A[símbolo de infinito]–coalgebra structures on the Zp-homology of Eilenberg-Mac Lane spaces

A∞–coalgebra structures on the Zp-homology of Eilenberg-Mac Lane spaces Ainhoa Berciano and Pedro Real May 2004 Abstract We study here the A(∞)-coalgebra structure of the homology H∗(K(π, n); Zp) of an Eilenberg-Mac Lane space K(π, n), where πis a finitely generated abelian group and nis a positive integer. Using diverse techniques of homological perturbation, we get that the components ∆i(p−2)+2 of degree i(p−2) (with i≥0) are the only (possibly) non-null morphisms of said structure. 1 Extended Abstract An Eilenberg-Mac Lane space K(π, n), depending on a group πand a non-negative integer nis a simplicial set having only two non-null integer homotopy groups π0(K(π, n)) = Zand πn(K(π, n)) = π(except the case n= 0, which only have one non-null homotopy group in degree 0 π0(K(π, 0)) = π). We only consider here K(π, n) spaces being πan finitely generated abelian group. In this case, the chain complex C∗(K(π, n)) canonically associated to an Eilenberg-Mac Lane space have a Hopf-algebra structure, that is, C∗(K(π, n)) is endowed with an algebra and a coalgebra structures that are compatible in some strong sense. Henri Cartan determines in his complete study [1] of the homology of Eilenberg-Mac Lane spaces four types of commutative differential graded augmented algebras, called elementary complexes, which can be made by using exterior and divided power algebras. The first is the exterior algebra E(u, 2n−1) with one generator uof degree 2n−1; the second is the divided power algebra Γ(v, 2n) with one “generator” vof degree 2n; the third is the twisted tensor product Γ(u, 2n)˜ ⊗ρ±prE(v, 2n+ 1) with differential operator defined by ρ±pr(v) = ±pru; and the fourth is the twisted tensor product E(u, 2n−1) ˜ ⊗ρ±prΓ(v, 2n) with differential operator defined by ρ±pr(v) = ±pru. Tensor products of these elementary complexes describe the integer homology algebra H∗(K(π, n)) for any πand n≥1. Alain Proute in his brilliant Ph. D. Thesis “Alg´ebres differentielles fortement homotopiquement associatives” [8] studies the Zp-homology algebra (the field Zpbeing the ground ring) of K(π, n) and starts the work of determining its A(∞)-coalgebra structure. In the case H∗(K(π, n); Z2), this comultiplicative structure is reduced to a simple coalgebra. Consequently, this homology has a Hopf-algebra structure. If p6= 2 (pprime), the situation changes dramatically. To the difficult problems to control as the A(∞)-structures by tensor product evolve, the fact 1 that the A(∞)-coalgebra structure explodes in the case p6= 2 joins up. Prout´e only gives partial results for elementary algebras embodied in this context, that they permit however to conjecture aspects of nature of this complicated structure. In this article, we intend to extend Prout´e’s work and to determine the total group of component morphisms possibly non zero of the A(∞)-coalgebra structure of H∗(K(π, n); Zp). Only tensor products of two elementary complexes appear in H∗(K(π, n); Zp). They are the exterior and divided power algebras. The Real’s work [10] is our starting point, where, using homological perturbation, specify a contraction c(π,n);p(particular homotopy equivalence) among C∗(K(π, n); Zp) and H∗(K(π, n); Zp). This strong compatibility is possible basically thanks to the following elementary contractions: •A contraction CW B “connecting” the normalized chain complex C∗(W(G)) of the geometric classifying space of an abelian group G, and the reduced bar construction B(C∗(G)) of the normalized chain complex of G[9]. •An isomorphism from the reduced bar construction of an exterior algebra to a divided power algebra. •A contraction from the reduced bar construction B(Γ(u, 2n)) to an infinite tensor product of algebras of the kind E(v, 2npi+ 1) ⊗Γ(w, 2np(i+1) + 2). Attending to the terminology delivered in [10], all these three contractions are semi-full algebra contractions and, in particular, that means that the algebra structure is transferred correctly and of simple manner to the homology H∗(K(π, n); Zp). Regarding coalgebra’s structures, two first contractions respect to this structure, being the last the one and only contraction that does not show a property of compatibility. The explosion of coalgebra’s structure in the homology is precisely provoked by this last contraction. As said contraction is generated starting from elementary contractions cBQ among B(Qp(u, 2n)) (being Qp(u, 2n) = P(u, 2n)/(up)) and the tensor product E(v, 2n+ 1) ⊗Γ(w, 2np + 2) , a first step to examine the strongly homotopy associativity of the homology H∗(K(π, n); Zp) is to deal first with this question in elementary contractions. Prout´e’s study was limited, for the previously reasons expounded, to the determination of the A(∞)-coalgebra’s structure of the 1-homology of Qp(u, 2n). Such structure (∆2,∆3,∆4, ...) only shows two non-null morphisms: ∆2and ∆p. Our strategy is to discuss this work from the perspective of Homological Perturbation Theory and to try to extend this technique to more complex algebras. First, given a contraction c={N, M, f, g, φ}among a differential graded coalgebra (N, dN,∆) and a chain complex (M, dM), the component morphisms of the A(∞)-coalgebra structure on Mfollow (up to signs) the following formulae (from now on, we use Real’s notation [10]): ∆i=f⊗i(∆[(i−1)]φ[c,i−1] · · · (∆[2]φ[c,2])∆g, ∀i≥1.(1) The development of this formula in terms of the coproduct ∆ : N→N⊗Nand the homotopy operator φ:N∗→N∗+1, gives us a sum of compositions 2 f⊗i(∆[i−1,ji−1]φ[c,i−1,ki−1])· · · (∆[2,j2]φ[c,2,k2])∆g, 1≤jr, kr≤r. (2) An elementary work is to prove ( we would be able to do it using banally Inversion’s Theory [10, 2]), the following rule: “The morphism (2) will be zero if jr6=kr, for some r.” Now, let us focus in the contraction cBQ ={B(Qp(u, 2n)), E(v, 2n+ 1) ⊗Γ(w, 2np + 2), fBQ, gBQ, φBQ}. We denote an element of B(Q(p)(u, 2n)) of the form [ur1|. . . |urm] by [r1|. . . |rm], where 0≤ri< p. From now on, E(v, 2n+ 1) ⊗Γ(w, 2np + 2) will be denoted by Hfor short. The explicit morphisms of cBQ are the following: fBQ[r1|t1|. . . |rm|tm] = {Qn k=1 δp,rk+tk}γm(w), fBQ[r1|t1|. . . |rm|tm|l] = δ1,l{Qn k=1 δp,rk+tk}v·γm(w), where the symbols δi,j are defined by: δi,j =0i6=j 1i=j The morphism gBQ :H→¯ B(Q(p)(u, 2n)) is defined over the generators as follows: gBQ(v) = [1], gBQ(γk(w)) = [1|p−1|k times . . . |1|p−1]. The homotopy operator φBQ is defined by: φBQ1 = 0; φBQ[1] = 0; φBQ[x] = −[1|x−1] 1 < x < p; φBQ[x|y] = −[1|x−1|y]; φBQ[x|y|z] = −[1|x−1|y|z]−δp,x+y[1|p−1|φ(z)] where z∈¯ B(Q(p)(u, 2n)). The coassociative coproduct on B(Q(p)(u, 2n)) is ∆([a1| · · · |ar]) = r X i=0 [a1| · · · |ai]⊗[ai+1| · · · |ar]. The following properties can be easily verified: If ∆(x) = P∆1(x)⊗∆2(x), being xan element of B(Q(p)(u, 2n)), φBQ(∆1)ig= 0, i ≥1 3 φBQ∆1φBQ = 0, i ≥1. Let us consider the submodule Sof B(Q(p)(u, 2n)) generated by the elements [a1|a2| · · · |ar] such that or else a2i+1 = 1 for all ior else a2i= 1 for all i. Again, it is easy to prove the following properties: φBQ(S)⊂S, ∆(S)⊂S⊗S, Im gBQ ⊂S. Now, we will use Inversion Theory to get an economical formulation ( in terms of number of summands embodied) of the morphisms ∆i:H→H⊗H.We will say that an element [a1|a2| · · · |ar]∈B(Q(p)(u, 2n)) has tinversions if there exists tdifferent couples (a2i−1, a2i) of the form (1, a), where a∈ {1,2, . . . , p −3, p −2}. With this definition in hand, we can verify the following properties: φBQ(element with i-inversions) = Xelements with at least (i+1)-inversions, ∆(element with i-inversions) = Xelements with at least (i−1)-inversions, fBQ(element with i-inversions) = 0,∀i≥1. Therefore, in the economical formula of ∆i(see (1)) derived from this technique, the coproduct ∆ of B(Q(p)(u, 2n)) will be only applied to elements with 1-inversions and the homotopy operator φBQ will be applied to elements with 0-inversions only. Using this argument, it is possible to deduce that only the morphisms ∆2and ∆pare non-zero in the A(∞)-coalgebra structure over H. This work can be extended without too many problems to reduced bar constructions of tensor products of Q(p)(u, 2m). In this way, we show the main result of the paper: Theorem. Let πbe a finitely generated abelian group, n≥1be a positive integer and p6= 2 be a prime. The (possibly) non-null morphisms of the A(∞)-coalgebra structure of H∗(K(π, n); Zp)are ∆i(p−2)+2,∀i≥0. References [1] Cartan H.: Alg`ebres d’ Eilenberg-MacLane. S´eminaire H. Cartan, 1954/55, (expos´e 2 al 11), 1956. [2] CHATA group. Computing “small” 1–homological models for CDGAs. Preprint Depto. Mat. Apl. I, Univ. de Sevilla. 4 [3] V. K. A. M. Gugenheim. On the chain complex of a fibration. Illinois J. Math. 3, pp. 398-414, 1972. [4] V. K. A. M. Gugenheim and L. Lambe. Perturbation theory in Differential Homological Algebra, I. Illinois J. Math 33, pp. 56-82, 1989. [5] V. K. A. M. Gugenheim, L. Lambe and J. Stasheff. Perturbation theory in Differential Homological Algebra, II. Illinois J. Math. 35 n. 3, pp. 357-373, 1991. [6] V. K. A. M. Gugenheim and J. Stasheff. On Perturbations and A∞–structures. Bull. Soc. Math. Belg., vol. 38, pp. 237-246, 1986. [7] J. Huebschmann and T. Kadeishvili. Small models for chain algebras. Math. Zeit. v. 207, pp. 245-280, 1991. [8] A. Prout´e. Alg`ebres diff´erentielles fortement homotopiquement associatives. Ph. D. thesis, Universit´e Paris VII, 1984. [9] Real P.: Algoritmos de c´alculo de homolog´ıa efectiva de los espacios clasificantes, Ph. D. thesis, Seville University, 1993. [10] P. Real. Homological Perturbation Theory and associativity, Homology, Homotopy and Applications, v. 2, pp. 51-88, 2000. Ainhoa Berciano Alcaraz Departamento de Matematica Aplicada, Estadistica e Investigacion Operativa. Facultad de Ciencia y Tecnologia. Universidad del Pais Vasco-Euskal Herriko Unibertsitatea. Barrio Sarriena s/n. 48940, Leioa (Spain) Tel´efono: 34.946.013.368; e-mail: mepb[email protected]u.es Pedro Real Jurado Dpto. Matem´atica Aplicada I Escuela T´ecnica Superior de Ingenier´ıa Inform´atica. Universidad de Sevilla Avda. Reina Mercedes, s/n, 41012, Sevilla (Spain) Tel´efono: 34.954.556.921; e-mail: [email protected] P´agina de red: http://www.us.es/gtocoma 5