A new lower bound for LS-category

Arabian Journal of Mathematics, Sep 2017

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

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A new lower bound for LS-category

A new lower bound for LS-category Mathematics Subject Classification 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∗( (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∗( (X ), K)) and eK(X ). We also define for any minimal Sullivan algebra ( 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 ( V , d) is a minimal Sullivan model of X , this invariant fulfills the relation r(X, K) = r( V , d). - 1 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 Ui , 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 eK(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 that each Hi (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)) ≤ eK(X ) if in addition evC∗(X,K) = 0 [ 2 ]. Recall from [ 6 ] that for any graded K-algebra G, depth(G) (possibly ∞) is the largest integer n such that ExtiG,∗(K, G) = 0, ∀i < n. Equivalently, depth(G) = inf{ p, Ext Gp,∗(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 E2p,q = Ext Hp,∗q( (X),K)(K, H∗( (X ), K)) ⇒ E x tCp∗+(q (X),K)(K, C∗( (X ), K)) (1.0.1) to define the integer r(X, K) = sup{ p ∈ N | E ∞p,∗ = 0} 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 x t is the differential-Ext of Eilenberg–Moore (see Sect. 2 for more details). Our first main theorem proves the following improvement of Bisiaux result: Theorem 1.1 Let X be a simply connected CW-complex such that each Hi (X, K) is finite dimensional over an arbitrary field K. If X is Gorenstein and evC∗(X,K) is nonzero, then depth(H∗( (X ), K)) ≤ r(X, K) ≤ eK(X ). 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: E x t(p,qV,dk )(K, ( V , dk )) ⇒ E x t(p+Vq,d)(K, ( V , d)) (1.0.2) which we call the Eilenberg–Moore spectral sequence of ( V , d). Throughout, the differential dk (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 r( A, d) := sup{ p ∈ N | E ∞p,∗[( V , d)] = 0}, where E ∞∗,∗[( V , d)] is the ∞ term of (1.0.2). 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 1. the two Eilenberg–Moore spectral sequence (1.0.1) and (1.0.2) are isomorphic. 2. r(X, K) = r( V , d) ≤ r( V , dk ) and the equality holds if dim(V ) < ∞. 2 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 (s V ) p = V p+1 (resp., (s−1V ) p = V p−1, resp., V ∨ = HomK(V , K)). 2.1 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 = ⊕ii==0∞V i which has a well ordered basis {xα}α∈I such that d xα ∈ V<α (V<α = span{vγ , γ < α}). Such algebra is said to be minimal if deg(xα) < deg(xβ ) ⇒ α < β, ∀α, β ∈ I. If V 0 = K and V 1 = 0, this is equivalent to say that d(V ) ⊆ ⊕ii==2∞ i V , where i V designates the subspace of V spanned by elements of word-length i . A minimal Sullivan model for a cdga ( A, d) is a minimal Sullivan algebra ( V , d) equipped with a quasi-isomorphism ( V , d) → ( A, d). By [12, Theorem 7.1], if H 0( A, d) = K, H 1( A, d) = 0 and dim(H i ( A, d)) < ∞ for all i ≥ 0, then ( A, d) always has a minimal Sullivan model. 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 AP 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 ∼= HomZ(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 . 2.2 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 ( A, d)-semi-free if it is the union of ( A, d)-submodules 0 = F−1 ⊂ F0 ⊂ F1 ⊂ · · · such that each Fi /Fi−1 is free. If in addition K ⊗A P is zero, (P, d) is called an ( A, d)-semi-free minimal module. Now, let (M, d) be an ( A, d)-module. A morphism (P, d)→(M, d) 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 A1 = 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 another ( A, d)-module (N , d) and let (P, d) → (M, d) an ( A, d)-semi-free resolution of (M, d). The ( A, d)-module Hom A(P, N ) = ⊕p≥0Hom Ap,∗(P, N ) where the p-component Hom Ap,∗(P, N ) = ⊕i≥0Hom A(Pi , N i+p), is provided with the differential defined by: D( f ) = d ◦ f − (−1)p f ◦ d, ∀ f ∈ Hom Ap,∗(P, N ). The Eilenberg–Moore functor E xt is defined as follows: E xt(A,d)((M, d), (N , d)) = H ∗(Hom A(P, N ), D). Analogously, if P ⊗A N = ⊕p≥0(P ⊗A N )p,∗ with (P ⊗A N )p,∗ = ⊕i≥0(Pi ⊗A Ni−p) is endowed with the following differential D( p ⊗ n) = d( p) ⊗ n + (−1)p p ⊗ d(n), ∀ p ⊗ n ∈ (P ⊗A N )p,∗, we obtain the Eilenberg–Moore functor T or defined by T or(A,d)((M, d), (N , d)) = H∗(P ⊗A N , D). 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 by A¯ = K er (ε) its ideal of augmentation. Recall that the reduced bar-construction ( A⊗ B( A), d) with coefficients in A, where B( A) = n≥0 T n(s A¯), 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 K of of K as an H ( A, d)-module [9, Proposition 20.11], there is an Eilenberg–Moore resolution ( P, d) → K as an ( A, d)-module. Therefore, there is an equivalence of ( A, d)-modules: ( P, d) → ( A ⊗ B( A), d) [9, Proposition 6.6]. Roughly speaking, one can suppose the E1 term of the spectral sequence induced by the filtration (Fq = ⊕n≤q A ⊗ T n(s A¯))q≥0 on A ⊗ B( A) to be an H ( A, d)-semi-free resolution of K. 2.3 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: (cf. [ 23 ] or [ 14 ], Prop. 1.6 (iii)) by or ∞ if such p doesn’t exists. 2.4 The Toomer invariant We assume that char(K) = 2 and consider the projection pn : V → V / ≥n+1V 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 eK( V , d) to be the smallest integer n (possibly ∞) such that pn 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 eK(X ) = eK( V , d), where eK(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 eK(X ) = eK( 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 eK(T (W ), d) with respect to the projection and showed that eK(X ) = eK(T (W ), d). Recall finally that an alternative version of the Toomer invariant is given in terms of the Milnor–Moore spectral sequence pn : T (W ) → T (W )/T ≥n+1W Ext Hp,∗q( (X),K)(K, K) ⇒ H p+q (X, K) eK(X ) = sup{ p ∈ N | E ∞p,∗ = 0} Hom(A,d)(( P, d), ( A, d)) −→ ( A, d) ev(A,d) : E x t(A,d)(K, ( A, d)) −→ H ∗( A, d). is given by f → f (z), where z ∈ P is a cocycle representing 1K. 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, evC∗(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 x tC∗(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). (2.3.1) (2.3.2) (2.4.1) 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: H p,q ( V , d2) ⇒ H p+q ( V , d), (2.4.2) are isomorphic from their second terms. Here d2 designates the quadratic part of the differential d. Therefore, whenever X satisfies the Poincaré duality property and denoting by ω its fundamental class, eK(X ) = eK( V , d) = sup{ p/ω can be represented by a cocycle in ≥ p V } [4, Lemma 10.1]. Similarly, when K is any field and (T (W ), d) is a minimal free model of X over K, by [2, Lemma 2.1], we have eK(T (W ), d) = sup{ p/ω can be represented bya cocycle in T ≥ p W }. 3 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. 3.1 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 = and k ≥ 2. The map d is an algebra derivation defined on V ; that is, i≥k di , with di (V ) ⊆ i V d(x y) = d(x )y + (−1)|x|x d(y), ∀x , y ∈ V . So, by extension, we have d( i V ) ⊆ ≥i+k−1V , ∀i ≥ 1. Hence each di is also a derivation defined on V and particularly, for any x ∈ V , dk2(x ) ∈ 2k−1V is by word-length reason the (2k − 1)−th homogeneous part of d2(x ). Whence dk2 = 0 and then ( V , dk ) is also a free cdga. As char(K) may be nonzero, in the sequel, we will use the divided power algebra (s V ) (see for instance [ 12 ]). If {vi }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 ⊗ (s V ) restricts to dV on V and on s V it is given by: D(svi ) = vi + φ, φ ∈ V<i ⊗ (s V<i ), ∀i ∈ I. [21, Remark 1.2]. So, by extension, we have D(γ p(sv)) = D(sv)γ p−1(sv); ∀ p ≥ 1, ∀sv ∈ (s V )even. ( V ⊗ (s V ), d) is a differential graded algebra and also an ( V , d)-semi-free module. Therefore, the projection ( V ⊗ (s V ), D) → K is a semi-free resolution of K called an acyclic closure of ( V , d) (cf. section 2 in [ 12 ]). When K = Q, (s V ) is replaced by the free cdga (s V ). Consider now on A = Hom( V,d)(( V ⊗ (s V ), D), ( V , d)) the filtration F p = { f ∈ Hom V ( V ⊗ (s V ), V ) | f ( (s V )) ⊆ ≥ p V }, ∀ p ≥ 0 and the differential defined by (3.1.1) (3.1.2) (3.1.3) D( f ) = d ◦ f + (−1)| f |+1 f ◦ D, ∀ f ∈ A. Lemma 3.1 The filtration (3.1.3) verifies the following: (i) (F p) p≥0 is decreasing, (ii) F 0(A) = A, (iii) D(F p(A)) ⊆ F p(A). Proof Properties (i) and (ii) are immediate. Moreover, the property (iii) follows from the definition of D on A, specially, the relation D(γ p(sv)) = (v + φ)γ p−1(sv) on (s V )even [cf. (3.1.1) and (3.1.2) above]. The general term of the spectral sequence induced by the filtration (3.1.3) is given by: p Er = { f ∈ F p, D( f ) ∈ F p+r } { f ∈ F p+1, D( f ) ∈ F p+r } + F p ∩ D(F p−r+1) . Moreover, Ar,s = (Homr V ( V ⊗ (s V ), V ))r+s = { f ∈ A, f ( (s V )) ⊆ (Fr /Fr+1)r+s . A straightforward calculation permits to prove that r V }r+s is isomorphic to Ek p,q ∼ = Ker[A p,q →Dk A p+k−1,q−k+2] Im[A p−k+1,q+k−2 Dk → A p,q ] , where Dk stands for the differential of Hom( V,dk )(( V ⊗ isomorphism of graded modules Ek∗,∗ ∼= ⊕ p,q≥0E x t(p,qV,dk )(K, ( V , dk )). This yields the spectral sequence (s V ), Dk ), ( V , dk )). Hence, we obtain the 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 {0} = F s (An) ⊆ F s−1(An) ⊆ · · · ⊆ F t+1(An) ⊆ F t (An) = An. 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). 2. Consider on ( V , d) the filtration defined by (3.1.4) (3.1.5) (3.1.6) An easy calculation shows that it induces the following spectral sequence: F p = ≥ p V = ∞ i= p i V . H p,q ( V , dk ) ⇒ H p+q ( V , d) 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) : E x t( V,d)(K, ( V , d)) −→ H ∗( 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). 3.2 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 = A ⊗ T n(s A¯), ∀q ≥ 0 n≤q and endow Hom A( A ⊗ B( A), A) with the following one: IF q = { f | f (IF k ) = 0, ∀k < q}, ∀q ≥ 0. Clearly IF 0 = Hom A( 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 (HomT (Z)(T (Z ) ⊗ (s Z ⊕ K), T (Z )), D) the filtration q IF (T (Z),d) = { f | f (T (Z ) ⊗ (s Z ⊕ K)) ⊆ T ≥q Z }, ∀q ≥ 0, we obtain the following convergent spectral sequence introduced by Bisiaux [ 2 ]: E x t(pT,q(Z),d2)(K, (T (Z ), d2)) ⇒ E x t(pT+(qZ),d)(K, (T (Z ), d)). Proof of Proposition 3.3 Let us denote by ( A) = (T (s−1 A∨), d) the dual of B( A). Thus ( A ⊗ B( A))∨ = ( A) ⊗ A∨ is a left ( A)-module and then the filtration F q = ⊕n≤q ( A) ⊗ An∨, ∀q ≥ 0 exhibits it as an ( A)-semi-free resolution of K [ 7 ]. Again, by Remark 2.1, we can assume that it is an Eilenberg–Moore semi-free resolution of K. Notice that ( A), denoted thereafter by (T (W ), d), is a finitetype free model of C ∗(X, K) [14, Proposition 1.6]. 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: (3.2.1) (3.2.2) (3.2.3) E x tC∗( (X),K)(K, C∗( (X ), K)) ∼= E x tC∗(X,K)(K, C ∗(X, K)) (3.2.4) is deduced from the following isomorphisms of complexes: ∼ϕA (Hom A( A ⊗ B( A), A), D) −→ (End A⊗B(A)( A ⊗ B( A)), [d, ]) −∼=→∨ (End (A)⊗A∨ ( ( A) ⊗ A∨), [d∨, ]) ∼←ϕ−(A) (Hom (A)( ( A) ⊗ A∨, ( A)), D). Here, ϕA( f ) = ( f ⊗ i dB(A)) ◦ (i dA ⊗ B(A)) and its inverse map is the projection ψA⊗B A(g) = (I dA ⊗ εB(A)) ◦ g. B(A) and εB(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 IF q and IF q(T (W ),d) = { f | f ( ( A) ⊗ A∨) ⊆ ( A)≥q }. If f ∈ IF q , then ϕA( f )( A ⊗ B( A)<q ) = 0 and ϕA( f )( A ⊗ B( A)≥q ) ⊆ A ⊗ B( A)≥q . So ∨ ◦ ϕA( f )( ( A)<q ⊗ A∨) = 0 and ∨ ◦ϕA( f )( ( A)≥q ⊗ A∨) ⊆ ( A)≥q ⊗ A∨. Hence ∨ ◦ ϕA( f )( ( A) ⊗ A∨) ⊆ ( A)≥q ⊗ A∨. Applying ϕ−1A, we conclude that ϕ−1A ◦ ∨ ◦ ϕA( f )( ( A)≥q ⊗ A∨) ⊆ ( A)≥q . Therefore, q q ϕ−1A ◦ ∨ ◦ ϕA(IF ) ⊆ IF (T (W ),d) 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 : (T (Z ), d) → ( V , d). Since (T (Z ), d) is also a free model of C ∗(X, Q), the spectral sequences of the form (3.2.3) induced by both (T (W ), d) and (T (Z ), d) are isomorphic. Now, by degree reason, ϕ preserves filtrations and then by [14, Proposition 3.6] it induces a quasiisomorphism E2(ϕ) : (T (Z ), d2) −→ ( V , d2), where d2 stands for the quadratic part of d. Also, ϕ induces on ( V ⊗ (s V ), D) the structure of a (T (Z ), d)-module and then [9, Proposition 6.4.] the following diagram is completed by a quasi-isomorphism (T (Z ) ⊗ (s Z ⊕ Q), D) −→ : (T (Z ) ⊗ (s Z ⊕ Q), D) −→ ( V ⊗ (s V ), D) ( V ⊗ (s V ), D) ↓ Q of (T (Z ), d)-modules between acyclic closures of (T (Z ), d) and ( V , d). We then provide HomT (Z)(T (Z ) ⊗ (s Z ⊕ Q), V ), D) with the filtration IH q = { f | f (T (Z ) ⊗ (s Z ⊕ Q)) ⊆ ( V )≥q } (3.2.5) and define the differential graded morphism of complexes : HomT (Z)(T (Z ) ⊗ (s Z ⊕ Q), T (Z )) → HomT (Z)(T (Z ) ⊗ (s Z ⊕ Q), V )) (resp., : Hom V ( V ⊗ (s V ), V )) → HomT (Z)(T (Z ) ⊗ (s Z ⊕ Q), V )) by putting ( f ) = ϕ ◦ f (resp., (g) = g ◦ ). These morphisms preserve filtrations (3.2.2) and (3.2.5) (resp., (3.1.3) and (3.2.5)) and induce morphisms of spectral sequences. Finally, since ( V ⊗ (s V ), D) yields an acyclic closure ( V ⊗ (s V ), D2) of ( V , d2) ([10, Prop. 3.12] and [21, Lemma 2.1]), the quasi-isomorphism E2(ϕ) aforementioned implies that the later morphisms are in fact isomorphisms of spectral sequences. Hence, the two spectral sequences (3.2.3) and (1.0.2) are isomorphic. Composing the obtained isomorphisms, we deduce that the two spectral sequences (1.0.1) and (1.0.2) are isomorphic. Proof of Theorem 1.1 The inequality depth(H∗( (X ), K)) ≤ r(X, K) comes immediately from the convergence of the Eilenberg-Moore spectral sequence (1.0.1) established in the first step of the proof of Proposition 3.3. Now, X being a Gorenstein space, we have dim E x t(T (W ),d)(K, (T (W ), d)) = 1. There exists then an unique pair ( p, q) of integers and an unique [ f ] of bi-degree ( p, q) [with respect to the filtration (3.2.2)] generating E x t(T (W ),d)(K, (T (W ), d)). It follows from the isomorphism between (1.0.1) and (3.2.3) that p = r(X, K). The hypothesis evK(T (W ), d) = 0 implies that ev([ f ]) = [ f (1)] = 0. But f (1) ∈ T ≥ p(W ), so eK(T (W ), d) ≥ p by the characterization given in Remark 2.2. Consequently, r(X, K) ≤ eK(X ). 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 , dk ) degenerates at its first term. Hence, r( V , d) ≤ r( V , dk ) is then a consequence of the convergence of (1.0.2) (relative to ( V , d)). Now suppose that dim V < ∞. Since d is decomposable, [21, Proposition 3.1] asserts that both ( V , dk ) and ( V , dk ) are Gorenstein algebras. Therefore, in (1.0.2), there exists a unique pair ( p, q) such that E x t(∗,∗V,dk )(K, ( V , dk )) = E x t(p,qV,dk )(K, ( V , dk )) = E∞ p,q = E x t(∗,∗V,d)(K, ( V , d)) 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. V , d ) = sup{ p | p . can be represented by a cocycle in F } 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 , dk ) is also elliptic, then e0( V , d ) = e0( V , dk,σ ), where dk,σ designates the pure differential associated to dk [ 13 ]. So the following natural question arises: Under what hypothesis, r( V , d ) = r( V , dσ )? r( By Theorem 1.2, since dσ,k = dk,σ , if the answer to this question is positive, we will have r( V , dσ,k ) = r( V , dk,σ ) when dim(V ) < ∞. V , d ) = Acknowledgements I am indebted to J. C. Thomas for very useful conversations which enabled me to improve my results significantly. I am also grateful to the reviewers whose comments, gave me the opportunity to improve the drafting of this work. 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Youssef Rami. A new lower bound for LS-category, Arabian Journal of Mathematics, 2017, 1-9, DOI: 10.1007/s40065-017-0181-5