A new lower bound for LS-category

Let X be a simply connected CW-complex of ﬁnite type and K an arbitrary ﬁeld. In this paper, we use the Eilenberg–Moore spectral sequence of C ∗ ((cid:2)( X ), K ) to introduce a new homotopical invariant r ( X , K ) . If X is a Gorenstein space with nonzero evaluation map, then r ( X , K ) turns out to interpolate depth ( H ∗ ((cid:2)( X ), K )) and e K ( X ) . We also deﬁne for any minimal Sullivan algebra ((cid:3) V , d ) a new spectral sequence and make use of it to associate to any 1-connected commutative differential graded algebra ( A , d ) a similar invariant r ( A , d ) . When ((cid:3) V , d ) is a minimal Sullivan model of X , this invariant fulﬁlls the relation r ( X , K ) = r ((cid:3) V , d ) .


Introduction
Lusternik-Schnirelman category (LS-category for short), originally introduced in [19], is an integer that gives a numerical measure of possible dynamics on a smooth closed manifold. In fact, it provides a lower bound on the number of critical points admitted by any smooth function on any closed manifold. If X is a topological space, cat(X ) is the least integer n such that X is covered by n + 1 open subsets U i , 0 ≤ i ≤ n, each of which is contractible in X . It is an invariant of homotopy type of the space. Though its definition seems easy, its computation turns out to be a tough task. In [23], Toomer has introduced e K (X ) (K being an arbitrary field) which proves to be a lower bound of cat(X ). Subsequently, Félix et al. proved the famous depth theorem which states that depth(H * ( (X ), K)) ≤ cat(X ) [6]. Recall that when X is a simply connected CW-complex such Y. Rami (B) Département de Mathématiques et Informatique, Université My Ismail, B. P. 11 201, Zitoune, Meknes, Morocco E-mail: yousfoumadan@gmail.com; y.rami@fs-umi.ac.ma that each H i (X, K) is finite dimensional, then H * ( (X ), K) is a positively graded connected finite type and cocommutative Hopf algebra. It is worthwhile reminding that the enveloping algebra U L of any homotopy Lie algebra L associated to a finite type minimal Sullivan algebra ( V, d) is a further example of a positively graded connected finite type and cocommutative Hopf algebra. If depth(U L) < ∞, then U L is left Noetherian if and only if L is finite dimensional [8,Example 1.3 and Theorem C]. In this context, Bisiaux improved the depth theorem by proving that depth(H * ( (X ), K)) ≤ e K (X ) if in addition ev C * (X,K) = 0 [2]. Recall from [6] that for any graded K-algebra G, depth(G) (possibly ∞) is the largest integer n such that Ext i, * G (K, G) = 0, ∀i < n. Equivalently, depth(G) = inf{ p, Ext p, * G (K, G) = 0} with the convention that G has infinite depth when Ext * , * G (K, G) ≡ 0. Here Ext p, * stands for the ( p, * ) component of the graded functor Ext (see §20 in [9] for instance).
The object of this paper is twofold. First, inspired by the definition of Toomer's invariant in terms of the Milnor-Moore spectral sequence, we use the Eilenberg-Moore spectral sequence to define the integer for any simply connected CW-complex X of finite type and any field K. If there is no such integer, we put r(X, K) = ∞. Here E xt is the differential-Ext of Eilenberg-Moore (see Sect. 2 for more details). Our first main theorem proves the following improvement of Bisiaux result: On the other hand, in order to make an algebraic study of r(X, K), our second goal is to associate to any minimal Sullivan algebra ( V, d) the following spectral sequence: which we call the Eilenberg-Moore spectral sequence of ( V, d). Throughout, the differential d k (cf. Sect. 3.1) is the first nonzero homogeneous part of the differential d. Now, given a 1-connected commutative differential graded algebra (A, d) and ( V, d) its minimal Sullivan model, we set Similarly, if there is no such integer, we put r(A, d) = ∞. In this perspective, our second main result reads the following: Theorem 1.2 Let K be a field whose char(K) = 2. If X is a simply connected CW-complex of finite type (or else in the range of Anick) and ( V, d) its minimal Sullivan model, then

Preliminaries
This section provides the tools and notions which are useful in the sequel. All graduations are written either as superscripts (for cohomology) or as subscripts (for homology) with the convention V k = V −K . A commutative differential graded algebra (resp., differential graded algebra, resp., differential graded Lie algebra) will be abbreviated by cdga (resp., dga, resp., dgl). The suspension (resp., desuspension, resp., dual) of a graded K-vector space V is defined by (sV ) p = V p+1 (resp., (s −1 V ) p = V p−1 , resp., V ∨ = Hom K (V, K)).

A minimal Sullivan model
Let K be a field of characteristic p = 2. A Sullivan algebra is a free cdga ( V, d), where V = Exterior(V odd ) ⊗ Symmetric(V even ), generated by the graded K-vector space V = ⊕ i=∞ i=0 V i which has a well ordered basis {x α } α∈I such that dx α ∈ V <α (V <α = span{v γ , γ < α}). Such algebra is said to be minimal if To define such a notion for spaces, we distinguish two cases. First, suppose that K has characteristic zero and let X be a finite type simply connected CW-complex. The minimal Sullivan model ( V, d) of X is by definition that of the cdga A P L (X ) of polynomial differential forms on X with coefficients in K [22]. It is unique (up to quasi-isomorphism) and its generator satisfies the isomorphism V ∼ = Hom Z (π * (X ), Q). Now, assume that char(K) = p > 2 and let X be an r -connected CW-complex with dim(X ) < r p (some r ≥ 1). For such a space, said to be in the range of Anick [1], the chain algebra (C * ( (X ), K) is quasi-isomorphic to the enveloping algebra U L of an appropriate finite type dgl L = L ≥1 . Denote by C * (L) = ( (s L) ∨ , d) the Cartan-Chevalley-Eilenberg complex of L [12]. This is a cdga which is related to C * (X, K) by a sequence of chain homotopies [1]. We still call its minimal Sullivan model, the minimal Sullivan model of X .

Eilenberg-Moore functors
Given (A, d) a connected augmented K-dga on an arbitrary field K and denote by A its underlying graded algebra. Following [7] (see also §6. in [9]), an (A, d)-module (P, d) is called free if it is free as an A -module on a basis of cocycles. It is said of degree zero of (A, d)-modules inducing an isomorphism in cohomology is called a quasi-isomorphism of A-modules. We denote it by (P, d) → (M, d) and call it an (A, d)-semi-free resolution (resp., minimal semi-free resolution) of (M, d) if (P, d) is (A, d)-semi-free (resp., (A, d)-semi-free and minimal). If A 1 = 0 and M = M ≥r (some r ∈ Z) such a resolution exists and may be chosen to be minimal [7,Lemma A.3].
We are now in a position to recall the definition of Eilenberg-Moore functors called also differential graded Tor and Ext, since they are introduced in the context of differential graded homological algebra, where semi-free resolutions replace ordinary free resolutions. Given , is provided with the differential defined by: The Eilenberg-Moore functor E xt is defined as follows: we obtain the Eilenberg-Moore functor T or defined by Remark 2.1 Now assume that char(K) = 2. Consider a dga (A, d) over K endowed with an augmentation ε : A → K and denote byĀ = K er(ε) its ideal of augmentation. Recall that the reduced bar-construction , is a semi-free resolution of K as an (A, d)module (cf. §19 in [9] or §2.2 in [18]). On the other hand, H (A, d) ⊗ B (H (A, d)) being a free resolution of K as an H (A, d)-module [9,Proposition 20.11], there is an Eilenberg-Moore resolution (P, d) → K of K as an (A, d)-module. Therefore, there is an equivalence of (A, d)-modules: [9,Proposition 6.6]. Roughly speaking, one can suppose the E 1 term of the spectral sequence induced by the

Evaluation map and Gorenstein spaces
Let (A, d) be an augmented dga over an arbitrary field K and ρ : (P, d) → (K, 0) any minimal semi-free resolution of K. A chain map: is given by f → f (z), where z ∈ P is a cocycle representing 1 K . Passing to the cohomology, we obtain the evaluation map of (A, d): Note that the definition of ev (A,d) is independent on the choice of (P, d) and z. Moreover, it is natural with respect to (A, d). As a particular case, ev C * (X,K) is called the evaluation map of X over K On the other hand, the authors of [7] introduced the concept of a Gorenstein space over K. It is a space X such that dim E xt C * (X,K) (K, C * (X, K)) = 1. In addition, if dim H * (X, K) < ∞, then X satisfies the Poincaré duality property over K and its fundamental class is closely related to the evaluation map (See, [11,16,21] for more details).

The Toomer invariant
We assume that char(K) = 2 and consider the projection of a minimal Sullivan algebra ( V, d) onto the quotient dga obtained by factoring out by the differential graded ideal generated by monomials of length at least n + 1. Thus, we define The Toomer invariant e K ( V, d) to be the smallest integer n (possibly ∞) such that p n induces an injection in cohomology. By [4], if char(K) = 0 and ( V, d) is a minimal Sullivan model of a simply connected finite type CW-complex X , then e K (X ) = e K ( V, d), where e K (X ) denotes the classical Toomer invariant introduced in [23]. If X is taken in the range of Anick, applying a similar argument yields in odd characteristic the coincidence e K (X ) = e K ( V, d).
Now, consider an arbitrary field K and denote by (T (W ), d) the free model of X introduced in [14]. In a similar way, Halperin and Lemaire defined the invariant e K (T (W ), d) with respect to the projection and showed that e K (X ) = e K (T (W ), d).
Recall finally that an alternative version of the Toomer invariant is given in terms of the Milnor-Moore spectral sequence (cf. [23] or [14], Prop. 1.6 (iii)) by or ∞ if such p doesn't exists.

Remark 2.2 In [4]
, it is shown that for any minimal Sullivan model ( V, d) of X , the Milnor-Moore spectral sequence (2.4.1) and the following one: are isomorphic from their second terms. Here d 2 designates the quadratic part of the differential d. Therefore, whenever X satisfies the Poincaré duality property and denoting by ω its fundamental class,

Main results
In this section, we first introduce the spectral sequence (1.0.2) and then use it, in conjunction with (1.0.1), to give the proofs of our main results.

Eilenberg-Moore spectral sequence of a free cdga
Let ( V, d) be a free cdga over a field K whose char(K) = 2 and assume that d = i≥k d i , with d i (V ) ⊆ i V and k ≥ 2. The map d is an algebra derivation defined on V ; that is, So, by extension, we have d( i V ) ⊆ ≥i+k−1 V, ∀i ≥ 1. Hence each d i is also a derivation defined on V and particularly, for any x ∈ V , d 2 k (x) ∈ 2k−1 V is by word-length reason the (2k − 1) −th homogeneous part of d 2 (x). Whence d 2 k = 0 and then ( V, d k ) is also a free cdga. As char(K) may be nonzero, in the sequel, we will use the divided power algebra (sV ) (see for instance [12]). If {v i } i∈I is a well ordered basis of V and V <i denotes the subspace generated by {v j , j < i}, the differential D on the product algebra V ⊗ (sV ) restricts to d V on V and on sV it is given by: ( V ⊗ (sV ), d) is a differential graded algebra and also an ( V, d)-semi-free module. Therefore, the projection ( V ⊗ (sV ), D) → K is a semi-free resolution of K called an acyclic closure of ( V, d) (cf. section 2 in [12]). When K = Q, (sV ) is replaced by the free cdga (sV ).

Consider now on
and the differential defined by  The general term of the spectral sequence induced by the filtration (3.
where D k stands for the differential of ( V, d k )). Hence, we obtain the isomorphism of graded modules E * , * ( V, d k )). This yields the spectral sequence which we call the Eilenberg-Moore spectral sequence of ( V, d).
Remark 3.2 1. When dim(V ) < ∞, the filtration (3.1.3) is clearly bounded in the sense of Theorem 2.6 in [20]; that is, for each dimension n, there exists s = s(n) and t = t (n) such that So the spectral sequence (3.1.4) is convergent. Furthermore, if V is of finite type, the convergence is a consequence of Theorem 1.2 (1).

Consider on ( V, d) the filtration defined by
An easy calculation shows that it induces the following spectral sequence: whose convergence is guaranteed if ( V, d) is a minimal Sullivan algebra (cf. §9. in [4]). Moreover, the chain map (2.3.1) is filtration-preserving, so that, the evaluation map ev ( V,d) : is a morphism of the spectral sequences (3.1.4) and (3.1.6). Notice that if k = 2, we find the spectral sequence (2.4.2).

Proofs of the main results
Let us note at the outset that the proofs of our main results are consequences of the following Proposition:

Proposition 3.3 Let X be a simply-connected CW-complex of finite type (or else in the range of Anick) and denote by ( V, d) its minimal Sullivan model. Then, the cohomological Eilenberg-Moore spectral sequence of C * ( (X ), K) is convergent and it is isomorphic to the one of ( V, d).
Before giving the proof of this proposition, we recall beforehand the construction of the Eilenberg-Moore spectral sequence associated with C * ( (X ), K) as well as that of the one introduced by Bisiaux in [2]. In fact, we will show that the latter is isomorphic to each of the two spectral sequences of Eilenberg-Moore, namely (1.0.1) and (1.0.2). Notice first that for any field K and any simply connected CW-complex X of finite type, the Adams-Hilton model A = (T (W ), d) → C * ( (X ), K) is a finite type free model of C * ( (X ), K) [14]. Now, consider on A ⊗ B(A), the filtration defined by IF q = n≤q A ⊗ T n (sĀ), ∀q ≥ 0 and endow Hom A (A ⊗ B(A), A) with the following one: (3.2.1) (A ⊗ B(A), A), (IF q ) q≥0 is decreasing and it is stable with respect to the differential of Hom A (A ⊗ B(A), A). So it induces a cohomological spectral sequence which is, by Remark 2.1, the Eilenberg-Moore spectral sequence (1.0.1) (see for instance [9], §20(d)).
On the other hand, let (B, d) = (K ⊕ B ≥2 , d) be a dga quasi-isomorphic to C * (X, K) (K being any field) and denote by (T (Z ), d) → (B, d) its free minimal model [14]. An acyclic closure of (T (Z ), d) has the form (T (Z ) ⊗ (s Z ⊕ K), D) [2]. Therefore, by taking on (Hom T (Z ) we obtain the following convergent spectral sequence introduced by Bisiaux [2]: (T (Z ), d)). The rest of the proof falls into two steps: Step 1. (In this step, we assume K an arbitrary field). Using [7, Remark 1.3], we will replace C * ( (X ), K) and C * (X, K), respectively, by A and (A). Thus, referring to [7, Theorem 2.1], the isomorphism: is deduced from the following isomorphisms of complexes: ) and its inverse map is the projection ψ A⊗B A (g) = (I d A ⊗ ε B(A) ) • g. B(A) and ε B(A) are, respectively, the diagonal and the co-unity of B A (see for instance [18,Proposition 1.5.14]).

We endow respectively Hom A (A ⊗ B(A), A) and Hom (A) ( (A) ⊗ A ∨ , (A)) with the filtrations
and then the composition isomorphism is one of filtered complexes. Consequently, the spectral sequence (1.0.1) is isomorphic to the one (3.2.3) and then it is convergent.
Step 2. (Here we assume K = Q and notice that the same proof remains valid in odd characteristic with X in the range in Anick (cf. Sect. 2.1). Let ( V, d) be a minimal Sullivan model of X and consider its minimal free model given by a quasi-isomorphism ϕ : is also a free model of C * (X, Q), the spectral sequences of the form ( is completed by a quasi-isomorphism of (T (Z ), d)-modules between acyclic closures of (T (Z ), d) and ( V, d).
We then provide Hom and define the differential graded morphism of complexes by putting ( f ) = ϕ • f (resp., (g) = g • ). These morphisms preserve filtrations (3.2.2) and (3.2.5) (resp., Proof of Theorem 1.2 It remains to prove the second assertion. By Proposition 3.3, we have r(X, K) = r ( V, d). Moreover, remark that the spectral sequence (1.0.2) relative to ( V, d k ) degenerates at its first term. Hence, r( V, d) ≤ r( V, d k ) is then a consequence of the convergence of (1.0.2) (relative to ( V, d)). Now suppose that dim V < ∞. Since d is decomposable, [ and then the desired equality follows.

Remark 3.4
Following the same approach of [7], we state below for any minimal Sullivan algebra ( V, d) some properties for r( V, d) similar to that listed in Remark 2.2.
1. r( V, d) is the largest integer p such that some non trivial class in E xt * ( V,d) (Q, ( V, d)) is represented by a cocycle in F p . Equivalently, it is the least integer p such that the projection A → A/F > p induces an injection in cohomology. 2. Suppose that dim(V ) < ∞; hence, by [7], ( V, d) is a Gorenstein algebra and denote by the unique generator of E xt * ( V,d) (Q, ( V, d)). Therefore, r( V, d) = sup{ p | can be represented by a cocycle in F p }.
We end this paper by asking the following question.
Question Recall first that a minimal Sullivan algebra ( V, d) is said to be elliptic, if both V and H ( V, d) are finite dimensional. The main result established by Lechuga and Murillo [17] states that if ( V, d k ) is also elliptic, then e 0 ( V, d) = e 0 ( V, d k,σ ), where d k,σ designates the pure differential associated to d k [13]. So the following natural question arises: Under what hypothesis, r( V, d) = r( V, d σ )?
By Theorem 1.2, since d σ,k = d k,σ , if the answer to this question is positive, we will have r( V, d) = r( V, d σ,k ) = r( V, d k,σ ) when dim(V ) < ∞.