I we ha e wo con ac ions ( i, gi, φi) om Ni o Mi, wi h i= 1,2 hen, he ollowing con ac ions can be cons uc ed (see [6]): •The enso p oduc con ac ion ( 1⊗ 2, g1⊗g2, φ1⊗g2 2+ 1N1⊗φ2) om N1⊗N2 o M1⊗M2. •I N2=M1, he composi ion con ac ion ( 2 1, g1g2, φ1+g1φ2 1) om N1 o M2. I pand qa e non-nega i e in ege s, a (p, q)-shu le (α, β) is a pa i ion o he se {0,1, . . . , p +q−1}o in ege s in o wo disjoin subse s, α1<· · · < αpand β1<· · · < βq, o pand qin ege s, espec i ely. The signa u e o he shu le (α, β) is de ined by sig(α, β) = p X i=1 αi−(i−1). A e hese p elimina ies, we a e able o desc ibe a e y impo an homo opy equi alence in Algeb aic Topology. This con ac ion ells us ha he associa ed chain complex CN ∗(X×Y) educes o he enso p oduc o chain complexes CN ∗(X) and CN ∗(Y). An Eilenbe g-Zilbe con ac ion [8] om CN ∗(X×Y) o CN ∗(X)⊗CN ∗(Y), whe e Xand Ya e gi en simplicial se s, is de ined by he iple (AW, EML, SHI) whe e: 6 •The Alexande -Whi ney ope a o AW :CN ∗(X×Y)−→ CN ∗(X)⊗CN ∗(Y) is de ined by: AW(am×bm) = m X i=0 ∂i+1 · · · ∂mam⊗∂0· · · ∂i−1bm. I X=Y,AW can be conside ed as a “simplicial app oxima ion” o he dia- gonal and his ope a o p o ides a me hod o cons uc ing he cup p oduc in cohomology. I we in e change he ac o s amand bmin he o mula, we ob ain a di e en app oxima ion. Compa ison o hese wo di e en app oxima ions o he diagonal leads o he S een od squa es. •The Eilenbe g-Mac Lane ope a o EML :CN ∗(X)⊗CN ∗(Y)−→ CN ∗(X×Y) is de ined by: EML(ap⊗bq) = X (α,β)∈{(p,q)−shu les} (−1)sig(α,β)sβq· · · sβ1ap×sαp· · · sα1bq. This ope a o can be seen as a p ocess o “ iangula ion” in he ca esian p oduc X×Y. •And he Shih ope a o SHI :CN ∗(X×Y)−→ CN ∗+1(X×Y) is de ined by: SHI(a0×b0) = 0; SHI(am×bm) = X(−1)¯m+sig(α,β)+1 sβq+ ¯m· · · sβ1+ ¯ms¯m−1∂m−q+1 · · · ∂mam ×sαp+1+ ¯m· · · sα1+ ¯m∂¯m· · · ∂m−q−1bm; whe e ¯m=m−p−q,sig(α, β) = p+1 X i=1 αi−(i−1), and he las sum is aken o e all he indices 0 ≤q≤m−1, 0 ≤p≤m−q−1 and (α, β)∈ {(p+ 1, q)-shu les}. A ecu si e o mula o he SHI ope a o has al eady been gi en by Eilenbe g and Mac Lane in [7]. The explici o mula gi en he e was s a ed by Rubio in [16] and p o ed by Mo ace in he appendix o [14]. In con as o he deep s udies ound in he li e a u e on he AW and EML ope a o s, i u ns o be qui e su p ising he lack o in e es shown up o now in he s udy o he homo opy ope a o in ol ed in an Eilenbe g-Zilbe con ac ion, no only om he poin o iew o ge ing i s explici o mula bu also o ob aining algeb aic p ese a ion esul s o his ope a o wi h ega d o he unde lying coalgeb a s uc u e on CN ∗(X×X). 7 Gi en a simplicial se Xand a posi i e in ege p, we can o m a con- ac ion ( , g, φ) om CN ∗(X×p imes · · · ×X) o CN ∗(X)⊗p imes · · · ⊗CN ∗(X), app o- p ia ely composing Eilenbe g-Zilbe con ac ions. I p= 2, hen ( , g, φ) is he con ac ion (AW, EML, SHI). The con ac ion om CN ∗(X×X×X) on o CN ∗(X)⊗CN ∗(X)⊗CN ∗(X) is de ined by he composi ion o he Eilenbe g-Zilbe con ac ion om CN ∗(X×X×X) o CN ∗(X)⊗CN ∗(X×X) and he enso p oduc con ac ion o he iden i y mo phism 1CN ∗(X)and he Eilenbe g-Zilbe con ac ion om CN ∗(X×X) on o CN ∗(X)⊗CN ∗(X). And so on. F om now on, he con ac ions ob ained in his way will also be called Eilenbe g- Zilbe con ac ions. Le pbe a posi i e in ege . Le us de ine se e al chain maps (o mo phisms) we use in his pape . We will omi in he no a ion o hese mo phism i s dependency on p. The diagonal map ∆ : CN ∗(X)→CN ∗(X×p imes · · · ×X) is de ined by ∆(x) = (x, p imes . . . , x). The ollowing au omo phisms a e also de ined: :CN ∗(X×p imes · · · ×X)→CN ∗(X×p imes · · · ×X), such ha (x1, x2, . . . , xp) = (x2, . . . , xp, x1) and T:CN ∗(X)⊗p imes · · · ⊗CN ∗(X)→CN ∗(X)⊗p imes · · · ⊗CN ∗(X) de ined by T(x1⊗x2⊗ · · · ⊗ xp) = (−1)|x1|(|x2|+···+|xp|)x2⊗ · · · ⊗ xp⊗x1. We now ou line he p oblem conce ning S een od squa es in which we a e in e- es ed. I is well-known ha he AW ope a o is no commu a i e, ha is, assuming ha X=Y,AW 6=AW. On he o he hand, his ope a o de e mines he cup p oduc in cohomology. I Gis a ing, gi en wo cochains c∈Ci(X;G) and c0∈Cj(X;G), and x∈CN i+j(X), he cup-p oduc o cand c0is de ined by: c^c0(x) = µ(< c ⊗c0, AW ∆(x)>) =µ(< c, ∂i+1 · · · ∂i+jx > ⊗< c0, ∂0· · · ∂i−1x >), whe e µis he homomo phism induced by he mul iplica ion on G. S een od in [19] de e mined ha he e exis s an in ini e sequence o mo phisms {Di}, called highe diagonal app oxima ion, which “measu es” his lack o commu a i i y. Mo e p eci- sely, he e is a sequence o g aded homomo phisms Di:CN ∗(X)→CN ∗(X)⊗CN ∗(X) o deg ee isuch ha : 8 D0=AW ∆ d⊗Di+1 + (−1)iDi+1 d=T Di+ (−1)i+1Di, whe e dand d⊗a e he di e en ials o CN ∗(X) and CN ∗(X)⊗CN ∗(X), espec i ely. Mo eo e , he mo phism Dican be exp essed in he o m Di=hi∆, whe e hi:CN ∗(X×X)→CN ∗(X)⊗CN ∗(X) is a homomo phism o deg ee i. In he li e- a u e, he exis ence o he owe o i e a ed Alexande -Whi ney ope a o s {hi}is gua an eed by he acyclic models me hod (see [5]). This echnique can be conside- ed as a cons uc i e me hod in he simplicial ca ego y and ecu si e o mulae o {Di}can be es ablished (see [13]). In [15], a di e en app oach is p esen ed. The s ong homo opy commu a i i y o AW is de e mined by making use o he explici Eilenbe g-Zilbe con ac ion (AW, EML, SHI). Hence, he o mula o a mo phism Diis gi en in e ms o he mo phisms AW,SHI, he diagonal ∆ and he au omo phism . Mo e p ecisely, he o mula o hiis AW ( SHI)i, o all i∈N. Now, he de ini ion o he cohomology ope a ion Sqi:Hj(X;Z2)→ Hj+i(X;Z2) (see (1)) akes he o m: Sqi(c)(x) = (µ(< c ⊗c, AW ( SHI)j−i(x, x)>), i ≤j 0, i > j (2) whe e c∈Cj(X;Z2) and x∈CN i+j(X). I is ob ious ha he o mulae o he mo phism hican be gi en in e ms o ace and degene acy ope a o s o X. Ou objec i e in he nex sec ion is o show how o “simpli y” hese o mulae and o ob ain an explici de ini ion o S een od squa es only in e ms o ace ope a o s o X. 3 An explici combina o ial desc ip ion o he cup-ip oduc s I is clea ha he image o hilies in CN ∗(X)⊗CN ∗(X). The e o e, i we exp ess he ac o s o he summands o he o mula o Diin a canonical way (see Lemma 2.1), hose summands o he simpli ied o mula o Diha ing a ac o wi h a degene acy ope a o in i s exp ession mus be elimina ed. 9 P oposi ion 4.1 Le pbe p ime wi h p≥2and ia non-nega i e in ege , hen φdγi· · · φγ1φ= (−1)i−1φγi· · · φγ1dφ + i−1 X k=1 (−1)i−kφγi· · · φγk+2φγφγk−1· · · φγ1φ. P oo We p o e he p oposi ion by induc ion on he pa ame e i. •I i= 1 hen, φdγ1φ=φγ1dφ. •I i= 2 hen, φdγ2φγ1φ=φγ2dφγ1φ=φγ2(g −1−φd)γ1φ =−φγφ −φγ2φγ1dφ. •In gene al, φdγiφγi−1· · · φγ1φ=φγidφγi−1· · · φγ1φ =φγi(g −1−φd)γi−1· · · φγ1φ =−φγφγi−2· · · φγ1φ−φγiφdγi−1· · · φγ1φ (by induc ion assump ion) =−φγφγi−2· · · φγ1φ−(−1)i−2φγiφγi−1· · · φγ1dφ − i−2 X k=1 (−1)i−1−kφγiφγi−1· · · φγk+2φγφγk−1· · · φγ1φ = (−1)i−1φγi· · · φγ1dφ + i−1 X k=1 (−1)i−kφγi· · · φγk+2φγφγk−1· · · φγ1φ. 2 Le Γi(k) = γiφγi−1· · · φγk+2φγφγk−1φ· · · φγ1φ∆. We can p o e P oposi ion 4.2 Le pbe p ime and ia non-nega i e in ege , hen γiφγi−1· · · φγ2φ∆ = γi−1φγi−2· · · φγ2φγ1φ∆ + i−1 X k=1 (−1)k+1Γi(k). 16 P oo Using γk+1 =γk+ (−1)k+1γ, o all 1 < k < i, we ha e γiφγi−1· · · φγk+2φγk+1φγk−1· · · φγ1φ∆ = γiφγi−1· · · φγk+2φγkφγk−1· · · φγ1φ∆+(−1)k+1Γi(k). I we use his ac successi ely, we ob ain he ollowing iden i y: γiφγi−1· · · φγ2φ∆ = γi−1φγi−2· · · φγ2φγ1φ∆ + i−1 X k=1 (−1)k+1Γi(k) 2 The main esul o his sec ion is he ollowing one. Theo em 4.3 Le pbe a p ime numbe , p≥2, and ia non-nega i e in ege . Then, he e exis s a sequence o mo phisms {Di}de ined by Di= γiφγi−1· · · φγ1φ∆, e i ying ha dDi+ (−1)i+1Did=αiDi−1, whe e α2j−1=T−1and α2j= 1 + T+T2+· · · +Tp−1. P oo Le us begin wi h he i s e m o he iden i y: dDi+ (−1)i+1Did=d γiφγi−1· · · φγ1φ∆+(−1)i+1 γiφγi−1· · · φγ1φ∆d = γidφγi−1· · · φγ1φ∆ +(−1)i+1 γiφγi−1· · · φγ1(g −1−dφ)∆ = γidφγi−1· · · φγ1φ∆ +(−1)i γiφγi−1· · · φγ2φ∆+(−1)i γiφγi−1· · · φγ1dφ∆ (by P oposi ion 4.1, we ge ) = γidφγi−1· · · φγ1φ∆+(−1)i γiφγi−1· · · φγ2φ∆ +(−1)i(−1)i−2 γiφdγi−1· · · φγ1φ∆ + i−2 X k=1 (−1)i(−1)k γiφγi−1· · · φγk+2φγφγk−1φ· · · φγ1φ∆ 17 = γi(dφ +φd)γi−1· · · φγ1φ∆ +(−1)i γiφγi−1· · · φγ2φ∆ + i−2 X k=1 (−1)i−kΓi(k) = γi(g −1)γi−1· · · φγ1φ∆ +(−1)i γiφγi−1· · · φγ2φ∆ + i−2 X k=1 (−1)i−kΓi(k) (le β2j=T+T2+· · · +Tp−1and β2j−1=T, hen we ha e) =βi γi−1φγi−2· · · φγ1φ∆ +(−1)i γiφγi−1· · · φγ2φ∆ + i−1 X k=1 (−1)i−kΓi(k) (P oposi ion 4.2 implies) =βi γi−1φγi−2· · · φγ1φ∆+(−1)i γi−1φγi−2· · · φγ1φ∆ + i−1 X k=1 (−1)i+k+1Γi(k) + i−1 X k=1 (−1)i−kΓi(k) =αi γi−1φγi−2· · · φγ1φ∆. 2 5 P oo o he main heo em The p oo consis s in inding ou he ac o s o he o mula (w i en in he canonical way) ha a e degene a ed and in elimina ing he summands ha ing hese ac o s. Fi s o all, no ice ha using he commu a i i y p ope ies o he ope a o s o a simplicial se (essen ially, (s3)), i is easy o see ha a ac o o he o mula whose exp ession begins (on he le ) by ∂j1· · · ∂j sk· · · (6) such ha 0 ≤j1<· · · < j < k, is degene a ed in i s simpli ied o m. Ha ing said ha , le us begin wi h he p oo o he heo em. Fo n= 0, we ob ain he explici o mula o he Alexande -Whi ney ope a o . Le us assume ha he o mula is ue o k≤nso, le us p o e ha he o mula is ue o he case n+ 1. 18 Le us conside ha nis e en (when nis odd, he p oo is simila ). In his case, by induc ion assump ion, he o mula o e an elemen o deg ee mis as ollows: AW ( SHI)n+1 =AW ( SHI)n( SHI) = m+1 X in=n · · · i1−1 X i0=0 X(−1)A(n)+B(n,m+1,¯ı)+C(n,m+1,¯ı)+D(n,¯ı)(−1)¯m+sig(α,β)+1 ∂i0+1 · · · · ∂m+1sαp+1+ ¯m· · · sα1+ ¯m∂¯m· · · ∂m−q−1 ⊗∂0· · · · ∂in−1sβq+ ¯m· · · sβ1+ ¯ms¯m−1∂m−q+1 · · · ∂m whe e ¯ı = (i0, i1, . . . , in), ¯m=m−p−q,sig(α, β) = p+1 X i=1 αi−(i−1), and he las sum is aken o e all he indices 0 ≤q≤m−1, 0 ≤p≤m−q−1 and (α, β)∈ {(p+ 1, q)-shu les}. Le us ecall ha i he o mula is in he no malized o m, he summands which ha e a degene a ed ac o mus be elimina ed. The ollowing cases a e conside ed: I in> m −p, we ha e o conside he ollowing cases: •I in=m−p+ and βq< q −1 + , wi h 1 ≤ ≤p hen, αp+1 =p+q > · · · > α +1 =q+ > α =q+ −1> βq. So, he i s ac o o hese summands is: ∂i0+1 · · · · ∂in−1−1∂m−p+ +1 · · · ∂m+1sm· · · sm−p+ −1sα −1· · · sα1∂¯m· · · ∂m−q−1 =∂i0+1 · · · · ∂in−1−1sm−p+ −1sα −1· · · sα1∂¯m· · · ∂m−q−1. Since in−1−1< in−1 = m−p+ −1, his ac o is degene a ed by (6). •I in=m−p+ and βq=q−1 + , wi h 1 ≤ ≤p hen, αp+1 =p+q > · · · > α +1 =q+ > βq=q+ −1. And hence, in his case, he exp ession o he i s ac o begins in he o m: ∂i0+1 · · · · ∂in−1−1∂m−p+ +1 · · · ∂m+1sm· · · sm−p+ sα + ¯m· · · =∂i0+1 · · · · ∂in−1−1sα + ¯m· · · . 19 Now, we ha e o conside wo di e en cases: –I in−1−1< α + ¯m hen, his ac o is degene a ed. –I in−1−1≥α + ¯m, le us deno e α =q+ −1−j, whe e 1 ≤j≤q+ −1. Then, αp+1 =p+q > · · · > α +1 =q+ , and βq=q+ −1>· · · > βq−j+1 =q+ −j > α =q+ −1−j. Hence, he second ac o o hese summands is: ∂0· · · · ∂in−2−1∂in−1+1 · · · ∂m−p+ −1sm−p+ −1· · · sm−p+ −jsβq−j+ ¯m· · · =∂0· · · · ∂in−2−1sin−1· · · sm−p+ −jsβq−j+ ¯m· · · Since in−2−1< in−1, his ac o is degene a ed. •I in=m−p+ and βq> q−1+ , wi h 1 ≤ ≤p hen in−1< βq+ ¯m. So, he second ac o o he summands has he o m (6) and hence, hese summands mus be elimina ed. •I in=m+1 and βq< p+q hen αp+1 =p+qand since in−1−1< in−1 = m hen, he i s ac o o he summands is degene a ed. •I in=m+ 1 and βq=p+q hen βq=p+q > · · · > βj+1 =p+j+ 1 > αp+1 =p+j wi h 0 ≤j≤q−1 and he i s ac o o he summands is: ∂i0+1 · · · · ∂in−1−1sm−q+jsαp+ ¯m· · · sα1+ ¯m∂¯m· · · ∂m−q−1. We ha e o conside wo di e en cases: –I in−1−1< m −q+j hen, his ac o is degene a ed. –I in−1−1≥m−q+j hen he second ac o o he summands is: ∂0· · · · ∂in−2−1∂in−1+1 · · · ∂msm· · · sm−q+j+1sβj+ ¯m· · · =∂0· · · · ∂in−2−1sin−1· · · sm−q+j+1sβj+ ¯m· · · , which is degene a ed due o he ac ha in−2−1< in−1. 20 I in< m −p, hen in−1< βq+ ¯m. So, hese summands ha e he second ac o in he o m (6) and hence, mus be elimina ed. I in=m−p, wo cases hold: •I βq> q −1, hen he second ac o o hese summands is degene a ed as abo e. •I βq=q−1 and in−1>¯m−2 hen αp+1 =p+q > · · · > α1=q > βq=q−1>· · · > β1= 0, and he second ac o o he enso p oduc is ∂0· · · · ∂in−2−1∂in−1+1 · · · ∂m−p−1sm−p−1· · · sm−p−q−1∂m−q+1 · · · ∂m =∂0· · · · ∂in−2−1sin−1· · · sm−p−q−1∂m−q+1 · · · ∂m; and since in−2−1< in−1, his ac o is degene a ed. Finally, i βq=q−1 and in−1≤¯m−2 hen he o mula (sa e o he signs) co esponding o AW ( SHI)n+1 is: ¯m−2 X in−1=n−1 · · · i1−1 X i0=0 m−1 X q=0 m−q−1 X p=0 ∂i0+1 · · · · ∂in−1−1∂m−p+1 · · · ∂m+1sm· · · sm−p∂¯m· · · ∂m−q−1 ⊗∂0· · · · ∂in−2−1∂in−1+1 · · · ∂m−p−1sm−p−1· · · s¯m−1∂m−q+1 · · · ∂m = m X i0 n+1=n+1 i0 n+1−1 X i0 n=n i0 n−1 X in−1=n−1 · · · i1−1 X i0=0 ∂i0+1 · · · · ∂in−1−1∂i0 n+1 · · · ∂i0 n+1−1 ⊗∂0· · · · ∂in−2−1∂in−1+1 · · · ∂i0 n−1∂i0 n+1+1 · · · ∂m; whe e i0 n= ¯m−1 and i0 n+1 =m−q. Now, le us s udy he signs o he o mulae in his las case. Keeping in mind ha we a e wo king wi h he exponen o (−1), all he iden i ies a e mod 2. This p oo is based on he ac ha i and only i in=m−pand βq=q−1, he summands o AW ( SHI)na e non-degene a ed. 21 We i s e i y he o mula in he case n= 1. The exponen o (−1) associa ed o each summand is: ¯m+sig(α, β) + 1 = ¯m+ p+1 X i=1 αi−(i−1) + 1 = ¯m+ p+1 X i=1 (q−1 + i−(i−1)) + 1 = ¯m+q(p+1)+1 (no e i0= ¯m−1 and i1=m−q) =i0+ (m+i1)(i0+i1) = A(1) + B(1, m,¯ı) + C(1,¯ı) + D(1, m,¯ı) whe e ¯ı = (i0, i1), as asse ed. In gene al, we ha e o p o e ha A(n) + B(n, m + 1,¯ı) + C(n,¯ı) + D(n, m + 1,¯ı) + ¯m+sig(α, β)+1 =A(n+ 1) + B(n+ 1, m,¯ı) + C(n+ 1,¯ı) + D(n+ 1, m,¯ı). •I nis e en hen D(n, m+1,¯ı) = 0 and he exponen o (−1) in each summand is: A(n) + B(n, m + 1,¯ı) + C(n,¯ı) + ¯m+sig(α, β)+1 =A(n) + B(n, m + 1,¯ı) + C(n−1,¯ı) +(m+p+in−1)(in−1+· · · +i0) + ¯m+q(p+1)+1 (since i0 n= ¯m−1, i0 n+1 =m−qand hese iden i ies a e mod 2 hen) =A(n) + B(n, m + 1,¯ı) + C(n−1,¯ı) +(m+1+i0 n+1 +i0 n+in−1)(in−1+· · · +i0) +i0 n+ (m+i0 n+1)(i0 n+1 +i0 n) =A(n) + B(n, m + 1,¯ı) + C(n−1,¯ı) + in−1+· · · +i0 +(i0 n+in−1)(in−1+· · · +i0)+(m+i0 n+1)(in−1+· · · +i0) +i0 n+ (m+i0 n+1)(i0 n+1 +i0 n) =A(n) + B(n, m + 1,¯ı) + i0 n+in−1+· · · +i0 +C(n+ 1,¯ı) + D(n+ 1, m,¯ı). 22 We ha e o dis inguish wo cases: –I n≡0 mod 4 hen A(n) = A(n+ 1) and A(n) + B(n, m + 1,¯ı) + i0 n+in−1+· · · +i0 =A(n+ 1) + n−2 2 X j=0 i2j+1 +i0 n+in−1+· · · +i0 =A(n+ 1) + n−2 2 X j=0 i2j+i0 n=A(n+ 1) + B(n+ 1, m,¯ı). –I n≡2 mod 4 hen A(n) = A(n+ 1) + 1 and A(n) + B(n, m + 1,¯ı) + i0 n+in−1+· · · +i0 =A(n+1)+1+ n−2 2 X j=0 i2j+m+p+i0 n+in−1+· · · +i0 =A(n+1)+1+ n−2 2 X j=0 i2j+m+i0 n+1 +i0 n+ 1 +i0 n+in−1+· · · +i0 =A(n+ 1) + n−2 2 X j=0 i2j+1 +i0 n+1 +m=A(n+ 1) + B(n+ 1, m,¯ı). •I nis odd hen he exponen o (−1) is: A(n) + B(n, m + 1,¯ı) + C(n,¯ı) + D(n, m + 1,¯ı) + ¯m+sig(α, β)+1 =A(n) + B(n, m + 1,¯ı) + C(n,¯ı) +(m+1+m+p)(m+p+in−1+· · · +i0) + ¯m+q(p+1)+1 (since i0 n= ¯m−1, i0 n+1 =m−qand hese iden i ies a e mod 2 hen) =A(n) + B(n, m + 1,¯ı) + C(n,¯ı) +(i0 n+1 +i0 n)(m+1+i0 n+1 +i0 n+in−1+· · · +i0) +i0 n+ (m+i0 n+1)(i0 n+1 +i0 n) =A(n) + B(n, m + 1,¯ı) + C(n,¯ı) + (i0 n+1 +i0 n)(i0 n+in−1+· · · +i0) +i0 n+1 +i0 n+ (i0 n+1 +i0 n)(m+i0 n+1) + i0 n+ (m+i0 n+1)(i0 n+1 +i0 n) =A(n) + B(n, m + 1,¯ı) + C(n,¯ı) +(i0 n+1 +i0 n)(i0 n+in−1+· · · +i0) + i0 n+1 =A(n) + B(n, m + 1,¯ı) + i0 n+1 +C(n+ 1,¯ı) + D(n+ 1, m,¯ı). 23 Since nis odd hen n≡1,3,5,7 mod 8 and, in hese cases, A(n) = A(n+ 1). 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