On the computability of the p-local homology of twisted cartesian products of Eilenberg-Mac Lane spaces

On the computability of the p-local homology of twisted cartesian products of Eilenberg-Mac Lane spaces V. ´ Alvarez∗J.A. Armario†P. Real∗ Dpto. de Matem´atica Aplicada I. Univ. de Sevilla. E-mails: valv[email protected], [email protected], [email protected] Abstract Working in the framework of the Simplicial Topology, a method for calculating the p-local homology of a twisted cartesian product X(π, m, τ, π0, n) = K(π, m)×τK(π0, n) of Eilenberg-Mac Lane spaces is given. The chief technique is the construction of an explicit homotopy equivalence between the normalized chain complex of Xand a free DGA-module of finite type M, via homological perturbation. If Xis a commutative simplicial group (being its inner product the natural one of the cartesian product of K(π, m) and K(π0, n)), then Mis a DGA-algebra. Finally, in the special case K(π, 1) ,→Xp →K(π0, n), we prove that Mcan be a small twisted tensor product. 1 Introduction In Simplicial Topology [14], fibre bundles simplify its own structure since they can be considered as twisted cartesian products (TCPs) of two simplicial sets (fibre) ×τ(base), where the function τproduces a “torsion” on the 0-face of the cartesian product. An example of principal TCP is X(π, m, τ, π0, n) = K(π, m)×τK(π0, n), where K(π, m) and K(π0, n) are Eilenberg-Mac Lane spaces. We are interested here in a PTCP of the form X(π, m, τ, π0, n) where πand π0are finitely generated abelian groups and m, n ∈N. More precisely, we want to calculate the p-local homology of X. Our approach is based essentially in the use of techniques ∗Facultad de Inform´atica y Estad´ıstica. Avda. Reina Mercedes S.N. 41012, Sevilla (Spain). †Escuela Universitaria de Ingenier´ıa T´ecnica Agr´ıcola. Ctra. Utrera Km. 1, 41013, Sevilla (Spain). First author is supported by a grant from the Junta de Andaluc´ ia 1 from Homological Perturbation Theory (see [8], [9], [11]). Working over Zlocalised over a prime pand combining results from the papers [18], [16] and [17], it is easy to establish an explicit contraction (special homotopy equivalence) from the normalized chain complex of a K(π, m) to a tensor product G(π, m) of elementary complexes defined by Cartan ([4], expos´e 11) corresponding to admissible sequences. With these data at hand, a “constructive” version of the Serre spectral sequence applied to this case, provides us that the p-local homology of Xis reduced to that of a small “modified” (from a differential point of view) tensor product M= (G(π, m)⊗G(π0, n), dGπ ⊗1 + 1⊗dGπ0+dτ), where the formula of dτis given by the Basic Perturbation Lemma. Let us note that the cartesian product K(π, m)×K(π0, n) shows a structure of commutative simplicial group in a natural way. If X(π, m, τ, π0, n), endowed with this same product, is a simplicial group, we see here that the general method can be substantially improved. More concretely, we obtain that Mis a DGA-algebra and that the morphism dτis completely determined by knowing its images on the generators of this DGA-algebra M. Finally, taking as point of departure X(π, 1, τ, π0, n), we prove that its p-local homology can be derived from that of a twisted tensor product G(π, 1)⊗t0¯ B(G(π0, n−1)), where ¯ B( ) is the reduced bar construction (see [13]) and t0is a Brown’s cochain. This paper is organized as follows: Section 2 describes the main algebraic and simplicial objects we intend to deal with and gives same important results which play an essential role in the sequel. Possibly, the main result in this section is: Working over Z(p), there is a semi-full algebra contraction from C(K(π, n)to a tensor product G(π, n) of Cartan elementary complexes corresponding to admissible sequences. Section 3 is devoted to describe a constructive version of the Serre spectral sequence for computing the p-local homology of X(π, m, τ, π0, n). If Xis a simplicial group (being its inner product the natural one of the cartesian product of the Eilenberg-Mac Lane simplicial groups, then we see that the previous method can be enormously improved. Finally, we analyze in detail the case X(π, 1, τ, π0, n), concluding that the p-local homology of Xis determined by a twisted tensor product G(π, 1) ⊗t0¯ B(G(π0, n −1)). 2 Preliminaries Let Λ be a commutative ring with 1 6= 0. A simplicial set Xis a collection of simplices, each one having a dimension (positive integer), endowed with a collection of face operators and a collection of degeneracy operators satisfying several conditions of compatibility [14]. If Xhas only one 0-simplex, then Xis called reduced. If Gis a simplicial set provided with a group structure, compatible in a natural sense with the simplicial structure, then Gis called a simplicial group. To define a twisted cartesian product, several ingredients are needed: a base space B, a fibre space F, both of which are simplicial sets, a structural group Gwhich is a simplicial group, acting on the fibre (g∗f∈F;g∈G, f ∈F) and finally a twisting operator (or 2 geometric torsion) τwhich looks like a function B→G, satisfying some coherence conditions. The total space is denoted F×τBand is essentially the product of Bby F twisted according to the τ-operator. The twisted cartesian product (TCP), F×τB, is a simplicial set where the n-simplices, face and degeneracy operators are the following ones: (F×τB)n=Fn×Bn; ∂0(f, b) = (τb ∗∂0f, ∂0); ∂i(f, b) = (∂ia, ∂i) for i > 0; sj(f, b) = (sjfsjb),for i≥0; In the case that F=G,F×τBis called a principal twisted cartesian product (PTCP). If πis an ordinary (discrete) abelian group, one can consider Gπ as a simplicial group, where (Gπ)n=π n ≥0 and the face and degeneracy operators are all the identity maps. Then the simplicial group K(π, n) is defined as follows: K(π, 0) = Gπ and K(π, n + 1) = ¯ WK(π, n), where ¯ Wdenotes the classifying construction [14]. An important property of the Eilenberg-MacLane spaces is that they have only one non null homotopy group , πn(K(π, n)) = π. Given a simplicial set X, there is a DG-module canonically associated to X, that we denote by C∗(X) (called chain complex of X) where the n-component is the free Λ-module Cn(X) = Λ[Xn] generated by the n-simplices of X, and the differential operator dn:Cn(X)→Cn−1(X) is the alternated sum dn=P0≤i≤n(−1)i∂i. The homology groups H∗(X) are the quotient groups Hn(X) = ker dn/Im dn+1. The subset consisting of all the degenerated simplices of C∗(X), s(C∗(X)) is a submodule verifying dn(s(C∗(X))n)⊂s(C∗(X))n−1, so that, C(X) = {C∗(X)/s(C∗(X))}is a DG- module, called the normalized DG-module canonically associated to X. It is verified that H∗(X) = H∗(C∗(X)) = H∗(C(X)). From a algebraic point view, our main objets are commutative (connected) DGA- (co)algebras. The differential, product (resp. coproduct), augmentation and coaugmentation of a DGA-algebra Awill be denoted respectively by dA,µA(resp. ∆A, Aand ηA). 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|. We need here the reduced bar construction or simply bar construction ¯ B(A) of a DGA-algebra Aand the cobar construction ¯ Ω(C) of a DGA-coalgebra C(see [13]). Recall that, the bar construction is a coalgebra, ¯ B(A) = Tc(S(Ker A))), where Tc( ) is the tensor coalgebra and S( ) is the suspension functor. If Ais commutative then a product ∗(called shuffle product) can be defined on ¯ B(A), such that the reduced bar construction has a Hopf algebra structure. Recall too, that the cobar contruction is an algebra, ¯ Ω(A) = Ta(S−1(Ker A))), where Ta( ) is the tensor algebra and S−1( ) is the desuspension functor. We deal with a special type of homotopy equivalence: a contraction (see [6], [12]) is a data set c:{N, M, f, g, φ}, (we denote by N⇒M) where f:N→M,g:M→N are morphisms of DGA-modules (called, respectively, projection and inclusion) and 3 φ:N→Nis a morphism of graded modules of degree +1 (called homotopy operator), and these data are required to satisfy the rules: (r1) fg = 1M,(r2) fφ = 0; (r3) φg = 0, (r4) φdN+dNφ+gf = 1Nand (r5) φφ = 0. For instance, let us consider the Eilenberg-Zilber contraction: Theorem 1 (Eilenberg-Zilber Theorem [7]) Let Xand Ybe simplicial sets. We will consider the morphisms: Alexander-Whitney, AW :C(X×Y)−→ C(X)⊗C(Y) Eilenberg-MacLane, EML :C(X)⊗C(Y)−→ C(X×Y) And Shih, SHI :C(X×Y)−→ C(X×Y) The morphisms are defined by the formulas: AW(an×bn) = n X i=0 ∂i+1 · · · ∂nan⊗∂0· · · ∂i−1bn, EML(ap⊗bq) = X (α,β)∈{(p,q)−shuffles} (−1)sg(α,β)(sβq· · · sβ1ap×sαp· · · sα1bq), SHI(an×bn) = =−X(−1)m+sg(α,β)(sβq+m· · · sβ1+msm−1∂n−q+1 · · · ∂nan× ×sαp+1+m· · · sα1+m∂m· · · ∂m+p−1bn); where m=n−p−q,sg(α, β) = Pp i=1(αi−(i−1)), and the last sum is taken over the indices 0≤q≤n−1,0≤p≤n−q−1and (α, β)∈ {(p+ 1, q)-shuffles}. Then, the data set EZX,Y :{C(X×Y), C(X)⊗C(Y), AWX,Y , EMLX,Y , SHIX,Y }(1) is a contraction. From now on, a contraction of the type (1) will be called Eilenberg-Zilber contraction. The first definition of the SHI operator is given in an inductive way in [6]. The above explicit formula for the SHI operator is given by Rubio in [19]. An appendix in [17] is devoted to prove that this explicit formula satisfies the inductive definition given in [6]. We recall the concept of a perturbation datum. Let Nbe a graded module and let f:N→Nbe a morphism of graded modules. The morphism fis pointwise 4 nilpotent if for all x,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 N is a morphism of graded modules δ:N→Nof degree −1, such that (dN+δ)2= 0 and ξNδ= 0. A perturbation datum of the contraction c:{N, M, f, g, φ}is a perturbation δof the DGA-module Nverifying the composition φδ is pointwise nilpotent. We now introduce the main tool in Homological Perturbation Theory: the Basic Perturbation Lemma ([8], [11],[9], [2]). Theorem 2 (BPL) Let c:{N, M, f, g, φ}be a contraction and δ:N→Na perturbation datum of r. Then, a new contraction rδ:{(N, dN+δ, N, ηN),(M, dM+dδ, M, ηM), fδ, gδ, φδ} is defined by the formulas: dδ=fδΣδ rg;fδ=f(1 −δΣδ rφ);gδ= Σδ rg;φδ= Σδ rφ; where Σδ r=X i≥0 (−1)i(φδ)i= 1 −φδ +φδφδ − · · · + (−1)i(φδ)i+· · · . Let us note that Σδ r(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 DGA-module (M, dM, M, ηM). Theorem 3 ( Brown’s theorem [3]) Let F×τBbe a twisted cartesian product with structural group G. It is possible to establish the following contraction: C(F×τB)⇒C(F)⊗tC(B) (2) In [20], Shih proved that the contraction (2) is obtained from C(F×B)⇒C(F)⊗ C(B) via perturbation, with δ(f, b) = (τb ∗∂0f, ∂0b)−(∂0f, ∂0b) as perturbation datum. Note 1 C(F)⊗tC(B)is a twisted tensor product in the Brown’s sense, that is, its differential dcan be obtained by the action of a Brown cochain t:C(B)n→C(G)n−1; therefore, d=dF⊗1+1⊗dB+ (µ⊗1)(1 ⊗t⊗1)(1 ⊗∆);µis the morphism induced by the operation ∗from Gto F, and ∆is the coproduct in CN ∗(B)[3], [20]. Theorem 4 [14] Let F×τBbe a PTCP. If τ(b) = e0∀b∈B1,then t(b) = 0 ∀b∈ B1non degenerated; the element e0denote the unit of G0. Theorem 5 [15] Let t:C→Abe a Brown’s cochain (Cis a DGA-coalgebra and Ais a DGA-algebra). Then tadmits a lifting (a DGA-algebra morphism) T:¯ Ω(C)→A. If T:¯ Ω(C)→Ais a DGA-algebra morphism, then T|Cis a Brown’s cochain. 5 The perturbation problem for more complex structures has been largely considered. The most significant results for DGA-algebras (or DGA-coalgebras) can be found in [11], [12] and [17]. First, we review several notions. Definition 1 [17] Let Aand A0be two DGA-algebras and c:{A, A0, f, g, φ}a contraction. The projection fis a quasi algebra projection if the following conditions hold: fµA(φ⊗φ)=0, fµA(φ⊗g) = 0, fµA(g⊗φ)=0, The homotopy operator φis a a quasi algebra homotopy if the following conditions hold: φµA(φ⊗φ)=0, φµA(φ⊗g)=0, φµA(g⊗φ)=0, Definition 2 [11] Let Aand A0be two DGA-algebras and c:{A, A0, f, g, φ}a contraction. The homotopy operator φis said to be an algebra homotopy if φµA=µA(1A⊗φ+φ⊗gf). Definition 3 [17] Let Aand A0be two DGA-algebras and c:{A, A0, f, g, φ}a contraction. We say that ris •a semi-full algebra contraction if fis a quasi algebra projection, gis a morphism of DGA-algebras and φis a quasi algebra homotopy. •an almost-full algebra contraction if fand gare morphisms of DGA-algebras and φis a quasi algebra homotopy. •a full algebra contraction if fand gare morphisms of DGA-algebras and φis an algebra homotopy. Obviously, full and almost-full algebra contractions are, in particular, semi-full algebra contractions. It is not difficult to prove that both sets of semi-full and almostfull algebra contractions are closed by composition and tensor product of contractions. Now, we introduce an important example of a semi-full algebra contraction. Recall that if Xand Yare simplicial groups then C(X) and C(Y) are DGA-algebras. An interesting problem is the multiplicative behaviour of (1) when Xand Yare simplicial groups. It is possible to enunciate the following theorem. 6 Theorem 6 [1] If Gand G0are simplicial groups, then the Eilenberg-Zilber contraction: EZG,G0:{C(G×G0), C(G)⊗C(G0), AWG,G0, EMLG,G0, SHIG,G0} is an almost-full algebra contraction. Eilenberg and MacLane in [6] showed that the projection (AWG,G0) and the inclusion (EMG,G0) are DGA-algebra morphisms if Gand G0are abelian. We show in [1] that AWG,G0and EMG,G0are DGA-algebra morphisms in the non-abelian case and the homotopy operator (SHIG,G0) is a quasi-homotopy operator. Definition 4 [9] Let Aand A0be two DGA-algebras and c:{A, A0, f, g, φ}a contraction. An algebra perturbation datum δof cis a perturbation datum of this contraction which is also a derivation. The following result tells us that the set of semi-full algebra contractions is closed by homological perturbation. This theorem plays a key role in the proof of the main theorem of this paper. Theorem 7 (SF-APL) ([17]) Taking as data a semi-full algebra contraction rand an algebra perturbation datum δof r, the perturbed contraction rδis an algebra contraction of the same type, where the product on A0δis the original product µA0. Using the theorem above and Theorem 3, it is possible to state the following result. Theorem 8 [1] Let Fand Bbe simplicial groups and let us suppose that the PTCP F×τB, endowed with the natural inner product of the cartesian product F×B, is a simplicial group. Then the contraction (2) is a semi-full algebra contraction. Now, we enunciate several results which allow us to determine the p-local homology of a PTCP of Eilenberg-Mac Lane spaces. First of all, we give various theorems from [17] and [18]. Theorem 9 [18] Let Gbe a commutative simplicial group. There exists a semi-full algebra contraction cwb from C(W(G)) to the bar construction B(C(G)). The inclusion of this contraction is a morphism of DGA-coalgebras (see [14]). Theorem 10 [17] Let cbe an algebra contraction from a commutative DGA-algebra Ato other commutative DGA-algebra A0, in which the inclusion is a DGA-algebra morphism. Then there exists a semi-full algebra contraction ¯ B(c)from ¯ B(A)to ¯ B(A0). 7 Furthermore, in [10] is proved that ¯ B(c) is a full coalgebra contraction, (where A and A0can be non commutative DGA-algebras). Analogous results can be established on the cobar construction. We are interested in the following result: Theorem 11 [12] Let cbe an algebra contraction from a DGA-coalgebra Cto other DGA-coalgebra C0,where the inclusion gis a DGA-coalgebra morphism. Then there exists a full algebra contraction ¯ Ω(c)from ¯ Ω(C)to ¯ Ω(C0). If we note ¯ Ω(g) the inclusion of the contraction ¯ Ω(c), the following identity holds: ¯ Ω(g)|C0=g. (3) The following “endogamic” process will allow us to get a complete determination of the p-local homology of an Eilenberg-Mac Lane space. Theorem 12 [17] Let us consider Λ = Z(p). Let Abe a Cartan’s elementary complex (see [4]). Then there exists a semifull algebra contraction from ¯ B(A)to a tensor product of Cartan’s elementary complexes. The starting points in order to construct an explicit contraction from C(K(π, n)), being πan finitely generated abelian group, are: Theorem 13 [6] •There is an almost full algebra contraction from B(Λ[Z]) to the exterior algebra E(u, 1). •There is an almost full algebra contraction from ¯ B(Λ[Zpr]) to the twisted tensor product E(u, 1) ⊗ρΓ(v, 2), where ρis a derivation defined by ρ(v) = pr·uand ρ(u) = 0. Now, combining these results, it is possible to set up the following chain of DGA- algebra contractions: C(K(π, 1)) ⇒C(¯ W(K(π, 0))) ⇒¯ WN(C(K(π, 0))) ⇒¯ B(Λ[π]) ⇒G(π, 1) and hence, C(K(π, n + 1)) ⇒C(¯ W(¯ Wn(K(π, 0)))) ⇒¯ WN(¯ Wn N(C(K(π, 0)))) ⇒¯ B(¯ Wn N(C(K(π, 0))) ⇒¯ B(¯ Bn(Λ[π])) ⇒¯ B(G(π, n)) ⇒G(π, n + 1) Thanks to them, the following theorem extracted from [16] can be proved without difficulty: 8 Theorem 14 Let Λ = Z(p)be the ground ring. Let πand nbe a finitely generated abelian group and a non-negative integer, respectively. There is a semi-full algebra contraction cπ,n from C(K(π, n)) to a tensor product G(π, n). The latter is composed by Cartan’s elementary complexes corresponding to admissible sequences. Obviously, the p-local homology of K(π, n) is the the p-local homology of G(π, n) and this last homology can be computed easily. Note 2 G(π, n)is a free DGA-algebra of finite type. Let us take remark that we use in the proof of the theorem above that there is a semifull algebra contraction from ¯ B(G(π, n)) to G(π, n + 1). It is not hard to deduce that in the case n= 1, we have Theorem 15 [17] Let Λbe a commutative ring with 16= 0 and πa finitely generated abelian group. There is an almost-full algebra contraction cπ,1from C(K(π, 1)) to the tensor product G(π, 1). The projection of the last contraction is a DGA-algebra morphism because of the facts that there is an algebra isomorphism between C(W(K(π0,0))) and the bar construction B(C(K(π0,0))) and Th. 13. Using the previous results, we prove the following Theorem which plays an essential role in the proof of Theorem 19 Theorem 16 Under the hypothesis of Theorem 14, there exists a full algebra contraction cn,n−1from ¯ Ω(K(π, n)) to ¯ Ω¯ B(G(π, n −1)). Proof First, we have the following isomorphism of DGA-coalgebras: C(K(π, n)⇒C(¯ W(K(π, n −1)) (4) By Theorem 9, we establish the following contraction, where the inclusion is a morphism of DGA-coalgebras: C(¯ W(K(π, n −1))) ⇒¯ BC(K(π, n −1)).(5) By theorem 14, there is a semi-full algebra contraction: C(K(π, n −1)) ⇒G(π, n −1) . 9