Cartan's contructions and the twisted Eilenberg-Zilber theorem

Cartan’s contructions and the twisted Eilenberg-Zilber theorem 1 V. ´ Alvarez, J.A. Armario, M.D. Frau and P. Real Dpto. Matematica aplicada I, Universidad de Sevilla, Spain Dedicated to Professor Tornike Kadeishvili for his 60 anniversary Abstract Let G×τG0be the principal twisted Cartesian product with fibre G, base Gand twisting function τ:G0 ∗→G∗−1where Gand G0are simplicial groups as well as G×τG0; and CN(G)⊗tCN(G0) be the twisted tensor product associated to CN(G×τG0) by the twisted Eilenberg-Zilber theorem. Here we prove that the pair (CN(G)⊗tCN(G0), µ) is a multiplicative Cartan’s construction where µis the standard product on CN(G)⊗ CN(G0). Furthermore, assuming that a contraction from CN(G0) to HG0exists and using techniques from homological perturbation theory, we extend the former result to other “twisted” tensor products of the form CN(G)⊗HG0. Key Words: Simplicial groups; Twisted cartesian product; Eilenberg-Zilber theorem; Cartan’s construction; Contraction; Homological perturbation lemma. MSC: 55R20; 18D99. 1 Introduction The twisted Eilenberg-Zilber theorem [Bro67, Shi62] establishes a contraction (a special chain homotopy equivalence) from the normalized canonical chain complex CN(F×τB) of the twisted cartesian product F×τBto the twisted tensor product (in the sense of [B59]) CN(F)⊗tCN(B). As a module, CN(F)⊗tCN(B) is the ordinary tensor product of CN(F) with CN(B); both of which are DGA-algebras, when Fand Bare simplicial groups. In a recent paper [AAFR07], the authors proved that if F, B and F×τBare groups, then CN(F)⊗tCN(B) is a DGA-algebra with respect to the module map µ:CN(F)⊗tCN(B)⊗CN(F)⊗tCN(B)→CN(F)⊗tCN(B) by µ= (µCN(F)⊗µCN(B))(1 ⊗T⊗1) where µCN(F)and µCN(B)are the products in CN(F) and CN(B) respectively, and T(x⊗y)=(−1)|x| |y|y⊗x. Cartan introduces in [Car56] the notion of construction as an important tool for homology computations and to the study of cohomology operations (for further details, see [Moo76]). It is well-known that if Fis a group then CN(F)⊗tCN(B) is a Cartan’s construction. Furthermore, Proute [Pro84] proved that if Gis an abelian simplicial group then CN(G)⊗tCN(W(G)), associated to normalized chain complex of the universal G-bundle 1Address correspondence to Prof. Jos´e Andr´es Armario, Departamento de Matem´atica Aplicada I, ETSII, Universidad de Sevilla, Avda. Reina Mercedes, S.N. 41012 Sevilla (Spain); Fax: +34-954557878; E-mail: [email protected] 1 G×τW(G) (see [May67, p.88]) by the twisted Eilenberg-Zilber theorem, is a multiplicative Cartan’s construction. Here we extend this result to a wider class of principal twisted cartesian product of simplicial groups (TCP). Before stating this result we recall the notion of TCP. Consider two simplicial sets F,Band a simplicial group Gwhich operates on F from the left. A Twisted Cartesian Product Ewith fibre F, base Band structural group G consists of a simplicial set En=Fn×Bnand ∂0(f, b)=(τb ∗∂0f, ∂0b) ∂i(f, b) = (∂if, ∂ib),for i > 0 si(f, b) = (sif, sib),for i≥0; as face and degeneracy operators. Here ∗:G×F→Fis the action of Gon Fand τis a twisting function, i.e., τn:Bn→Gn−1, n ≥1 satisfies ∂0τ(b)=[τ(∂0b)]−1·τ(∂1b) ∂iτ(b) = τ(∂i+1b),for i > 0 siτ(b) = τ(si+1b),for i≥0 τ(s0b) = en, where endenotes the identity element of the corresponding group Gn. We write E=F×τB. If F=Gthen we say that this PCT is principal. Here are our main results. Theorem 1.1. Let Fand Bbe simplicial groups and τ:B→Fbe a twisting function, such that, the principal twisted Cartesian product (PTCP), F×τB, with fibre Fand base Bis a simplicial group. Then, the pair (CN(F)⊗tCN(B), µ) associated to CN(F×τB)by the twisted Eilenberg-Zilber theorem, is a multiplicative Cartan’s construction. If we also suppose that Fis reduced and there exist a contraction cfrom CN(B) to a DGA-module HB, c:φ:CN(B)f * ) gH(B). Then, using, the techniques of homological perturbation theory, it is able to construct (see [LS87, AAFR09]) a contraction φ:CN(F×τB)f * ) g(CN(F)⊗HB, D) where Ddenotes the differential of the complex on the left and has the form d⊗1+1⊗d+dt∩ (“terms of higher order”). In the case that HB is small enough so that the computation of its homology can actually be carried out, we say that the pair (c, HB) is a homological model for B. Let us observe that, in this situation, (CN(F)⊗HB, D) is similar to the dual of Hirsch complex [Hir53]. 2 Theorem 1.2. Under the hypotheses of the theorem 1.1 and assuming that Fis reduced, HB is a DGA-algebra and cis a semi-full algebra contraction (a notion of contraction between algebras weaker than algebra contraction). It is able to state that CN(F)⊗H(B), D is a multiplicative Cartan’s construction as well. 2 The proof of the theorem 1.1 Firstly, we will quickly review basic notions of Homological Algebra, and introduce the notation and terminology that we use throughout the remainder of this article. More details can be found in [McL95]. Let Λ be a commutative ring with non zero unit, taken henceforth as ground ring and fixed throughout, and Abe an augmented differential graded algebra over Λ, briefly a DGA-algebra. The differential, product, augmentation and coaugmentation of Awill be denoted respectively by dA,µA,Aand ηA. Nevertheless, we will sometimes write them simply as d,µ,and ηwhen no confusion can arise. In what follows, the Koszul sign conventions will be used. A morphism ρ:A∗→A∗−1is called derivation if it is compatible with the algebra structures on A. The degree of an element a∈Ais denoted by |a|. In addition, we recall that if Bis also a DGA-algebra, then A⊗Bhas canonically associated an algebra structure by means of the morphism µA⊗B= (µA⊗µB)(1A⊗T⊗1B), where T(b⊗a) = (−1)|b| |a|a⊗b. If the DG-algebra Ais connected, that is A0= Λ and d1:A1→A0is zero, then there is a canonical augmentation A= 1Λ:A0→Λ. If A is a DG-algebra,then A]will denote the graded algebra obtained from Aby setting the differential of Aequal to zero (i.e. forgetting the differential), and if Mis an A-module, then M]will denote A]-module obtained by setting the differential equal to zero. We will use here the twisted tensor product structure. Let Abe a DG-algebra and C be a DG-coalgebra (we denote by ∆Cits coproduct). A twisting cochain is a morphism of graded modules t:C∗→A∗−1such that dAt+tdC+t∪t= 0, At= 0, tηC= 0; where t∪t=−µA(t⊗t)∆C. It is well-known that dt=dA⊗1 + 1 ⊗dC+t∩is a differential on A⊗C, where the morphism t∩is defined by: t∩= (µA⊗1)(1 ⊗t⊗1)(1 ⊗∆C).(1) The DG-module (A⊗C, dt) is called the twisted tensor product (or TTP) of Aand C along t. We will also use the notation A⊗tCfor such a DG-module. A construction is a triple (A, N, M) where 1. Ais a DGA-algebra. 2. Mis an augmented A-module. 3 3. Nis a DGA-module such that N=M= Λ ⊗AM=M/I(A)Mwhere I(A) is the augmentation ideal of A. satisfying that M]=A]⊗N]. Example 2.1. The twisted tensor product A⊗tCgives rise to a construction (A, C, A⊗tC). Amultiplicative construction is a construction (A, N, M) together with the structure of an algebra on Mand also on Nsuch that A]⊗N]→M]is an isomorphism of algebras. Hence, the proof of Theorem 1.1 follows at once from the fact that the morphism µ= (µCN(F)⊗µCN(B))(1 ⊗T⊗1) endows to CN(F)⊗tCN(B) of a DGA-algebra structure (see [AAFR07, Theorem 3.9.]). 3 The proof of the theorem 1.2 We assume throughout this section that Mand Ndenote two DGA-modules such that a contraction from Nto Mexists. We recall that a contraction (see [EM53], [HK91]) is a data set c:{N, M, f, g, φ}where f:N→Mand g:M→Nare morphisms of DGA-modules (respectively, called the projection and the inclusion) and φ:N→Nis a morphism of graded modules of degree +1 (called the homotopy operator). These data are required to satisfy the rules: (c1) fg = 1M, (c2) φdN+dNφ+gf = 1N(c3) φφ = 0, (c4) φg = 0 and (c5) fφ = 0. The last three are called the side conditions [LS87]. In fact, these may always be assumed to hold, since the homotopy φcan be altered to satisfy these conditions [GL89]. These formulas imply that both chain complexes Nand Mhave the same homology. We will also denote a contraction cby φ:Nf * ) gM. The Eilenberg-Zilber theorem [EZ53] provides the most classic example of a contraction of chain complexes. Now we add an additional structure: Nis a DGA-A-module with product µN:A⊗N→ N. No such assumption is made on M(Mis a DGA-module) but the question will arise if (M, µM) where µM=φµN(1 ⊗g): A⊗M→Mbecomes a DGA-A-module. Under the hypothesis that φµN(1 ⊗g) = 0 we give an affirmative answer. Moreover, the injection gis A-lineal. The proof of this result is a simple inspection. Now, we recall the concept of a perturbation datum. let f:N→Nbe a morphism of graded modules. The morphism fis pointwise nilpotent if for all x∈N(x6= 0), a positive integer nexists (in general, the number ndepends on the element x) such that fn(x) = 0. A perturbation of a DGA-module Nis a morphism of graded modules δ:N→N of degree −1, such that (dN+δ)2= 0 and Aδ1= 0. A perturbation datum of the contraction c:{N, M, f, g, φ}is a perturbation δof the DGA-module Nverifying that the composition φδ is pointwise nilpotent. 4 ATransference Problem consists of a contraction c:{M, N, f, g, φ}together with a perturbation δof the DGA-module N. The problem is to determine new morphisms dδ, fδ, gδ and φδsuch that cδ:{(N, dN+δ),(M, dM+dδ), fδ, gδ, φδ}is a contraction. The Basic Perturbation Lemma ([Bro67, GL89, GLS91, Rea00]) gives an explicit solution to the Transference Problem, assuming that δis a perturbation datum of c. Theorem 3.1. (BPL) Let c:{N, M, f, g, φ}be a contraction and δ:N→Na perturbation datum of c. Then, a new contraction cδ:{(N, dN+δ),(M, dM+dδ), fδ, gδ, φδ} is defined by the formulas: dδ=fδΣδ cg;fδ=f(1 −δΣδ cφ);gδ= Σδ cg;φδ= Σδ cφ; where Σδ c=X i≥0 (−1)i(φδ)i= 1 −φδ +φδφδ − · · · + (−1)i(φδ)i+· · · . Let us note that Σδ c(x) is a finite sum for each x∈N, because of the pointwise nilpotency of the composition φδ. Moreover, it is obvious that the morphism dδis a perturbation of the DG-module (M, dM). The twisted Eilenberg-Zilber theorem can be seen as an important example of the usefulness of this lemma (see [Shi62]). It solves the Transference Problem for twisted cartesian products. In the theorem below we assume that Nis a DGA-A-module. This theorem gives conditions under which the BPL works preserving the DGA-A-module category. Theorem 3.2. Let δ:N→Nbe a perturbation datum of φ:Nf * ) gMsuch that δis compatible with the A-module structure on N(i.e., µN(1 ⊗δ) = δµN). If φµN(1 ⊗g) = 0 and φµN(1 ⊗φ)=0, then the DGA-module Mδ= (M, d+dδ), obtained by applying BPL, is a DGA-A-module with regards to the module map µMδ=fµN(1 ⊗g)and the injection of the perturbed contraction, gδ, is A-lineal. Proof. This is again seen by inspection. Let us recall that the DGA-module A⊗Nhas a trivial structure of A-module with regards to the module map µA⊗N:A⊗(A⊗N)→A⊗N a1⊗(a2⊗n)→µA(a1⊗a2)⊗n. From φ:Nf * ) gM, it is well-known that we can establish this new contraction 1⊗φ:A⊗N(1⊗f) * ) (1⊗g)A⊗M. (2) 5 It may be readily verified that the following identities hold: (1 ⊗φ)µA⊗N(1 ⊗(1 ⊗g)) = 0,(1 ⊗φ)µA⊗N(1 ⊗(1 ⊗φ)) = 0. µA⊗M= (1 ⊗f)µA⊗N(1 ⊗(1 ⊗g)). With these identities at hand, we can write the following consequence of Theorem 3.2. Corollary 3.3. If δ:A⊗N→A⊗Nis a perturbation datum for (2) such that it is compatible with the A-module structure on A⊗N, then the DGA-module (A⊗M)δ= (A⊗M, d +dδ), obtained by applying BPL, is a DGA-A-module with regards to the module map µA⊗Mand the injection of the perturbed contraction, (1 ⊗g)δ, is A-lineal. In the sequel, we focus on the case A=CN(F) and N=CN(B) where Fand Bare simplicial groups. If we assume that a contraction, c, φ:CN(B)f * ) gH(B) (3) exists, then we can establish the following contraction CN(F)⊗CN(B)* )CN(F)⊗HB. (4) Let τ:B→Fbe a twisting function, such that, the principal twisted Cartesian product (PTCP), F×τB, with fibre Fand base Bis a simplicial group. The complex CN(F)⊗t CN(B) denotes the TTP associated to CN(F×τB) by the twisted Eilenberg-Zilber theorem. Lemma 3.4. The morphism t∩= (µCN(F)⊗1)(1 ⊗t⊗1)(1 ⊗∆CN(B))(see (1)) satisfies the following properties: 1. If Fis reduced, then t∩is a perturbation datum of the contraction (4). 2. t∩is compatible with the CN(F)-module structure on CN(F)⊗tCN(B)(i.e., µF⊗B(1⊗ t∩) = t∩µF⊗B). Proof. 1. See [LS87, Lemma 3.4] or [AAFR09, proposition 5.3]. 2. This is again seen by inspection. Hence, if Fis reduced, we can perturb the contraction (4) using t∩as a perturbation datum. With these inputs, the BPL gives as output the following contraction: CN(F)⊗tCN(B)* )(CN(F)⊗HB, D).(5) where Ddenotes its differential. Now, we can state: 6 Proposition 3.5. If Fis reduced, then (CN(F)⊗HB, D)becomes a DGA-CN(F)-module with regards to the trivial module structure on CN(F)⊗HB. Proof. The proof follows from Corollary 3.3 and lemma 3.4. Focusing only on the underlying graded module structure. We have the following identity: I(CN(F))(CN(F)⊗HB) = I(CN(F)) ⊗HB. It may be readily verified that the following properties hold: 1. t∩(I(CN(F)) ⊗CN(B)) ⊆I(CN(F)) ⊗CN(B). 2. The correspondent restrictions of the morphisms composing the contraction (5) form the following contraction: I(CN(F)) ⊗CN(B)* )I(CN(F)) ⊗HB. The formula for the differential of the complex (CN(F)⊗HB, D), given by the BPL, is D=d⊗1+1⊗d+dt∩where dt∩= (1 ⊗f)t∩(1 ⊗g)−(1 ⊗f)t∩(1 ⊗φ)t∩(1 ⊗g) + · · · (6) As immediate consequence of this formula and the properties above, we have D(I(CN(F)) ⊗HB)⊆I(CN(F)) ⊗HB. Hence, (I(CN(F)) ⊗HB, D) is a DGA-CN(F)-submodule of (I(CN(F)) ⊗HB, D). Proposition 3.6. We have the following identity of DGA-modules HB = (CN(F)⊗HB, D)/(I(CN(F))(CN(F)⊗HB, D)) Proof. On the one hand, it is easy to see that this two complexes are isomorphic as graded module. On the other hand, taking into account that tvanishes on one-simplices since Fis reduced (see [May67]) and the formula (6), we have dt∩(f⊗b) = 0|b| ≤ 1 0 mod I(CN(F)) ⊗HB |b|>1 Hence, we have that the differential of both complexes are the same. Proposition 3.7. Under the hypotheses of the Preposition 3.5 and 3.6. We can state that (CN(F)⊗HB, D)is a construction. 7 Finally, If we assume that the contraction (3) is semi-full, we will get the desired result. Now, we recall that cis a semi-full algebra contraction ([Rea00]) if the injection, g, is a morphism of DGA-algebras and the projection, f, and the homotopy operator, φ, satisfy the following properties: fµC(B)(φ⊗φ)=0, fµC(B)(φ⊗g)=0, fµC(B)(g⊗φ)=0, φµC(B)(φ⊗φ)=0, φµC(B)(φ⊗g)=0, φµC(B)(φ⊗φ) = 0. Under these conditions we can say that these contractions (4) and (5) are semi-full (see [Rea00, Theorem 4.18]). Moreover, (CN(F)⊗HB, D) is a DGA-algebra with regards to the standard product CN(F)⊗HB. From this fact, we conclude that (CN(F)⊗HB, D) is a multiplicative construction. 4 Conclusions In this paper we have faced the problem of transferring the multiplicative construction structure up to contraction. The problem of transferring other structures (e.g. algebra, coalgebra, TTP) up to contraction has been widely treated in the literature [GLS91, Rea00, AAFR05]. In general, we have that (co)algebra and TTP become A∞-(co)algebra and A∞- TTP via contraction, respectively. We proved that the property of being a multiplicative construction on CN(F)⊗tCN(B) has been transferred to (CN(F)⊗HB, D) via contraction. However, if we focus on the structure of TTP, we only have that (CN(F)⊗HB, D) is an A∞-twisted tensor product of CN(F) (algebra) and HB (A∞-coalgebra) along ¯ t=tg (A∞-twisted cochain), see [AAFR05, Theorem 2.1]. 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