On LS-Category of a Family of Rational Elliptic Spaces II

Transcription

1 E extracta mathematicae Vol. 31, Núm. 2, On LS-Category of a Family of Rational Elliptic Spaces II Khalid Boutahir, Youssef Rami Département de Mathématiques et Informatique, Faculté des Sciences, Université My Ismail, B. P Zitoune, Meknès, Morocco khalid.boutahir@edu.umi.ac.ma y.rami@fs-umi.ac.ma Presented by Antonio M. Cegarra Received June 24, 2016 Abstract: Let X be a finite type simply connected rationally elliptic CW-complex with Sullivan minimal model ΛV, d and let k 2 the biggest integer such that d = i k di with d iv Λ i V. If ΛV, d k is moreover elliptic then catλv, d = catλv, d k = dimv even k 2 + dimv odd. Our work aims to give an almost explicit formula of LScategory of such spaces in the case when k 3 and when ΛV, d k is not necessarily elliptic. Key words: Elliptic spaces, Lusternik-Schnirelman category, Toomer invariant. AMS Subject Class. 2010: 55P62, 55M Introduction The Lusternik-Schirelmann category c.f. [7], catx, of a topological space X is the least integer n such that X can be covered by n + 1 open subsets of X, each contractible in X or infinity if no such n exists. It is an homotopy invariant c.f. [3]. For X a simply connected CW complex, the rational L-S category, cat 0 X, introduced by Félix and Halperin in [2] is given by cat 0 X = catx Q catx. In this paper, we assume that X is a simply connected topological space whose rational homology is finite dimensional in each degree. Such space has a Sullivan minimal model ΛV, d, i.e. a commutative differential graded algebra coding both its rational homology and homotopy cf. 2. By [1, Definition 5.22] the rational Toomer invariant of X, or equivalently of its Sullivan minimal model, denoted by e 0 ΛV, d, is the largest integer s for which there is a non trivial cohomology class in H ΛV, d represented by a cocycle in Λ s V, this coincides in fact with the Toomer invariant of the fundamental class of ΛV, d. As usual, Λ s V denotes the elements in ΛV of wordlength s. For more details [1], [3] and [14] are standard references. In [4] Y. Felix, S. Halperin and J. M. Lemaire showed that for Poincaré duality spaces, the rational L-S category coincides with the rational Toomer 235

2 236 k. boutahir, y. rami invariant e 0 X, and in [9] A. Murillo gave an expression of the fundamental class of ΛV, d in the case where ΛV, d is a pure model cf. 2. Let then ΛV, d be a Sullivan minimal model. The differential d is decomposable, that is, d = i k d i, with d i V Λ i V and k 2. Recall first that in [8] the authors gave the explicit formula catλv, d = dim V odd + k 2 dim V even in the case when ΛV, d k is also elliptic. The aim of this paper is to consider another class of elliptic spaces whose Sullivan minimal model ΛV, d is such that ΛV, d k is not necessarily elliptic. To do this we filter this model by F p = Λ k 1p V = i=k 1p Λ i V. 1 This gives us the main tool in this work, that is the following convergent spectral sequence cf. 3: H p,q ΛV, δ H p+q ΛV, d. 2 Notice first that, if dimv < and ΛV, δ has finite dimensional cohomology, then ΛV, d is elliptic. This gives a new family of rationally elliptic spaces. In the first step, we shall treat the case under the hypothesis assuming that H N ΛV, δ is one dimensional, being N the formal dimension of ΛV, d cf. [5]. For this, we will combine the method used in [8] and a spectral sequence argument using 2. We then focus on the case where dim H N ΛV, δ 2. Our first result reads: Theorem 1. If ΛV, d is elliptic, ΛV, d k is not elliptic and H N ΛV, δ = Q.α is one dimensional, then cat 0 X = catλv, d = sup{s 0, α = [ω 0 ] with ω 0 Λ s V }. Let us explain in what follow, the algorithm that gives the first inequality, catλv, d sup{s 0, α = [ω 0 ] with ω 0 Λ s V } := r. i Initially we fix a representative ω 0 Λ r V of the fundamental class α with r being the largest s such that ω 0 Λ s V.

3 on ls-category of a family of rational elliptic spaces ii 237 ii A straightforward calculation gives successively: with ω 0 = ω ω ω l 0 ω i 0 = ω i,0 0, ωi,1 0,..., ωi,k 2 0 Λ k 1p+i V Λ k 1p+i+1 V Λ k 1p+i+k 2 V. Using δω 0 = 0 we obtain dω 0 = a a a0 t+l with a 0 i = a 0,0 i, a 0,1 i,..., a 0,k 2 i Λ k 1p+i V Λ k 1p+i+1 V iii We take t the largest integer satisfying the inequality: t Λ k 1p+i+k 2 V. 1 N 2k 1p + l 2k k 1 Since d 2 = 0, it follows that a 0 2 = δb 2 for some k 2 b 2 Λ k 1p+2 k 1+j V. j=0 iv We continue with ω 1 = ω 0 b 2. v By the imposition iii, the algorithm leads to a representative ω t+l 1 Λ r V of the fundamental class of ΛV, d and then e 0 ΛV, d r. Now, dimv < imply dim H N ΛV, δ <. Notice also that the filtration 1 induces on cohomology a graduation such that H N ΛV, δ = p+q=n H p,q ΛV, δ. There is then a basis {α 1,..., α m } of H N ΛV, δ with α j H p j,q j ΛV, δ, 1 j m. Denote by ω 0j Λ r j V a representative of the generating class α j with r j being the largest s j such that ω 0j Λ s j V. Here p j and q j are filtration degrees and r j {p j k 1,..., p j k 1+k 2}. The second step in our program is given as follow: Theorem 2. If ΛV, d is elliptic and dim H N ΛV, δ = m with basis {α 1,..., α m }, then, there exists a unique p j, such that cat 0 X = sup{s 0, α j = [ω 0j ] with ω 0j Λ s V } := r j.

4 238 k. boutahir, y. rami Remark 1. The previous theorem gives us also an algorithm to determine LS-category of any elliptic Sullivan minimal model ΛV, d. Knowing the largest integer k 2 such that d = i k d i with d i V Λ i V and the formal dimension N this one is given in terms of degrees of any basis elements of V, one has to check for a basis {α 1,..., α m } of H N ΛV, δ which is finite dimensional since dimv <. The NP-hard character of the problem into question, as it is proven by L. Lechuga and A. Murillo cf [12], sits in the determination of the unique j {1,..., m} for which a represent cocycle ω 0j of α j survives to reach the E term in the spectral sequence Basic facts We recall here some basic facts and notation we shall need. A simply connected space X is called rationally elliptic if dim H X, Q < and dimx Q <. A commutative graded algebra H is said to have formal dimension N if H p = 0 for all p > N, and H N 0. An element 0 ω H N is called a fundamental class. A Sullivan algebra [3] is a free commutative differential graded algebra cdga for short ΛV, d where ΛV = ExteriorV odd SymmetricV even generated by the graded K-vector space V = i= i=0 V i which has a well ordered basis {x α } such that dx α ΛV <α. Such algebra is said minimal if degx α < degx β implies α < β. If V 0 = V 1 = 0 this is equivalent to saying that dv i= i=2 Λi V. A Sullivan model [3] for a commutative differential graded algebra A,d is a quasi-isomorphism morphism inducing isomorphism in cohomology ΛV, d A, d with source, a Sullivan algebra. If H 0 A = K, H 1 A = 0 and dimh i A, d < for all i 0, then, [6, Th.7.1], this minimal model exists. If X is a topological space any minimal model of the polynomial differential forms on X, A P L X, is said a Sullivan minimal model of X. ΛV, d or X is said elliptic, if both V and H ΛV, d are finite dimensional graded vector spaces see for example [3]. A Sullivan minimal model ΛV, d is said to be pure if dv even = 0 and dv odd ΛV even. For such one, A. Murillo [9] gave an expression of a cocycle representing the fundamental class of HΛV, d in the case where ΛV, d is elliptic. We recall this expression here: Assume dim V <, choose homogeneous basis {x 1,..., x n }, {y 1,..., y m }

5 on ls-category of a family of rational elliptic spaces ii 239 of V even and V odd respectively, and write dy j = a 1 jx 1 + a 2 jx a n 1 j x n 1 + a n j x n, j = 1, 2,..., m, where each a i j is a polynomial in the variables x i, x i+1,..., x n, and consider the matrix, a 1 1 a2 1 a n 1 a 1 2 A = a2 2 a n 2. a 1 m a 2 m a n m For any 1 j 1 < < j n m, denote by P j1...j n the determinant of the matrix of order n formed by the columns i 1, i 2,..., i n of A: a 1 j 1 a n j 1. a 1 j n Then see [9] if dim H ΛV, d <, the element ω ΛV, ω = 1 j 1 < <j n m a n j n 1 j 1+ +j n P j1...j n y 1... ŷ j1... ŷ jn... y m, 3 is a cocycle representing the fundamental class of the cohomology algebra. 3. Our spectral sequence Let ΛV, d be a Sullivan minimal model, where d = i k d i with d i V Λ i V and k 2. We first recall the filtration given in the introduction: F p = Λ k 1p V = i=k 1p Λ i V. 4 F p is preserved by the differential d and satisfies F p ΛV F q ΛV F p+q ΛV, p, q 0, so it is a filtration of differential graded algebras. Also, since

6 240 k. boutahir, y. rami F 0 = ΛV and F p+1 F p this filtration is decreasing and bounded, so it induces a convergent spectral sequence. Its 0 th -term is E p,q F p p+q Λ k 1p V p+q 0 = F p+1 = Λ k 1p+1. V Hence, we have the identification: E p,q 0 = Λ pk 1 V Λ pk 1+1 V Λ pk 1+k 2 V p+q, 5 with the product given by: u 0, u 1,..., u k 2 u 0, u 1,..., u k 2 = v 0, v 1,..., v k 2 for all u 0, u 1,..., u k 2, u 0, u 1,..., u k 2 Ep,q 0 with v m = i+j=m u iu j and m = 0,..., k 2. The differential on E 0 is zero, hence E p,q 1 = E p,q 0 and so the identification above gives the following diagram: E p,q 1 = Λ k 1p V Λ k 1p+1 V Λ k 1p+k 2 V p+q δ E p+1,q 1 = d d k d k+1 d 2k 1 1 d k d 2k 1 k δ Λ k 1p+1 V Λ k 1p+1+1 V Λ k 1p+1+k 2 V p+q+1 with δ defined as follows, δu 0, u 1,..., u k 2 = w k, w k+1,..., w 2k 2 with w k+j = i+i =j i =0,...,k 2 d k+i u i. Let E p 1 = Ep, 1 = q 0 Ep,q 1 and E 1 = p 0 Ep, 1 = ΛV as a graded vector space. In this general situation, the 1 st -term is the graded algebra ΛV provided with a differential δ, which is not necessarily a derivation on the set V of generators. That is, ΛV, δ is a commutative differential graded algebra, but it is not a Sullivan algebra. This gives, consequently, our spectral sequence: E p,q 2 = H p,q ΛV, δ H p+q ΛV, d. 6 Once more, using this spectral sequence, the algorithm completed by proves of claims that will appear, will give the appropriate generating class of H N ΛV, δ that survives to the term. Accordingly, the explicit formula of LS category for this general case, is expressed in terms of the greater filtering degree of a represent of this class.

7 on ls-category of a family of rational elliptic spaces ii Proof of the main results 4.1. Proof of Theorem 1. Recall that ΛV, d is assumed elliptic, so that, catλv, d = e 0 ΛV, d [4]. Notice also that the subsequent notations imposed us sometimes to replace a sum by some tuple and vice-versa The first inequality. In what follows, we put: r = sup{s 0, α = [ω 0 ] with ω 0 Λ s V }. Denote by p the least integer such that pk 1 r < p + 1k 1 and let then ω 0 Λ r V. We have ω 0 Λ k 1p V Λ k 1p+k 2 V Λ k 1p+k 1 V Λ k 1p+2k 3 V Since ω 0 = N and dim V <, there is an integer l such that with ω0 0 0 and i = 0,..., l, ω 0 = ω ω ω l 0 ω i 0 = ω i,0 0, ωi,1 0,..., ωi,k 2 0 Λ k 1p+i V Λ k 1p+i+k 2 V. We have successively: δω0 i = δ ω i,0 0, ωi,1 0,..., ωi,k 2 0 = d k ω i,0 0, 0, i +i =2 0,..., 0, δω 0 = = l i=0 δ ω i,0 0, ωi,1 0,..., ωi,k 2 0 l d k ω i,0 0, i=0 0 0, i +i =2 0,...,

8 242 k. boutahir, y. rami Also, we have dω 0 = dω0 0 + dω dωl 0, with: dω0 0 = d ω 0,0 0, ω0,1 0,..., ω0,k 2 0 = d k ω 0,0 0, d k+i ω 0,i 0,..., d k+i ω 0,i 0 + 2k 3 dω0 1 = d = k =k 1 Λ k 1p+k V ω 1,0 0, ω1,1 0,..., ω1,k 2 0 d k ω 1,0 0, 3k 4 dω0 i = d = k =2k 2 Λ k 1p+k V d k+i ω 1,i 0,..., ω i,0 0, ωi,1 0,..., ωi,k 2 0 d k ω i,0 0, k 1p+i+2k i+3 k =k 1p+i+1k i+1 Therefore l dω 0 = d k ω i,0 0, + i=0 l i=0 0,..., 0,..., Λ k 1p+k V d k+i ω 1,i d 2k 2 ω i,1 0 + d 2k 2 + d 2k 3 ω i, d 2k 2 + d 2k d k+1 ω i,k d k ω 0 k >2k 2

9 on ls-category of a family of rational elliptic spaces ii 243 that is: dω 0 = δω 0 + l i=0 As δω 0 = 0, we can rewrite: d 2k 2 ω i,1 0 + d 2k 2 + d 2k 3 ω i, d 2k d k+1 ω i,k d k ω 0. k >2k 2 dω 0 = a a a 0 t+l with a 0 i = a 0,0 i k 2,..., a 0,k 2 Λ k 1p+i+j V. Note also that t is a fixed integer. Indeed, the degree of a 0 t+l is greater than or equal to 2 k 1p + t + l + k 2 and it coincides with N + 1, N being the formal dimension of ΛV, d. Then N k 1p + t + l + k 2. Hence t i j=0 1 N 2k 1p + l + 5 2k. 2k 1 In what follows, we take t the largest integer satisfying this inequality. Now, we have: with d 2 ω 0 = da da da 0 t+l = da 0,0 2, a0,1 2,..., a0,k da 0,0 3, a0,1 3,..., a0,k da 0,0 t+l, a0,1 t+l,..., a0,k 2 t+l, da 0,0 2, a0,1 2,..., a0,k 2 2 = d k a 0,0 2, a0,1 2,..., a0,k d k+1 a 0,0 2, a0,1 2,..., a0,k = d k a 0,0 2, d k+i a 0,i 2,..., + d 2k 1 a 0,0 2 + d 2k 2 a 0,1 2 +,... + d k+i a 0,i 2

10 244 k. boutahir, y. rami da 0,0 3, a0,1 3,..., a0,k 2 3 = d k a 0,0 3, a0,1 3,..., a0,k d k+1 a 0,0 3, a0,1 3,..., a0,k = d k a 0,0 3, d k+i a 0,i 3,..., + d 2k 1 a 0,0 3 + d 2k 2 a 0,1 3 +,... + d k+i a 0,i 3 It follows that: d 2 ω 0 = Since d 2 ω 0 = 0, we have d k a 0,0 2, d k a 0,0 2, d k+i a 0,i 2,..., + d 2k 1 a 0,0 2 + d 2k 2 a 0,1 2 +, d 2k 1 a 0,0 3 + d 2k 2 a 0,1 3 +,... + d k+i a 0,i 2,..., d k+i a 0,i 2 d k+i a 0,i 2 = δa 0 2 = 0 with a 0 2 = a0,0 2,..., a0,k 2 2 k 2 j=0 Λk 1p+2+j V. Consequently a 0 2 is a δ-cocycle. Claim 1. a 0 2 is an δ-coboundary. Proof. Recall first that the general r th -term of the spectral sequence 6 is given by the formula: E p,q r = Zr p,q /Z p+1,q 1 r 1 + B p,q r 1, where and B p,q r Z p,q r = { x [F p ΛV ] p+q dx [F p+r ΛV ] p+q+1} = d [F p r ΛV ] p+q 1 F p ΛV = d Z p r+1,q+r 2 r 1.

11 on ls-category of a family of rational elliptic spaces ii 245 Recall also that the differential d r : Er p,q Er p+r,q r+1 in Er, is induced from the differential d of ΛV, d by the formula d r [v] r = [dv] r, v being any representative in Zr p,q of the class [v] r in Er p,q. We still assume that dim H N ΛV, δ = 1 and adopt notations of Notice then ω 0 Z p,q 2 and it represents a non-zero class [ω 0 ] 2 in E p,q 2. Otherwise ω 0 = ω 0 + dω 0, where ω 0 Zp+1,q 1 1 and ω 0 Bp,q 1, so that α = [ω 0 ] = [ω 0 d δω 0 ]. But ω 0 d δω 0 Λ r+1 is a contradiction to the definition of ω 0. Now, using the isomorphism E, 2 = H, ΛV, δ, we deduce that, [ω 0 ] 2 E p,q 2 being the only generating element must survive to E p,q 3, otherwise, the spectral sequence fails to converge. Whence d 2 [ω 0 ] 2 = [a 0 2 ] 2 = 0 in E p+2,q 1 2, i.e., a 0 2 Zp+3,q B p+2,q 1 1. However a 0 2 k 2 j=0 Λk 1p+2+j V, so a 0 2 Bp+2,q 1 1, that is a 0 2 = dx, x k 2 j=0 Λk 1p+1+j V. By wordlength argument, we have necessary a 0 2 = δx, which finishes the proof of Claim 1. Notice that this is the first obstruction to [ω 0 ] to represent a non zero class in the term E, 3 of 6. The others will appear progressively as one advances in the algorithm. Let then b 2 k 2 j=0 Λk 1p+2 k 1+j V such that a 0 2 = δb 2 and put ω 1 = ω 0 b 2. Reconsider the previous calculation for it: with dω 1 = dω 0 db 2 = a a a 0 t+l d kb 2 + d 4 b 2 +, k 2 d k b 2 = d k b 0 2, b 1 2,..., b k 2 2 = d k b 0 2, d k b 1 2,..., d k b k 2 2 Λ k 1p+2+j V, j=0 d k+1 b 2 = d k+1 b 0 2, b 1 2,..., b k 2 2 k 2 = d k+1 b 0 2, d k+1 b 1 2,..., d k+1 b2 k 2 Λ k 1p+2+j+1 V, j=0

12 246 k. boutahir, y. rami This implies that dω 1 = a a a 0 t+l d k b 0 2, and then: So, d 2k 1 b 0 2 +,... d k+i b i 2,..., = a 0 2 δb 2 + a 0 3 d 2k 1 b 0 2 +,... + = a 0 3 d 2k 1 b 0 2 +,... +, dω 1 = a a a 1 t+l, with k 2 a1 i Λ k 1p+i+j V. j=0 d k+i b i 2 d 2 ω 1 = da da da 1 t+l = d k a 1,0 3, d k+i a 1,i 3,..., + d 2k 1 a 1,0 3 +,... + Since d 2 ω 1 = 0, by wordlength reasons, d k a 1,0 3, d k+i a 1,i 3,..., d k+i a 1,i 3 d k+i a 1,i 3 = δa 1 3 = 0. We claim that a 1 3 = δb 3 and consider ω 2 = ω 1 b 3. We continue this process defining inductively ω j = ω j 1 b j+1, j t+l 2 such that: k 2 dω j = a j j+2 + aj j aj t+l, with aj i Λ k 1p+i+h V and a j j+2 a δ-cocycle. h=0

13 on ls-category of a family of rational elliptic spaces ii 247 Claim 2. a j j+2 is a δ-coboundary, i.e., there is k 2 b j+2 Λ k 1p+j+2 k 1+j V j=0 such that δb j+2 = a j j+2 ; 1 j t + l 2. Proof. We proceed in the same manner as for the first claim. Indeed, we have clearly for any 1 j t+l 2, ω j = ω j 1 b j+1 = ω 0 b 2 b 3 b j+1 Z p,q j+2 and it represents a non zero class [ω j] j+2 in E p,q j+2 which is also one dimensional. Whence as in Claim 1, we conclude that, a j j+2 is a δ-coboundary for all 1 j t + l 2. Consider ω t+l 1 = ω t+l 2 b t+l, where δb t+l = a t+l 2 t+l. Notice that dω t+l 1 = dω t+l 2 = N +1, but by the hypothesis on t, we have dω t+l 2 = and then a t+l 2 t+l dω t+l 2 b t+l = a t+l 2 t+l δb t+l d δb t+l = d δb t+l > N + 1. It follows that dω t+l 1 = 0, that is ω t+l 1 is a d-cocycle. But it can t be a d-coboundary. Indeed suppose that ω t+l 1 = ω0 0 + ω ωl 0 b 2 + b b t+l, were a d-coboundary, by wordlength reasons, ω0 0 would be a δ-coboundary, i.e., there is x k 2 j=0 Λk 1p k 1+j V such that δx = ω0 0. Then ω 0 = δx + ω ω0. l Since δω 0 = 0, we would have δω ωl 0 = 0 and then [ω 0] = [ω ω0 l ]. But ω ωl 0 Λ>r V, contradicts the property of ω 0. Consequently ω t+l 1 represents the fundamental class of ΛV, d. Finally, since ω t+l 1 Λ r V we have e 0 ΛV, d r For the second inequality. Denote s = e 0 ΛV, d and let ω Λ s V be a cocycle representing the generating class α of H N ΛV, d. Write ω = ω 0 + ω ω t, ω i Λ s+i V. We deduce that: dω = d k ω 0 + d k+i ω i + + d k+i ω i + d k ω k 1 + d 2k 1 ω 0 + i+i =1 = δω 0, ω 1,..., ω k 2 + i+i =k 2

14 248 k. boutahir, y. rami Since dω = 0, by wordlength reasons, it follows that δω 0, ω 1,..., ω k 2 = 0. If ω 0, ω 1,..., ω k 2, were a δ-boundary, i.e., ω 0, ω 1,..., ω k 2 = δx, then ω dx = ω 0, ω 1,..., ω k 2 δx + ω k ω t d δx = ω k ω t d δx, so, ω dx Λ s+k 1 V, which contradicts the fact s = e 0 ΛV, d. Hence ω 0, ω 1,..., ω k 2 represents the generating class of H N ΛV, δ. But ω 0, ω 1,..., ω k 2 Λ s V implies that s r. Consequently, e 0 ΛV, d r. Thus, we conclude that e 0 ΛV, d = r Proof of Theorem 2. It suffices to remark that since ΛV, d is elliptic, it has Poincaré duality property and then dim H N ΛV, d = 1. The convergence of 6 implies that dim E, = 1. Hence there is a unique p, q such that p+q = N and E, = E p,q. Consequently only one of the generating classes α 1,..., α m had to survive to E. Let α j this representative class and p j, q j its pair of degrees. Example 1. Let d = i 3 d i and ΛV, d be the model defined by V even = < x 2, x, 2 >, V odd =< y 5, y 7, y, 7 >, dx 2 = dx, 2 = 0, dy 5 = x 3 2, dy 7 = x,4 2 and dy, 7 = x2 2 x,2 2, in which subscripts denote degrees. For k 3, l 0, we have For k 4, l 0, Clearly we have x k 2x,l 2 = xk 3 2 x 3 2x,l 2 = d y 5 x k 3 2 x,l 2. x,k 2 xl 2 = x l 2x,k 4 2 x,4 2 = d x l 2x,k 4 2 y 7. dim HΛV, d < and dim HΛV, d 3 =. Using A. Murillo s algorithm cf. 2 the matrix determining the fundamental class is: x A = 0 x,3 2, x 2 x,2 2 0

15 on ls-category of a family of rational elliptic spaces ii 249 so, ω = x 2 2 x,3 2 y, 7 + x 2x,5 2 y 5 Λ 6 V is a generator of this fundamental cohomology class. It follows that e 0 ΛV, d = 6 m + nk 2. Example 2. Let d = i 3 d i and ΛV, d be the model defined by V even = < x 2, x, 2 >, V odd =< y 5, y 9, y, 9 >, dx 2 = dx, 2 = 0, dy 5 = x 3 2, dy 9 = x,5 2 and dy, 9 = x3 2 x,2 2. Clearly we have dim HΛV, d < and dim HΛV, d 3 =. Using A. Murillo s algorithm cf. 2 the matrix determining the fundamental class is: x A = 0 x,4, 2 x 2 2 x,2 2 0 so, ω = x 2 2 x,4 2 y, 9 + x2 2 x,6 2 y 5 Λ 7 V is a generator of this fundamental cohomology class. It follows that e 0 ΛV, d = 7 m + nk 2. References [1] O. Cornea, G. Lupton, J. Oprea, D. Tanré, Lusternik-Schnirelmann Category, Mathematical Surveys and Monographs 103, American Mathematical Society, Providence, RI, [2] Y. Félix, S. Halperin, Rational LS-category and its applications, Trans. Amer. Math. Soc , [3] Y. Félix, S. Halperin, J.-C. Thomas, Rational Homotopy Theory, Graduate Texts in Mathematics 205, Springer-Verlag, New York, [4] Y. Félix, S. Halperin, J. M. Lemaire, The rational LS-category of products and Poincaré duality complexes, Topology , [5] Y. Félix, S. Halperin, J.-C. Thomas, Gorenstein Spaces, Adv. in Math , [6] S. Halperin, Universal enveloping algebras and loop space homology, J. Pure Appl. Algebra , [7] I. M. James, Lusternik-Schnirelmann category, in Handbook of Algebraic Topology, North-Holland, Amsterdam, 1995, [8] L. Lechuga, A. Murillo, A formula for the rational LS-category of certain spaces, Ann. Inst. Fourier Grenoble ,

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