Cartan's contructions and the twisted Eilenberg-Zilber theorem

Ca an’s con uc ions and he wis ed Eilenbe g-Zilbe heo em 1 V. ´ Al a ez, J.A. A ma io, M.D. F au and P. Real Dp o. Ma ema ica aplicada I, Uni e sidad de Se illa, Spain Dedica ed o P o esso To nike Kadeish ili o his 60 anni e sa y Abs ac Le G×τG0be he p incipal wis ed Ca esian p oduc wi h ib e G, base Gand wis ing unc ion τ:G0 ∗→G∗−1whe e Gand G0a e simplicial g oups as well as G×τG0; and CN(G)⊗ CN(G0) be he wis ed enso p oduc associa ed o CN(G×τG0) by he wis ed Eilenbe g-Zilbe heo em. He e we p o e ha he pai (CN(G)⊗ CN(G0), µ) is a mul iplica i e Ca an’s cons uc ion whe e µis he s anda d p oduc on CN(G)⊗ CN(G0). Fu he mo e, assuming ha a con ac ion om CN(G0) o HG0exis s and using echniques om homological pe u ba ion heo y, we ex end he o me esul o o he “ wis ed” enso p oduc s o he o m CN(G)⊗HG0. Key Wo ds: Simplicial g oups; Twis ed ca esian p oduc ; Eilenbe g-Zilbe heo em; Ca an’s cons uc ion; Con ac ion; Homological pe u ba ion lemma. MSC: 55R20; 18D99. 1 In oduc ion The wis ed Eilenbe g-Zilbe heo em [B o67, Shi62] es ablishes a con ac ion (a special chain homo opy equi alence) om he no malized canonical chain complex CN(F×τB) o he wis ed ca esian p oduc F×τB o he wis ed enso p oduc (in he sense o [B59]) CN(F)⊗ CN(B). As a module, CN(F)⊗ CN(B) is he o dina y enso p oduc o CN(F) wi h CN(B); bo h o which a e DGA-algeb as, when Fand Ba e simplicial g oups. In a ecen pape [AAFR07], he au ho s p o ed ha i F, B and F×τBa e g oups, hen CN(F)⊗ CN(B) is a DGA-algeb a wi h espec o he module map µ:CN(F)⊗ CN(B)⊗CN(F)⊗ CN(B)→CN(F)⊗ CN(B) by µ= (µCN(F)⊗µCN(B))(1 ⊗T⊗1) whe e µCN(F)and µCN(B)a e he p oduc s in CN(F) and CN(B) espec i ely, and T(x⊗y)=(−1)|x| |y|y⊗x. Ca an in oduces in [Ca 56] he no ion o cons uc ion as an impo an ool o ho- mology compu a ions and o he s udy o cohomology ope a ions ( o u he de ails, see [Moo76]). I is well-known ha i Fis a g oup hen CN(F)⊗ CN(B) is a Ca an’s con- s uc ion. Fu he mo e, P ou e [P o84] p o ed ha i Gis an abelian simplicial g oup hen CN(G)⊗ CN(W(G)), associa ed o no malized chain complex o he uni e sal G-bundle 1Add ess co espondence o P o . Jos´e And ´es A ma io, Depa amen o de Ma em´a ica Aplicada I, ETSII, Uni e sidad de Se illa, A da. Reina Me cedes, S.N. 41012 Se illa (Spain); Fax: +34-954557878; E-mail: [email p o ec ed] 1 G×τW(G) (see [May67, p.88]) by he wis ed Eilenbe g-Zilbe heo em, is a mul iplica- i e Ca an’s cons uc ion. He e we ex end his esul o a wide class o p incipal wis ed ca esian p oduc o simplicial g oups (TCP). Be o e s a ing his esul we ecall he no ion o TCP. Conside wo simplicial se s F,Band a simplicial g oup Gwhich ope a es on F om he le . A Twis ed Ca esian P oduc Ewi h ib e F, base Band s uc u al g oup G consis s o a simplicial se En=Fn×Bnand ∂0( , b)=(τb ∗∂0 , ∂0b) ∂i( , b) = (∂i , ∂ib), o i > 0 si( , b) = (si , sib), o i≥0; as ace and degene acy ope a o s. He e ∗:G×F→Fis he ac ion o Gon Fand τis a wis ing unc ion, i.e., τn:Bn→Gn−1, n ≥1 sa is ies ∂0τ(b)=[τ(∂0b)]−1·τ(∂1b) ∂iτ(b) = τ(∂i+1b), o i > 0 siτ(b) = τ(si+1b), o i≥0 τ(s0b) = en, whe e endeno es he iden i y elemen o he co esponding g oup Gn. We w i e E=F×τB. I F=G hen we say ha his PCT is p incipal. He e a e ou main esul s. Theo em 1.1. Le Fand Bbe simplicial g oups and τ:B→Fbe a wis ing unc ion, such ha , he p incipal wis ed Ca esian p oduc (PTCP), F×τB, wi h ib e Fand base Bis a simplicial g oup. Then, he pai (CN(F)⊗ CN(B), µ) associa ed o CN(F×τB)by he wis ed Eilenbe g-Zilbe heo em, is a mul iplica i e Ca an’s cons uc ion. I we also suppose ha Fis educed and he e exis a con ac ion c om CN(B) o a DGA-module HB, c:φ:CN(B) * ) gH(B). Then, using, he echniques o homological pe u ba ion heo y, i is able o cons uc (see [LS87, AAFR09]) a con ac ion φ:CN(F×τB) * ) g(CN(F)⊗HB, D) whe e Ddeno es he di e en ial o he complex on he le and has he o m d⊗1+1⊗d+d ∩ (“ e ms o highe o de ”). In he case ha HB is small enough so ha he compu a ion o i s homology can ac ually be ca ied ou , we say ha he pai (c, HB) is a homological model o B. Le us obse e ha , in his si ua ion, (CN(F)⊗HB, D) is simila o he dual o Hi sch complex [Hi 53]. 2 Theo em 1.2. Unde he hypo heses o he heo em 1.1 and assuming ha Fis educed, HB is a DGA-algeb a and cis a semi- ull algeb a con ac ion (a no ion o con ac ion be ween algeb as weake han algeb a con ac ion). I is able o s a e ha CN(F)⊗H(B), D is a mul iplica i e Ca an’s cons uc ion as well. 2 The p oo o he heo em 1.1 Fi s ly, we will quickly e iew basic no ions o Homological Algeb a, and in oduce he no a ion and e minology ha we use h oughou he emainde o his a icle. Mo e de ails can be ound in [McL95]. Le Λ be a commu a i e ing wi h non ze o uni , aken hence o h as g ound ing and ixed h oughou , and Abe an augmen ed di e en ial g aded algeb a o e Λ, b ie ly a DGA-algeb a. The di e en ial, p oduc , augmen a ion and coaugmen a ion o Awill be deno ed espec i ely by dA,µA,Aand ηA. Ne e heless, we will some imes w i e hem simply as d,µ,and ηwhen no con usion can a ise. In wha ollows, he Koszul sign con en ions will be used. A mo phism ρ:A∗→A∗−1is called de i a ion i i is compa ible wi h he algeb a s uc u es on A. The deg ee o an elemen a∈Ais deno ed by |a|. In addi ion, we ecall ha i Bis also a DGA-algeb a, hen A⊗Bhas canonically associa ed an algeb a s uc u e by means o he mo phism µA⊗B= (µA⊗µB)(1A⊗T⊗1B), whe e T(b⊗a) = (−1)|b| |a|a⊗b. I he DG-algeb a Ais connec ed, ha is A0= Λ and d1:A1→A0is ze o, hen he e is a canonical augmen a ion A= 1Λ:A0→Λ. I A is a DG-algeb a, hen A]will deno e he g aded algeb a ob ained om Aby se ing he di e en ial o Aequal o ze o (i.e. o ge ing he di e en ial), and i Mis an A-module, hen M]will deno e A]-module ob ained by se ing he di e en ial equal o ze o. We will use he e he wis ed enso p oduc s uc u e. Le Abe a DG-algeb a and C be a DG-coalgeb a (we deno e by ∆Ci s cop oduc ). A wis ing cochain is a mo phism o g aded modules :C∗→A∗−1such ha dA + dC+ ∪ = 0, A = 0, ηC= 0; whe e ∪ =−µA( ⊗ )∆C. I is well-known ha d =dA⊗1 + 1 ⊗dC+ ∩is a di e en ial on A⊗C, whe e he mo phism ∩is de ined by: ∩= (µA⊗1)(1 ⊗ ⊗1)(1 ⊗∆C).(1) The DG-module (A⊗C, d ) is called he wis ed enso p oduc (o TTP) o Aand C along . We will also use he no a ion A⊗ C o such a DG-module. A cons uc ion is a iple (A, N, M) whe e 1. Ais a DGA-algeb a. 2. Mis an augmen ed A-module. 3 3. Nis a DGA-module such ha N=M= Λ ⊗AM=M/I(A)Mwhe e I(A) is he augmen a ion ideal o A. sa is ying ha M]=A]⊗N]. Example 2.1. The wis ed enso p oduc A⊗ Cgi es ise o a cons uc ion (A, C, A⊗ C). Amul iplica i e cons uc ion is a cons uc ion (A, N, M) oge he wi h he s uc u e o an algeb a on Mand also on Nsuch ha A]⊗N]→M]is an isomo phism o algeb as. Hence, he p oo o Theo em 1.1 ollows a once om he ac ha he mo phism µ= (µCN(F)⊗µCN(B))(1 ⊗T⊗1) endows o CN(F)⊗ CN(B) o a DGA-algeb a s uc u e (see [AAFR07, Theo em 3.9.]). 3 The p oo o he heo em 1.2 We assume h oughou his sec ion ha Mand Ndeno e wo DGA-modules such ha a con ac ion om N o Mexis s. We ecall ha a con ac ion (see [EM53], [HK91]) is a da a se c:{N, M, , g, φ}whe e :N→Mand g:M→Na e mo phisms o DGA-modules ( espec i ely, called he p o- jec ion and he inclusion) and φ:N→Nis a mo phism o g aded modules o deg ee +1 (called he homo opy ope a o ). These da a a e equi ed o sa is y he ules: (c1) g = 1M, (c2) φdN+dNφ+g = 1N(c3) φφ = 0, (c4) φg = 0 and (c5) φ = 0. The las h ee a e called he side condi ions [LS87]. In ac , hese may always be assumed o hold, since he homo opy φcan be al e ed o sa is y hese condi ions [GL89]. These o mulas imply ha bo h chain complexes Nand Mha e he same homology. We will also deno e a con ac ion cby φ:N * ) gM. The Eilenbe g-Zilbe heo em [EZ53] p o ides he mos classic example o a con ac ion o chain complexes. Now we add an addi ional s uc u e: Nis a DGA-A-module wi h p oduc µN:A⊗N→ N. No such assump ion is made on M(Mis a DGA-module) bu he ques ion will a ise i (M, µM) whe e µM=φµN(1 ⊗g): A⊗M→Mbecomes a DGA-A-module. Unde he hypo hesis ha φµN(1 ⊗g) = 0 we gi e an a i ma i e answe . Mo eo e , he injec ion gis A-lineal. The p oo o his esul is a simple inspec ion. Now, we ecall he concep o a pe u ba ion da um. le :N→Nbe a mo phism o g aded modules. The mo phism is poin wise nilpo en i o all x∈N(x6= 0), a posi i e in ege nexis s (in gene al, he numbe ndepends on he elemen x) such ha n(x) = 0. A pe u ba ion o a DGA-module Nis a mo phism o g aded modules δ:N→N o deg ee −1, such ha (dN+δ)2= 0 and Aδ1= 0. A pe u ba ion da um o he con ac ion c:{N, M, , g, φ}is a pe u ba ion δo he DGA-module N e i ying ha he composi ion φδ is poin wise nilpo en . 4 AT ans e ence P oblem consis s o a con ac ion c:{M, N, , g, φ} oge he wi h a pe - u ba ion δo he DGA-module N. The p oblem is o de e mine new mo phisms dδ, δ, gδ and φδsuch ha cδ:{(N, dN+δ),(M, dM+dδ), δ, gδ, φδ}is a con ac ion. The Basic Pe u ba ion Lemma ([B o67, GL89, GLS91, Rea00]) gi es an explici solu ion o he T ans e ence P oblem, assuming ha δis a pe u ba ion da um o c. Theo em 3.1. (BPL) Le c:{N, M, , g, φ}be a con ac ion and δ:N→Na pe u ba ion da um o c. Then, a new con ac ion cδ:{(N, dN+δ),(M, dM+dδ), δ, gδ, φδ} is de ined by he o mulas: dδ= δΣδ cg; δ= (1 −δΣδ cφ);gδ= Σδ cg;φδ= Σδ cφ; whe e Σδ c=X i≥0 (−1)i(φδ)i= 1 −φδ +φδφδ − · · · + (−1)i(φδ)i+· · · . Le us no e ha Σδ c(x) is a ini e sum o each x∈N, because o he poin wise nilpo ency o he composi ion φδ. Mo eo e , i is ob ious ha he mo phism dδis a pe u ba ion o he DG-module (M, dM). The wis ed Eilenbe g-Zilbe heo em can be seen as an impo an example o he use- ulness o his lemma (see [Shi62]). I sol es he T ans e ence P oblem o wis ed ca esian p oduc s. In he heo em below we assume ha Nis a DGA-A-module. This heo em gi es condi ions unde which he BPL wo ks p ese ing he DGA-A-module ca ego y. Theo em 3.2. Le δ:N→Nbe a pe u ba ion da um o φ:N * ) gMsuch ha δis compa ible wi h he A-module s uc u e on N(i.e., µN(1 ⊗δ) = δµN). I φµN(1 ⊗g) = 0 and φµN(1 ⊗φ)=0, hen he DGA-module Mδ= (M, d+dδ), ob ained by applying BPL, is a DGA-A-module wi h ega ds o he module map µMδ= µN(1 ⊗g)and he injec ion o he pe u bed con ac ion, gδ, is A-lineal. P oo . This is again seen by inspec ion. Le us ecall ha he DGA-module A⊗Nhas a i ial s uc u e o A-module wi h ega ds o he module map µA⊗N:A⊗(A⊗N)→A⊗N a1⊗(a2⊗n)→µA(a1⊗a2)⊗n. F om φ:N * ) gM, i is well-known ha we can es ablish his new con ac ion 1⊗φ:A⊗N(1⊗ ) * ) (1⊗g)A⊗M. (2) 5 I may be eadily e i ied ha he ollowing iden i ies hold: (1 ⊗φ)µA⊗N(1 ⊗(1 ⊗g)) = 0,(1 ⊗φ)µA⊗N(1 ⊗(1 ⊗φ)) = 0. µA⊗M= (1 ⊗ )µA⊗N(1 ⊗(1 ⊗g)). Wi h hese iden i ies a hand, we can w i e he ollowing consequence o Theo em 3.2. Co olla y 3.3. I δ:A⊗N→A⊗Nis a pe u ba ion da um o (2) such ha i is compa ible wi h he A-module s uc u e on A⊗N, hen he DGA-module (A⊗M)δ= (A⊗M, d +dδ), ob ained by applying BPL, is a DGA-A-module wi h ega ds o he module map µA⊗Mand he injec ion o he pe u bed con ac ion, (1 ⊗g)δ, is A-lineal. In he sequel, we ocus on he case A=CN(F) and N=CN(B) whe e Fand Ba e simplicial g oups. I we assume ha a con ac ion, c, φ:CN(B) * ) gH(B) (3) exis s, hen we can es ablish he ollowing con ac ion CN(F)⊗CN(B)* )CN(F)⊗HB. (4) Le τ:B→Fbe a wis ing unc ion, such ha , he p incipal wis ed Ca esian p oduc (PTCP), F×τB, wi h ib e Fand base Bis a simplicial g oup. The complex CN(F)⊗ CN(B) deno es he TTP associa ed o CN(F×τB) by he wis ed Eilenbe g-Zilbe heo em. Lemma 3.4. The mo phism ∩= (µCN(F)⊗1)(1 ⊗ ⊗1)(1 ⊗∆CN(B))(see (1)) sa is ies he ollowing p ope ies: 1. I Fis educed, hen ∩is a pe u ba ion da um o he con ac ion (4). 2. ∩is compa ible wi h he CN(F)-module s uc u e on CN(F)⊗ CN(B)(i.e., µF⊗B(1⊗ ∩) = ∩µF⊗B). P oo . 1. See [LS87, Lemma 3.4] o [AAFR09, p oposi ion 5.3]. 2. This is again seen by inspec ion. Hence, i Fis educed, we can pe u b he con ac ion (4) using ∩as a pe u ba ion da um. Wi h hese inpu s, he BPL gi es as ou pu he ollowing con ac ion: CN(F)⊗ CN(B)* )(CN(F)⊗HB, D).(5) whe e Ddeno es i s di e en ial. Now, we can s a e: 6 P oposi ion 3.5. I Fis educed, hen (CN(F)⊗HB, D)becomes a DGA-CN(F)-module wi h ega ds o he i ial module s uc u e on CN(F)⊗HB. P oo . The p oo ollows om Co olla y 3.3 and lemma 3.4. Focusing only on he unde lying g aded module s uc u e. We ha e he ollowing iden- i y: I(CN(F))(CN(F)⊗HB) = I(CN(F)) ⊗HB. I may be eadily e i ied ha he ollowing p ope ies hold: 1. ∩(I(CN(F)) ⊗CN(B)) ⊆I(CN(F)) ⊗CN(B). 2. The co esponden es ic ions o he mo phisms composing he con ac ion (5) o m he ollowing con ac ion: I(CN(F)) ⊗CN(B)* )I(CN(F)) ⊗HB. The o mula o he di e en ial o he complex (CN(F)⊗HB, D), gi en by he BPL, is D=d⊗1+1⊗d+d ∩whe e d ∩= (1 ⊗ ) ∩(1 ⊗g)−(1 ⊗ ) ∩(1 ⊗φ) ∩(1 ⊗g) + · · · (6) As immedia e consequence o his o mula and he p ope ies abo e, we ha e D(I(CN(F)) ⊗HB)⊆I(CN(F)) ⊗HB. Hence, (I(CN(F)) ⊗HB, D) is a DGA-CN(F)-submodule o (I(CN(F)) ⊗HB, D). P oposi ion 3.6. We ha e he ollowing iden i y o DGA-modules HB = (CN(F)⊗HB, D)/(I(CN(F))(CN(F)⊗HB, D)) P oo . On he one hand, i is easy o see ha his wo complexes a e isomo phic as g aded module. On he o he hand, aking in o accoun ha anishes on one-simplices since Fis educed (see [May67]) and he o mula (6), we ha e d ∩( ⊗b) = 0|b| ≤ 1 0 mod I(CN(F)) ⊗HB |b|>1 Hence, we ha e ha he di e en ial o bo h complexes a e he same. P oposi ion 3.7. Unde he hypo heses o he P eposi ion 3.5 and 3.6. We can s a e ha (CN(F)⊗HB, D)is a cons uc ion. 7 Finally, I we assume ha he con ac ion (3) is semi- ull, we will ge he desi ed esul . Now, we ecall ha cis a semi- ull algeb a con ac ion ([Rea00]) i he injec ion, g, is a mo phism o DGA-algeb as and he p ojec ion, , and he homo opy ope a o , φ, sa is y he ollowing p ope ies: µC(B)(φ⊗φ)=0, µC(B)(φ⊗g)=0, µC(B)(g⊗φ)=0, φµC(B)(φ⊗φ)=0, φµC(B)(φ⊗g)=0, φµC(B)(φ⊗φ) = 0. Unde hese condi ions we can say ha hese con ac ions (4) and (5) a e semi- ull (see [Rea00, Theo em 4.18]). Mo eo e , (CN(F)⊗HB, D) is a DGA-algeb a wi h ega ds o he s anda d p oduc CN(F)⊗HB. F om his ac , we conclude ha (CN(F)⊗HB, D) is a mul iplica i e cons uc ion. 4 Conclusions In his pape we ha e aced he p oblem o ans e ing he mul iplica i e cons uc ion s uc u e up o con ac ion. The p oblem o ans e ing o he s uc u es (e.g. algeb a, coalgeb a, TTP) up o con ac ion has been widely ea ed in he li e a u e [GLS91, Rea00, AAFR05]. In gene al, we ha e ha (co)algeb a and TTP become A∞-(co)algeb a and A∞- TTP ia con ac ion, espec i ely. We p o ed ha he p ope y o being a mul iplica i e cons uc ion on CN(F)⊗ CN(B) has been ans e ed o (CN(F)⊗HB, D) ia con ac ion. Howe e , i we ocus on he s uc u e o TTP, we only ha e ha (CN(F)⊗HB, D) is an A∞- wis ed enso p oduc o CN(F) (algeb a) and HB (A∞-coalgeb a) along ¯ = g (A∞- wis ed cochain), see [AAFR05, Theo em 2.1]. Acknowledgmen We would like o hank K is een Cheng o he eading o his pape . This wo k was pa ially suppo ed by he PAICYT esea ch p ojec FQM–296 om Jun a de Andaluc´ıa (Spain). Re e ences [AAFR05] ´ Al a ez, V., A ma io, J.A., F au, M.D., Real, P. (2005). T ans e ing TTP- s uc u es ia con ac ion. Homology, Homo opy and Applica ions 7(2):41–54. [AAFR07] ´ Al a ez, V., A ma io, J.A., F au, M.D., Real, P. (2007). Algeb a s uc u es on he wis ed Eilenbeg-Zilbe heo em. Comm. Alg. 35:3273–3291. [AAFR09] ´ Al a ez, V., A ma io, J.A., F au, M.D., Real, P. 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