Cone-length and Lusternik-Schnirelmann category

Octavian Cornea University of Rochester Rochester, NY 14627

November 23, 1993

Abstract

For a simply connected, finite type CW − complex X we introduce a geometric notion of cone-length extending the rational one introduced by Lemaire and Sigrist and we show that it is larger by at most one than the L.S. category. We also prove that the rational version agrees with the ”homotopic nilpotency” of the algebra of P.L. forms on X .

1 Introduction

For a topological space X, the Lusternik-Schnirelmann category , catX, is defined to be the least natural number n such that there are n + 1 open subsets of X which cover X and are (each) contractible in X.

This number was introduced in [10] where it was showed that, when X is a smooth manifold, catX+1 gives a lower bound for the number of critical points of a smooth function on X. The definition we use in this paper for the L.S.-cat assigns to a point the category 0 and not 1 as it appears sometimes in the literature. It is easy to see that catX is an invariant of the homotopy type of X.

From now on, unless otherwise specified, we will work inside the pointed category of spaces of the homotopy type of pointed CW − complexes .

For path-connected spaces of this kind there are two other equivalent definitions of cat.

The first one, introduced by Whitehead, asserts that catX≤ n iff the diagonal Δ^{n+1} : X ⟶ X^{n+1} factors through the fat wedge T^{n+1}X ⊆ X^{n+1}.

The second, belonging to Ganea, states that catX≤ n iff the fibration ΩX^{∗(n+1)} ⟶^{i_{n}} B_{n}ΩX ⟶ X has a section. Here ΩX^{∗(n+1)} is the (n + 1)- fold join of ΩX and E_{n}ΩX = ΩX^{∗(n+1)} ⟶^{i_{n}} B_{n}ΩX is the n-th filtration in Milnor’s construction for the classifying space of ΩX.

Very often B_{n}ΩX is called the n-th space of Ganea associated to X, [3]. Remark. The L.S.-cat is an interesting notion for spaces more general

than the ones we concentrate on here (see [8]). Following Dold, for a pathconnected and well pointed space X we may define the category as 1+ the least number of sets in an open, numerable covering of X with subsets containing ∗ and contractible in X. This definition is equivalent with both the pointed and unpointed version of Whitehead’s definition.

For simply-connected rational spaces of finite type there is still another description of the category given by Felix and Halperin [4]. Let’s first define a related notion (all our rational differential graded algebras (dga’s) will be commutative).

Definition 1 Let A be a rational, augmented dga. The nil-length of A ,nilA, is the least natural number n such that there is another rational, augmented dga B, quasi-isomorphic to A, such that \overline{B} is nilpotent of order n + 1.If no such n exists, then nilA= ∞.

Felix and Halperin’s description of the category asserts that a rational space X′ has catX′≤ n if a free simply connected dga of finite type, representing the homotopy type of X′, is a homotopy retract of a dga of nil≤ n. This description is based on the fact that the n-th Ganea space is represented by such a nilpotent dga.

For further use let nilX′=nil(M(X′)) (where M(X′) is Sullivan’s minimal model of X′, [12]). If X′ is the rationalization of a simply connected CW − complex X, we may interpret nilX′ as the ”homotopic nilpotency” of the algebra of P.L. forms on X.

The starting point for the present paper is a study by Lemaire and Sigrist [9] of the cat of rational spaces. They define, for a rational free dgL (differential graded Lie algebra ) L, its cone-length ClL to be the least m such that there is a free dgL, (L′, d), quasi-isomorphic to L and a filtration on L′,

0 = F_{0}L′ ⊆ F_{1}L′ ⊆ … ⊆ F_{m}L′ = L′

such that F_{i}L′ = L(T_{i}), with dT_{i} ⊂ F_{i−1}L′, i ≥ 1 (if no such m exists ,then, of course, ClL= ∞; L(T_{i}) is the free Lie algebra on T_{i}). The cone-length of X′ is defined to be the cone-length of its Quillen model. Geometrically this means that X′ has ClX′≤ n iff there is a sequence of cofibration sequences:

H_{i} ⟶ Y_{i} ⟶ Y_{i+1} , 0 ≤ i < n

with H_{i}, Y_{i} rational spaces, H_{i} a bouquet of spheres for i > n, Y_{0} ≃ ∗ and Y_{n} ≃ X′. In the same paper it was showed that Cl=cat for formal and coformal spaces and it was conjectured that the equality holds for all rational spaces.

A few years later it was proved by Felix and Thomas, [5], that this conjecture is valid for spaces of category 2.

Remark. It is worth mentioning here that a relation between cat and a notion of cone-length appeared earlier in the work of Ganea. Indeed, let X be a path-connected space. Ganea , [6], introduced the notion of strong category of X : CatX=the least natural number n for which there is a covering of a space with the homotopy type of X by n + 1 open, contractible subsets. He proved that CatX≤ n iff there are cofibration sequences:

L_{i} ⟶ X_{i} ⟶ X_{i+1} , 0 ≤ i < n

with X_{0} ≃ ∗ and X_{n} ≃ X. It is clear that CatX≥catX. It was later proved by Ganea and Takens, [11], that CatX≤catX+1.

In this paper we introduce a new type of cone-length:

Definition 2 Let X be of the homotopy type of a path-connected, pointed CW −complex . The cone-length of X, ClX, is the least n for which there are cofibration sequences:

Σ^{i}Z_{i} ⟶ X_{i} ⟶ X_{i+1} , 0 ≤ i < n  (1)

with X_{0} ≃ ∗ and X_{n} ≃ X (for notational convenience the 0-suspension of a space is taken to be the space itself, Σ^{0}Z = Z). If no such n exists, ClX= ∞.

Remarks. a. Initially we intended to use a ”milder” cone-length in whose definition appears, for each i > 0, a single suspension ΣZ_{i} (instead of Σ^{i}Z_{i}). The notion obtained in this way will be denoted by clX. The definition of ClX above was suggested to us by Dieter Puppe.

b. It is obvious that ClX≥clX ≥CatX≥catX. Any co-H-space which is not a suspension is an example of space with Cat= 2 and cat= 1.

c. A priori we have, for rational spaces, two invariants denoted by ClX: one obtained using the rational variant of the above definition, and the other given by the original definition of Lemaire and Sigrist. In fact they agree because rational suspensions are wedges of spheres and, rationally, (1) means just that the i+1-level in a cone decomposition of X is obtained by attaching to the i-level cells of dimension at least i + 1. But this happens in any reasonable ”short” cone decomposition of X.

In the second section of the paper we will prove:

Theorem 1.1 For X as above, if catX= n, then there is a space Z such that Cl(X ∨ Σ^{n}Z)≤ n.

Some interesting consequences are:

Corollary 1.2 ClX≤catX+1

Corollary 1.3 If k >catX, then cat(B_{k}ΩX)=catX.

Remarks. a. It is a wellknown fact that for k ≤catX, cat(B_{k}ΩX)= k. That together with our result gives a precise description of cat(B_{k}ΩX) for all integers k.

b. The above result is stronger than a recent one of Dold’s, obtained by different methods, which asserts that k >catX implies cat(B_{k}ΩX)≤catX+1.

c. Incidentally we obtain, if X is a manifold, that the minimal number of critical points of a smooth function on X is bounded from below by ClX.

d. Recently Doeraene, [2], showed that the notion of L.S.-catagory can be formulated in the more general context of a type of category similar to a Quillen closed model category, but satisfying a suplimentary, important, axiom:

If, in the commutative, cube diagram below, the lateral faces are (homotopy) pull-backs and the bottom face is a push-out, then the top is a homotopy push-out.

[figure: D′; C′; A′; B′; D; C; A; B]

He called this type of category a J-category. Then he showed that inside a J-category, Whitehead’s and Ganea’s definitions of the L.S.-cat coincide. It was also proved by Hess and Lemaire [7] that, in this context, the original definition of cat can be formulated and that it agrees with the other two ones.

Theorem 1.1 and its corollaries are valid inside a J-category. Even if we prefer not to use the language of J-categories, we will write the proofs in such a way that a translation into that language will be more or less automatic.

There is considerable interest for product formulas for cat and Cat ([11], [1]).We will prove here such a formula for the cone-length.

Proposition 1.4 Cl(X × Y)≤ClX+ClY.

In the third section we will turn to the rational context.

First we will analyze the case of the spaces of cat=2. Using the geometric considerations of the second section we present a new proof for the result of Felix and Thomas giving the equality cat=Cl in this case. After another related construction concerning these spaces we go to the main result of the section:

Theorem 1.5 For any simply-connected rational space of finite type X′ we have ClX′=nilX′.

Remarks. a.This equality was believed to hold for some time ([4], [9]). b.It seems to be reasonable to expect that, in fact, nilX′=CatX′ (this is

certainly true for formal, coformal, and L.S.-category 2 spaces).

c.As all the homotopic invariants (that we are studying here) of the rationalization of a space are smaller or equal than the corresponding invariants of the initial space, a trivial consequence of the theorem above is that the minimal number of critical points of a smooth function on a (simply-connected) manifold is bounded from below by the ”homotopic nilpotency” of its algebra of P.L. forms. This suggests a more dirrect (and possibly geometric) relation between critical point theory and some form of nilpotency of the (P.L.) de Rham algebra of the manifold.

This work is part of the author’s doctoral dissertation. I would like to thank Joe Neisendorfer, my advisor, for his help, suggestions and support which made me go on with this project. Thanks are due to John Moore for his encouragement and for the interest he showed for these problems, to Steve Halperin for some very useful discussions and also to Albrecht Dold, Dieter Puppe, Kathryn Hess and Jean Paul Doeraene for sharing some of their ideas with me.

2 Cone-length. Geometric properties

The proof of Theorem 1.1 is based on the following proposition (which has some interest in itself):

Proposition 2.1 Let F ⟶^{i} E ⟶ B be a fibration with B, E path connected. Then Cl(E/F)≤ClB (here E/F is, up to homotopy, the cofibre of

F ⟶^{i} E ).

Proof.In the following each commutativity, pull-back or push-out diagram is understood to be valid up to homotopy.We will use the following two facts:

Lemma 2.2 Suppose that, in the commutative, cube diagram below, all lateral faces are pull-backs and that the bottom face is a push-out.

Denote by F the common fibre of the vertical edges. The induced square:

[figure: D′/F; C′/F; A′/F; B′/F]

is a push-out. Proof.The square is clearly a push-out and, in general, if we have a commutative diagram like:

[figure: D′; C′; A′; B′]

[figure: D′′; C′′; A′′; B′′; D′; C′; A′; B′; D′′′; C′′′; A′′′; B′′′]

with the vertical edges being cofibration sequences and the top and middle level being push-out squares, then the bottom level is a push out diagram also. In our case A′′ = B′′ = C′′ = D′′ = F. Lemma 2.3

((Σ^{n}X) × Y)/(∗ × Y) ≃ Σ^{n}((X × Y)/(∗ × Y))

Proof.In the commutative, cube diagram below the bottom is a push-out and the lateral faces are pull-backs.

[figure: X × Y; Y; Y; (ΣX) × Y; X; ∗; ∗; ΣX]

By applying the previous lemma we get a push-out square:

[figure: (X × Y)/(∗ × Y); ∗; ∗; (ΣX × Y)/(∗ × Y)]

That means Σ((X ×Y)/(∗×Y)) ≃ (ΣX ×Y)/(∗×Y). This is the statement for n = 1. The cases n > 1 follow immediately by induction.

We return now to the proof of the proposition.

Suppose ClB= n. Then there are cofibration sequences

Σ^{i}Z_{i} ⟶ X_{i} ⟶ X_{i+1} , 0 ≤ i < n

with X_{0} ≃ ∗ and X_{n} ≃ B. Denote by v_{i} : X_{i} ⟶ B the obvious inclusions. Consider the pull-back fibrations induced by the v_{i}’s:

F ⟶ E_{i} ⟶ X_{i}

We want to prove by induction that for 0 ≤ i ≤ n, Cl(E_{i}/F)≤ i. Clearly for i = 0 this is indeed the case as E_{0} = F and, hence, E_{0}/F ≃ ∗. Suppose now that Cl(E_{k}/F)≤ k < n. We have the following commutative diagram

[figure: (Σ^{k}Z_{k}) × F; F; E_{k}; E_{k+1}; Σ^{k}Z_{k}; ∗; X_{k}; X_{k+1}]

Where the bottom face is a push-out and the lateral faces are pull-backs. Apply now Lemma 2.2 .We get a push-out square:

[figure: (Σ^{k}Z_{k} × F)/(∗ × F); ∗; E_{k}/F; E_{k+1}/F]

But by Lemma 2.3 (Σ^{k}Z_{k} ×F)/(∗×F) ≃ Σ^{k}((Z_{k} ×F)/(∗×F)). This implies Cl(E_{k+1}/F)≤ k + 1.

Remark. The previous construction gives a very explicit cone decomposition of E/F in terms of one for B.

We start now the proof of Theorem 1.1 .

We have first to recall some of the properties of the spaces of Ganea (B_{n}ΩX). For a space X we have the fibrations

E_{n}ΩX = (ΩX)^{∗(n+1)} ⟶^{i_{n}} B_{n}ΩX ⟶^{t_{n}} X

The property that interests us here is that B_{n+1}ΩX is, up to homotopy, the cofibre of i_{n} and that the map t_{n+1} is induced in the obvious way (by sending E_{n}ΩX to a point) by t_{n}. Also, under the equivalence BΩX ≃ X, t_{n} coresponds just to the inclusion B_{n}ΩX ⊆ BΩX. Clearly B_{0}ΩX ≃ ∗. Recall that A ∗ B = Σ(A ∧ B), as a consequence of this:

(ΩX)^{∗(n+1)} = Σ^{n}(∧^{(n+1)}ΩX)  (2)

Suppose that catX≤ n. That means, by Ganea’s description of cat, that t_{n} : B_{n}ΩX ⟶ X admits a section s. This implies the existence of the commutative diagram below, whose rows and columns are fibrations:

[figure: ΩE_{n}ΩX; ∗; E_{n}ΩX; ΩE_{n}ΩX; X; id; B_{n}ΩX; j; ∗; X; X; id]

It follows that j is homotopically trivial, hence X/Ω(E_{n}ΩX) ≃ X∨ΣΩ(E_{n}ΩX). Making use of the formula:

ΣΩΣA ≃ Σ(∨_{k≥1}(∧^{(k)}A))  (3)

we get that ΣΩ(E_{n}ΩX) ≃ Σ^{n}Z for some Z (which can be determined exactly from (2) and (3) ).

Apply now the Proposition 2.1 to the fibration

ΩE_{n}ΩX ⟶ X ⟶ B_{n}ΩX

This will give Cl(X ∨ Σ^{n}Z)≤ n.

The proof of Corollary 1.2 is very easy now. Indeed X is the cofibre of the inclusion: Σ^{n}Z ⟶ X ∨ Σ^{n}Z.

Notice that the proof of the Theorem 1.1 gives a precise cone decomposition of X ∨ Σ^{n}Z and, consequently, of X. In this context it is useful to have:

Lemma 2.4 Let F ⟶ E ⟶ B be as before. Suppose Cl(B ∨ H)≤ n, then Cl((E/F) ∨ (H × F)/(∗ × F))≤ n.

Proof.Consider the fibration F ⟶ G ⟶ B ∨ H induced by the map B ∨ H ⟶ B which collapses H to the base point. Clearly this fibration is trivial over H. Hence we have the following commutative diagram:

[figure: F; H × F; E; G; ∗; H; B; B ∨ H]

The conditions needed for applying Lemma 2.2 are satified and this gives us the commutative push-out square:

[figure: ∗; (H × F)/(∗ × F); E/F; G/F]

That means G/F ≃ E/F ∨ (H × F)/(∗ × F). By Proposition 2.1 we also have Cl(G/F)≤ n.

For the Corollary 1.3, remark that E/F is a retract of G/F, hence cat(E/F)≤ cat(G/F)≤ n. Using Theorem 1.1 we get cat(E/F)≤catB and we may apply this to the fibration E_{k}ΩX ⟶ B_{k}ΩX ⟶ X.

Remarks a.The inequality cat(E/F)≤catB can also be obtained directly by using Whitehead’s definition of cat. Using this definition catX≤ n is equivalent to the existence of a covering of B with closed sets A_{i}, 0 ≤ i ≤ n, such that there are pointed homotopies h^{i} : B × [0, 1] ⟶ B, h_{0}^{i} = id, h_{1}^{i}(A_{i}) = ∗. Lift h^{i} to H^{i} : E × [0, 1] ⟶ E such that H_{0}^{i} = id. Let T_{i} ⊂ E be the preimage of A_{i}. Then H_{1}^{i}(T_{i}) ⊂ F and H_{t}^{i}(F) ⊂ F for all t ∈ [0, 1]. We get induced homotopies \overline{H}^{i} : E/F × [0, 1] ⟶ E/F and A_{i}/F, 1 ≤ i ≤ n will cover E/F, \overline{H}_{0}^{i} = id, \overline{H}_{1}^{i}(A_{i}/F) = ∗.

We will prove now the product formula (Proposition 1.4) : Cl(X × Y)≤ ClX + ClY

Proof.Let ClX= m, ClY = n. There are cofibration sequences:

U_{i} ⟶^{f_{i}} A_{i} ⟶ A_{i+1}, V_{j} ⟶^{g_{j}} B_{j} ⟶ B_{j+1}

with 0 ≤ i < m, 0 ≤ j < n, A_{m} ≃ X, B_{n} ≃ Y, U_{i} = Σ^{i}Z_{i}, V_{j} = Σ^{j}W_{j}, A_{0} = B_{0} = ∗.We may, of course, assume A_{i+1} = A_{i}∪_{f_{i}}CU_{i}, B_{j+1} = B_{j}∪g_{j}CU_{j} (CX being the cone over X).This gives an increasing filtration for X and one for Y:

X = A_{m} ⊃ A_{m−1} ⊃ … ⊃ A_{0} = ∗
Y = B_{n} ⊃ B_{n−1} ⊃ … ⊃ B_{0} = ∗

such that A_{i+1} − A_{i} = CU_{i} − U_{i}, B_{j+1} − B_{j} = CV_{j} − V_{j}. Here U_{i} (resp. V_{j}) is identified with the ”boundary” of the cone CU_{i} (resp. CV_{j}). Define T_{ij} = (A_{i} − A_{i−1}) × (B_{j} − B_{j−1}), \overline{T_{ij}} the closure of T_{ij}, D_{ij} = CU_{i−1} × CV_{j−1}, S_{k} = ⋃_{i+j=k} T_{ij}, and R_{t} = ⋃_{k≤t} S_{k}. Here 0 < i ≤ m, 0 < j ≤ n, 0 ≤ k ≤ m + n, 0 ≤ t ≤ m + n and we consider A_{k} and B_{k} empty when k < 0.

Notice that X × Y = R_{m+n} ⊃ R_{m+n−1} ⊃ … ⊃ R_{0} = ∗ is a filtration for X × Y and R_{t} − R_{t−1} = S_{t}, also T_{ij} ∩ T_{i′j′} is empty when (i, j) ≠ (i′, j′). Moreover if i + j = t, then \overline{T_{ij}} − T_{ij} ⊂ R_{t−1}

Each D_{ij} is attached to R_{i+j−1} following a map

h_{ij} : U_{i} ∗ V_{j} = (CU_{i−1} × V_{j−1}) ⋃_{(U_{i−1}×V_{j−1})}(U_{i−1} × CV_{j−1}) ⟶ R_{i+j−1}

The map h_{ij} is obtained from the push-out square:

[figure: U_{i−1} × V_{j−1}; p_{1}; V_{j−1}; p_{2}; U_{i−1}; U_{i−1} ∗ V_{j−1}]

(where p_{1} and p_{2} are projections) by h_{ij} = u ∗ v with

u : U_{i−1} ⟶^{f_{i−1}} A_{i−1} ⟶ A_{i−1} × B_{j} ↪ R_{i+j−1}

v : V_{j−1} ⟶^{g_{j−1}} B_{j−1} ⟶ A_{i} × B_{j−1} ↪ R_{i+j−1}

The previous remarks imply that we have a push-out diagram:

[figure: ⋁_{i+j=k} U_{i−1} ∗ V_{j−1}; ⋁_{i+j=k} D_{ij}; R_{k−1}; R_{k}]

Recall that D_{ij} ≃ ∗ and U_{i−1} ∗ V_{j−1} = Σ^{i−1}Z_{i−1} ∗ Σ^{j−1}W_{j−1} = Σ^{k−1}(Z_{i−1} ∧ W_{j−1}) fori + j = k.

This gives a cofibration sequence: Σ^{k−1}(Z_{i−1} ∧ W_{j−1}) ⟶ R_{k−1} ⟶ R_{k} for 0 < k ≤ n + m.

Remark. It is clear that the preceding results have analogues involving the ”milder” notion of cone-length, clX. For this invariant the same type of arguments as those used in proving Theorem 1.1 lead to:

If X is a retract of Y, then there exists ΣZ such that cl(X ∨ ΣZ)≤ clY and, in particular, clX≤ clY +1.

3 The rational case

By contrast with the precedent section here we will pursue an algebraic study of the cone-length of rational spaces. All objects in this section will be rational and all spaces will be of finite type.

We start by making more precise the definition we will use . We will denote by L(X) = L(x_{1}, …, x_{i}, …) the free Lie algebra on the set of generators X = {x_{i}}_{i∈I}.

Let (L, d) = (L(X), d) be a free dgL and let F= {F_{k}X} be an increasing filtration on X. We will say it is a differential filtration if:

a. 0 = F_{0}X, ⋃ F_{k}X = X

b. d(F_{n}X) ⊂ L(F_{n−1}X)

Clearly such a filtration induces one on L by F_{n}L = L(F_{n}X). For x ∈ L denote by fd(x) the filtration degree of x. Also denote by F (F, X) the length of F (this may also be infinite).

It is easy to see that ClL= min{F (F′, X′) : (L(X′), d′) ≃ (L, d) and F′ is a differential filtration on X′}.

Remark. As exemplified in [9] different systems of generators of the same dgL may admit differential filtrations of distinct length.

We begin by looking at some special results concerning the spaces of category 2.

A. If catX= 2 then ClX=catX. (This is the result of Felix and Thomas mentioned in the introduction.)

By Theorem 1.1 there is a bouquet of spheres Z = ⋁_{i} S^{t_{i}} with finitely many spheres in each dimension such that Cl(X ∨ Z)≤ 2. Consequently it will be enough to know:

Lemma 3.1 If Cl(X ∨ S^{n})= 2, then ClX= 2.

Proof.Let L = (L(G), d), G = {g_{i}}_{i∈I} be a dgL of the homotopy type of a Quillen model for X ∨ S^{n} and 0 ⊂ G_{1} ⊂ G_{2} = G a length 2 differential filtration. We may choose G in such a way that d(∑ u_{i}g_{i}) = 0 with u_{i} ∈ Q, u_{i} ≠ 0 implies g_{i} ∈ G_{1}. Clearly we have d(G_{1}) = 0 and d(G_{2}) ⊂ [G_{1}, G_{1}]. We know that H_{∗}(L(G), d) ≃ π_{∗+1}(X ∨ S^{n}) and H_{∗}(L(G)/[L(G), L(G)]) ≃ H_{∗+1}(X ∨ S^{n}). If h : L(G) ⟶ L(G)/[L(G), L(G)] is the projection, then h_{∗} : H_{∗}(L(G), d) ⟶ H_{∗}(L(G)/[L(G), L(G)]) is identified with the Hurewicz homomorphism.

Denote by α ∈ π_{n}(X ∨ S^{n}) the class of S^{n} ↪ X ∨ S^{n}. Remark that h_{∗}(α) ≠ 0. Choose x ∈ L of minimal filtration degree among the elements verifying dx = 0 and h_{∗}([x]) = h_{∗}(α).

Consider L_{1} = (L ∗ L(y), d_{1}), d_{1} = d on G, d_{1}(y) = x. Notice that L_{1} represents the homotopy type of X. If fd(x) = 1, then define fd(y) = 2 and we get an obvious length 2 differential filtration on G ∪ {y}.

Suppose fd(x) = 2. Write x = g+m where g = ∑ a_{i}g_{i}, a_{i} ∈ Q, m ∈ [L, L]. If fd(m) = 1 we obtain d(g) = 0 which implies fd(g) = 1 and this contradicts the choice of x. Hence fd(g) = 2 =fd(m). Let L_{2} = L_{1}/(y, dy), then, as the ideal (y, dy) is acyclic, L_{1} ≃ L_{2}. It is easy to see that L_{2} is free. Indeed a way to think about L_{2} is to choose a g_{h} ∈ G with a_{h} ≠ 0 and replace it by g as a generator of L. That is, let G′ = (G−{g_{h}})∪{g, y}. Then L_{2} ≃ L(G′ −{g}) and the projection p : L_{1} ⟶ L_{2} corresponds to a map sending t → t for t ∈ G′, g → −m and y → 0. This map induces also a filtration on L_{2} by fd(p(t)) =fd(t). It is clearly possible to choose the replaced g_{h} such that fd(g_{h}) = 2. In this case fd(g_{h}) =fd(g) =fd(m) = 2 and it is straightforward to check that this length 2 filtration is differential.

B. A special ”short” model for spaces of category 2.

Recall the functor C : DGL ⟶ DGCC_{0} where DGL is the category of dgL’s and DGCC_{0} is the category of differential, graded, cocomutative, connected coalgebras (dgc) [12]. For a dgl, (L, d), C(L) is, as a graded vector space, Λ(sL) ( s is the suspension; Λ(X) is the free, graded, commutative, connected algebra on X).The coproduct is induced by the coalgebra structure of the tensorial coalgebra. On C(L) the differential δ is described by: δ = δ_{I} + δ_{E}, δ_{I}(sx) = −sd(x), δ_{E}(sx_{1} ∧ sx_{2}) = (−1)^{|x_{1}|}s[x_{1}, x_{2}].

Denote by {C_{i}(L)} the primitive filtration on C(L).

Lemma 3.2 If a finite type, connected dgL, L, admits a differential filtration of length less or equal than 2, then there is a dgc C ↪^{i} C(L) such that C ⊆ C_{2}(L) and i is a quasi-isomorphism.

Proof.Again let L = (L(G), d) and let the differential filtration be 0 ⊂ G_{1} ⊆ G_{2} = G. If it is of length 1 assume G_{2} = G_{1}.Remark that in this case, as d is trivial, if we take Q(G) as a dgc with trivial differential and coproduct, the inclusion Q(G) ↪ C(L) is a quasi-isomorphism.

Suppose now the length of the filtration to be 2. We can assume that d is decomposable (dG ⊂ [L, L]). Indeed d(G_{1}) = 0 and if g ∈ G_{2} − G_{1}, then dg = ∑ a_{i}g_{i} + m with g_{i} ∈ G, m ∈ [L, L] and fd(g_{i}) = fd(m) = 1. If some a_{s} ≠ 0, by a standard method we may eliminate g and g_{s} from L and it is easy to see that on the resulting dgL we can define (as in the proof of the previous Lemma) a length 2 differential filtration. Because of this assumption each of the elements of G will represent a homology class of C(L) and, also, each such homology class is represented by an element of G. Denote by |x| the degree of x ∈ L, we will indicate this degree by an upper index.

Suppose that for k ≤ n we have proved that L_{k} = L(G_{1} ∪ (G_{2})^{≤k}, d) satisfies our claim. That is, there exists C_{k} ⊂ C_{2}(L_{k}) and the inclusion C_{k} ⊂ C(L_{k}) is a quasi-isomorphism. We will show that for any g ∈ G_{2} − G_{1}, |g| = k + 1, there is a dgc, C” ⊂ C_{2}((L_{k} ∗ L(g), d)), containing both C_{k} and sg and such that the inclusion C” ⊂ C((L_{k} ∗L(g), d)) is a quasi-isomorphism.

The construction of C” goes as follows:

Let dg = m. We want first to construct a dgc C′ such that: C′ ↪ C(L_{k}) induces a quasi-isomorphism, sm ∈ C′, C_{k} ⊂ C′ and C′ ⊂ C_{2}(L_{k}). If sm ∈ C_{k}, then C′ = C_{k}.

If sm ∉ C_{k}, express m = ∑ u_{i}[a_{i}, b_{i}], a_{i} ∈ L, b_{i} ∈ L, u_{i} ∈ Q. We know fd(a_{i}) =fd(b_{i}) = 1. Let J be such that s[a_{i}, b_{i}] ∉ C_{k} iff i ∈ J. Take C′_{1} = C_{k} ⊕ ⨁_{i∈J}(Q(s[a_{i}, b_{i}])). But s[a_{i}, b_{i}] represents a nontrivial homology class in C′_{1}. In fact, the s[a_{i}, b_{i}]’s, i ∈ J, will represent the kernel of H_{∗}(C′_{1} ↪ C(L_{k})). Indeed, in C(L), s[a_{i}, b_{i}] = δ((sa_{i}) ∧ (sb_{i})). Consequently, we have to add to C′_{1} the elements (sa_{i}) ∧ (sb_{i}). But for preserving the coalgebra structure we must have there also the sa_{i}’s and sb_{i}’s (for i ∈ J). So let C′_{2} = C′_{1} + ∑_{i∈J} Q(sa_{i} ∧ sb_{i}) + ∑_{i∈J} Q(sa_{i}) + ∑_{i∈J} Q(sb_{i}).

Remark that the possible kernel of H_{∗}(C′_{2} ↪ C(L_{k})) appears in degree at most max{|a_{i}|, |b_{i}| : i ∈ J} < |[a_{i}, b_{i}]| = |m|. That means that our procedure (applied this time to sm) moves the kernel of the inclusion C′_{∗} ⊂ C(L_{k}) downwards (with respect to |.|). Using it repeatedly, because of the finite type conditions, we will get C′ with the required properties.

Take C” = C′ ⨁ Q(sg) with d(sg) = sm.

It is clear that the construction applied to g can now be applied to the other elements of degree k + 1. Consequently, our claim will follow by induction.

We start now proving the Theorem 1.5 .

First the inequality ClX≤nilX.

Recall the functor L_{∗}: L_{∗} = L ∘ # where L : DGCC_{0} ⟶ DGL applied to a dgc, (C, δ), gives L(C) = (L(s^{−1}C), d) with d = d_{1} + d_{2}, d_{1}(s^{−1}c) = −s^{−1}(δc), d_{2}(s^{−1}c) = −1/2 ∑(−1)^{|c′_{i}|}[s^{−1}c′_{i}, s^{−1}c_{i}”] where c ∈ C ,\overline{Δ}c = ∑ c′_{i} ⨂ c_{i}” [12]; # is a dualization functor # : DGA_{0} ⟶ DGCC_{0}.

The idea for the proof comes from a simple remark: Let (ΛX, d) be a free dga with d decomposable. Choose a vector space basis, B, for ΛX which consists of monomials and take all duals with respect to this basis. Suppose that for b ∈ B we define fd(s^{−1}b^{∗}) = length of b as a monomial. It is easy to check that this filtration is differential.

The outline of the proof is the following :

1. Look for a certain type of filtration on a free Lie algebra of finite type which can be easely transformed into a differential one without increasing its length.

2. Show that for any finite type connected dga A, with \overline{A}^{(n+1)} = 0 , we can get on L_{∗}(A) a filtration like the one discussed at the previous point with length ≤ n.

3. Show that if nilX≤ n , then X can be represented by a finite type, simply connected dga A′ with \overline{A′}^{(n+1)} = 0.

In this case L_{∗}(A′) is a Quillen model of X and the previous two steps give a differential filtration of length less than n on L_{∗}(A′).

Proposition 3.3 Suppose L = L(X), X = {x_{i}}_{i∈I} of finite type. If F= {F_{∗}X} is an increasing filtration on X with the properties:

a. 0 = F_{0}X, ∪F_{i}X = X

b. For each x_{i} write dx_{i} = ∑ a_{i}^{h}x_{h} + m_{i}, where i,h ∈ I and m_{i} ∈ [L, L]. Assume :

fd(x_{i}) > fd(m_{i})  (4)

Then ClL≤ F (F,L).

Proof.It is clearly enough to find a free dgL, L′, and a quasi-isomorphismf : L ⟶ L′ such that f(X) contains a free system of generators of L′ and with the property that if we define

fd(f(x_{i})) = fd(x_{i})  (5)

then the filtration induced on L′ is differential.

The dgL L′ (together with the morphism f) is obtained by iterating the following procedure.

Suppose that for each x_{i} with i ∈ I and |x_{i}| < k we have fd(x_{i}) >fd(dx_{i}) and suppose there is an x ∈ X which verifies |x| = k and fd(x) ≤fd(dx) (if for all k no such x exists we are done).

We will construct a dgL, L_{1}, together with a quasi-isomorphismf_{1} : L ⟶ L_{1} such that:

1. The set f_{1}(X) contains a free system of generators of L_{1}, Y.

2. The filtration induced on L_{1} by (5) satisfies conditions a. and b. .

3. For some x_{α} with fd(x_{α}) ≤fd(dx_{α}), |x_{α}| = k, we have f_{1}(x_{α}) = 0 and for y ∈ Y, |y| < k, fd(y) >fd(dy).

Consider the set A = {i ∈ I : |x_{i}| = k, fd(x_{i}) ≤ fd(dx_{i})}. Recall that for each u ∈ I we have dx_{u} = ∑ a_{u}^{h}x_{h} + m_{u}. Let α ∈ A be such that fd(m_{α}) = min{fd(m_{j}) : j ∈ A}. Select also a β such that fd(x_{β}) = max{fd(x_{v}) : a_{α}^{v} ≠ 0}. Let L_{1} = L/(x_{α}, dx_{α}).

Let f_{1} : L ⟶ L_{1} be the projection. Denote y_{i} = f_{1}(x_{i}) for i ∈ I′ = I − {α, β}. Notice that, as the ideal (x_{α}, dx_{α}) is acyclic, f_{1} is a surjective quasi-isomorphism; it also satisfies :

1. f_{1}(x_{β}) = (−f_{1}(m_{α}) − ∑_{h≠α} a_{α}^{h}y_{h})/a_{α}^{β}.

2. f_{1}(x_{α}) = 0

Clearly Y = {y_{i}} is a free system of generators for L_{1}.

Denote m′_{i} = f_{1}(m_{i}). Let i ∈ I′. We have dy_{i} = ∑_{h∈I′} a_{i}^{h}y_{h} + m′_{i} + a_{i}^{β}(−m′_{α} − ∑_{h≠α} a_{α}^{h}y_{h})/a_{α}^{β}. Define now fd(y_{i}) =fd(x_{i}) .

We have to check if b. is satified. First notice fd(x_{β}) ≥fd(m_{α}+∑_{h≠α} a_{α}^{h}x_{h}) because fd(dx_{α}) ≥fd(x_{α}) ≥fd(m_{α}) and because of the choice of β. Consequently, fd(f_{1}(x_{β})) ≤fd(x_{β}). This implies fd(m′_{i}) ≤fd(m_{i}) for all i ∈ I′. Thus for all such i’s we have fd(y_{i}) ≥fd(m′_{i}). This proves b. when a_{i}^{β} = 0. When a_{i}^{β} ≠ 0 we need also fd(y_{i}) ≥fd(m′_{α}). Clearly |x_{i}| = k. There are two possibilities: first fd(x_{i}) >fd(dx_{i}); if this happens we have fd(y_{i}) =fd(x_{i}) >fd(dx_{i}) ≥fd(x_{β}) (because a_{i}^{β} ≠ 0) ≥fd(m_{α}) =fd(m′_{α}). We are left with the case fd(x_{i}) ≤fd(dx_{i}). But, by the choice of α, we have fd(m′_{i}) =fd(m_{i}) ≥fd(m_{α}).

Now we will see what kind of filtration we need on a dga A such that on L_{∗}(A) we have a filtration of the type discussed in the previous proposition.

Lemma 3.4 Let A be a finite type, connected dga and let B = {g_{i}}_{i∈I} be a basis (as a linear space) for A containing 1 ∈ Q. Suppose there is an increasing filtration on B, {B_{i}}, 0 ≤ i ≤ n, such that B_{0} = 1, B_{n} = B and for each g_{i} ∈ B we have

fd(g_{i}) > max{fd(g_{h}), fd(g_{k})}  (6)

whenever u^{i} ≠ 0 in g_{h}g_{k} = ∑ u^{s}g_{s}, where both g_{h} and g_{k} are different from 1. Then on L_{∗}(A) we can find a filtration with the properties in the previous proposition and having length n.

Proof.We dualize everything with respect to B. Denote by x_{i} = s^{−1}(g_{i}^{∗}) for g_{i} ≠ 1. The set X = {x_{i}}_{i∈I} is a set of free generators for L_{∗}(A). Define fd(x_{i}) =fd(g_{i}). Checking that this is the right type of filtration is straightforward (of course, we add 0 as the 0-filtration).

The second step is concluded by :

Lemma 3.5 Let A be a finite type, connected dga. If \overline{A}^{(t+1)} = 0, then there is a basis and a filtration of length at most t with the properties in the previous lemma.

Proof.Take {x_{1}, … x_{l}, …} a minimal system of generators for A as an algebra. Denote by A^{n} the elements of degree n in A. The construction of our basis will be done by induction. Define fd(1) = 0.

Suppose that for A^{t} with t < n we have constructed a basis consisting of monomials and a filtration satisfying the condition in the previous lemma and such that for the elements of this basis, b_{i}^{t} , i ∈ I_{t}, t < n, we have fd(b_{i}^{t}) =length of b_{i}^{t} as a monomial in the x_{i}’s. When we speak about the length of a monomial b ∈A we understand the longest expression of b written as a monomial in the x_{i}’s. Notice that, because of the finite type assumption, there are just finitely many such expressions for b, moreover, because \overline{A}^{(t+1)} = 0, all such expressions are shorter or equal than t. In this way the length is well defined (otherwise it may not be because A is not free).

Now A^{n} is generated (as a linear space) by monomials of the type m = Πx_{i}^{h_{i}}, h_{i} ∈ N and ∑ h_{i}|x_{i}| = n. Choose the longest such monomial, that is the one for which l(m) = ∑ h_{i} is maximal. Denote by b_{1}^{n} this monomial and define fd(b_{1}^{n}) = l(b_{1}^{n}). It is clear that the condition (6) is satisfied in the case of b_{1}^{n}. Now suppose we have constructed b_{1}^{n}, …, b_{k}^{n} linearly independent elements in A^{n}, monomials, such that fd(b_{i}^{n}) = l(b_{i}^{n}) ≥ l(b_{i+1}^{n}) =fd(b_{i+1}^{n}) and such that condition (6) is satisfied for each i ≤ k.

Take b_{k+1}^{n} to be the longest monomial linearly independent with respect to {b_{≤k}^{n}}. Define fd(b_{k+1}^{n}) = l(b_{k+1}^{n}).

Let’s check our critical condition. Take b_{u}^{s}, b_{v}^{m} with s + m = n. Express b_{u}^{s}b_{v}^{m} = ∑_{i=1}^{k+1} h^{i}b_{i}^{n}+g where g is linearly independent with respect to {b_{≤k+1}^{n}}, h^{k+1} ≠ 0. We have l(b_{u}^{s}b_{v}^{m}) > max{l(b_{u}^{s}), l(b_{v}^{m})} = max{fd(b_{u}^{s}), fd(b_{v}^{m})}. Thus if fd(b_{k+1}^{n}) = l(b_{k+1}^{n}) ≥ l(b_{u}^{s}b_{v}^{m}) we have nothing more to prove. If this is not the case, because of the choice of b_{k+1}^{n}, that means that b_{u}^{s}b_{v}^{m} is linearly dependent with respect to (b_{1}^{n}, …, b_{k}^{n}). But that leads to a contradiction because it implies that (b_{1}^{n}, …, b_{k}^{n}, b_{k+1}^{n}) are linearly dependent.

The third step is given by the following construction.

Proposition 3.6 Let A and B be two dga’s, A of finite type and simplyconnected and B augmented such that there is a morphism f : A ⟶ B of augmented algebras inducing an isomorphism in cohomology. If \overline{B}^{(n+1)} = 0, n > 0, then there exists a simply-connected finite type dga A′, \overline{A′}^{(n+1)} = 0, and a factorization of f:

A ⟶^{g} A′ ⟶^{f′} B

with g being a quasi-isomorphism.

Proof.Clearly f factors A ⟶^{g} A/ \overline{A}^{(n+1)} ⟶^{h} B. If n = 1, then we have an algebra isomorphism ⊕_{i}Qx_{i} ≃ A/ \overline{A}^{2} for a certain set of x_{i}’s; the sum is finite in each dimension. Let K ⊆ ⊕Qx_{i} be the ideal generated by a system of representatives for KerH^{∗}(h). Obviously, we may consider A/ \overline{A}^{2} = K ⊕ ((A/ \overline{A}^{2})/K). It follows that we get a map A ⟶ (A/ \overline{A}^{2})/K ⟶ B which is a quasi-isomorphism.

For the case when n > 1 we will use the next lemma.

Lemma 3.7 Let G be a connected, finite type dga. Suppose f : G ⟶ H is a quasi-isomorphism which factors G ⟶^{p} G/J ⟶^{f′} H where J is an ideal of G. There exists a twisted tensor product G/J ⊗_{d} ΛY and maps such that the diagram below is commutative:

[figure: G; f; G/J; f′; G/J ⊗_{d} ΛY; f”; p; h; H]

with h ∘ p, f” quasi-isomorphisms. Moreover the connectivity of Y is at least that of J − 2 and Y is of finite type.

Proof.We construct ΛY by induction. First let Y_{0} = 0 and f_{0} = f′. Clearly H^{0}(f_{0}) is an isomorphism and H^{∗}(f_{0}) is epi for ∗ ≥ 0. Suppose f_{i} : G/J ⊗_{d} ΛY ⟶ H constructed such that H^{∗}(f_{i}) is an epimorphism for all ∗ and it is an isomorphism for j ≤ i. Let {x_{i}}, x_{i} ∈ (G/J ⊗_{d} ΛY_{i})^{i+1} be a system of representatives for a basis of KerH^{i+1}(f_{i}). Let Y_{i+1} = Y_{i} ⊕ {s^{−1}x_{i}}. Take on G/J ⊗_{d} ΛY_{i+1} the differential which extends the differential on G/J ⊗_{d} ΛY_{i} by d(s^{−1}x_{i}) = x_{i}. Extend f_{i} to f_{i+1} : G/J ⊗_{d} ΛY_{i+1} ⟶ H by f_{i+1}(s^{−1}x_{i}) = t_{i}, where t_{i} ∈ H verifies dt_{i} = f_{i}(x_{i}). Passing to the limit we get our claim. Remark also that the connectivity of Y is one less the connectivity of f′ and this one is at least the connectivity of J-1. As KerH^{∗}(f_{i}) is finite dimensional for each ∗, Y is of finite type.

Return now to the proposition. We apply repeatedly the lemma. We

start with A ⟶^{g} A/ \overline{A}^{(n+1)} ⟶^{h} B. Let A_{1} = A/ \overline{A}^{(n+1)} ⊗_{d} ΛY^{1} given by

the lemma together with the quasi-isomorphisms A ⟶^{t_{1}} A_{1} ⟶^{f_{1}} B. Now f_{1} factors again A_{1} ⟶^{g_{1}} A_{1}/ \overline{A}_{1}^{(n+1)} ⟶^{h_{1}} B and we may apply again the lemma. At the i-th step we get A ⟶^{t_{i}} A_{i} ⟶^{f_{i}} B with t_{i} and f_{i} quasi-isomorphisms and A_{i} = A_{i−1} ⊗_{d} ΛY^{i}. Clearly, A′ = lim A_{i} is nilpotent of the right order and we have a diagram like in the statement. To conclude, we just have to see that A′ is of finite type. For that it is enough to show that the connectivity of the Y^{i} ’s is strictly increasing. By the lemma the connectivity of Y^{i} is at least that of \overline{A}_{i−1}^{(n+1)}-2. But A_{i−1} = A_{i−2} ⊗_{d} ΛY^{i−1}. Hence the connectivity of \overline{A}_{i−1}^{(n+1)} is at least 4+that of Y^{i−1}, (n > 1).

Remarks. a. If A in the previous proposition is the minimal model of a simply-connected, finite type space X we get a series of ”obstructions” in order for X to be of nil≤ n. The first step is to ask for the projection g_{0} = g : A ⟶ A/ \overline{A}^{(n+1)} to admit a retract up to homotopy (this is equivalent to catX≤ n). To formulate the n + 1 obstruction suppose g_{m} : A ⟶ A_{m}/ \overline{A}_{m}^{(n+1)} constructed and suppose it admits a retract up to homotopy. Then we may construct A_{m+1} and a map g_{m+1} : A ⟶ A_{m+1}/ \overline{A}_{m+1}^{(n+1)}. The m + 1 obstruction requires for g_{m+1} to admit a retract up to homotopy.

b. The inequality nil≥Cl together with the next lemma gives another proof (in the rational context) of the fact that Cl≤cat+1.

Lemma 3.8 If F ⟶ E ⟶ B is a fibration of finite type, simply-connected spaces, then nil(E/F)≤ nilB.

Proof.Suppose M(B) ⟶ B is a quasi-isomorphism with B simply-connected of finite type and \overline{B}^{(n+1)} = 0. We may represent the inclusion F ↪ E by a projection p : B ⊗_{d} F ⟶ F where F is a minimal model for F. Clearly Kerp ⊕ Q represents E/F; on the other hand Kerp = \overline{B} ⊗_{d} F, hence it is nilpotent of the wanted order.

Applying this to the fibration ΩE_{n}(ΩX) ⟶ X ⟶ B_{n}(ΩX), when n =catX, we get that nil(X ⋁ ΣΩE_{n}(ΩX)) ≤ n. Along with Cl≤nil, this implies cat+1 ≥ Cl.

We pass now to the inequality nil≤Cl. The main part in proving it consists in showing that we can model a map ΣZ ⟶ X in term of ”short” models.

Proposition 3.9 Let A be a free, simply-connected dga of finite type and let f : A ⟶ A_{1} be a quasi-isomorphism such that A_{1} is simply-connected of finite type and \overline{A}_{1}^{(n+1)} = 0. Let B be an augmented dga with \overline{B}^{2} = 0. If g : A ⟶ B is a morphism of augmented dga’s, then there exists a dga A_{2} of finite type, simply-connected with \overline{A}_{2}^{(n+2)} = 0 and maps making the next diagram commutative:

[figure: A; g; A_{2}; \overline{g}; A_{1}; h; e; B]

where both h and e are quasi-isomorphisms and e ∘ h = f.

Proof.Write A = A_{0} ⊗ F with A_{0} minimal and F free acyclic. If A_{0} = (Λ{x_{i}}, d) let A′ = (Λ{s^{−1}x_{i}}, 0). By the standard construction of a K.S. model for the multiplication μ : A ⊗ A ⟶ A ([3 ]) we get a twisted tensor product T = (A ⊗ A) ⊗_{d} A′ which fits into a commutative diagram:

[figure: A ⊗ A; μ; (A ⊗ A) ⊗_{d} A′; ψ; A]

where ψ is a quasi-isomorphism. Clearly, we have inclusions i_{1} : A ⟶ T, i_{1}(x) = x ⊗ 1 ⊗ 1, i_{2} : A ⟶ T, i_{2}(x) = 1 ⊗ x ⊗ 1; both of them are quasiisomorphisms. Recall also d(s^{−1}x_{i}) = x_{i}⊗1⊗1−1⊗x_{i}⊗1 mod (\overline{A ⊗ A ⊗ A′} ^{2}) Let p be the projection (A ⊗ A) ⊗_{d} A′ ⟶ A ⊗_{d} A′ which sends 1 ⊗ x ⊗ 1 to zero for x ∈ A.

The inclusion i_{1} makes T into an A module. Let K = A_{1} ⊗_{A} T = (A_{1} ⊗ A) ⊗_{d} A′. Let p_{1} = A_{1} ⊗ p, ψ_{1} = A_{1} ⊗ ψ. Remark that ψ_{1} : (A_{1} ⊗ A) ⊗_{d} A′ ⟶ A_{1} is a quasi-isomorphism. The morphism g makes B into an A module. Consider K as an A module via the inclusion i′_{2} given by the composition of the map T ⟶ K with i_{2}. Let S = B ⊗_{A} K = (A_{1} ⊗B)⊗_{d} A′. Denote p′_{1} = B ⊗ p_{1}.

We get the following diagram:

[figure: A; i′_{2}; B; i_{2}”; g; K; ψ_{1}; S; g′; A_{1}]

Here both i′_{2} and i_{2}” are quasi-isomorphisms , g′ : K = (A_{1}⊗A)⊗_{d}A′ ⟶ (A_{1} ⊗ B) ⊗_{d} A′ = S is the obvious map and ψ_{1} ∘ i′_{2} = f.

We also have the following commutative diagram:

[figure: 0; (A_{1} ⊗ \overline{A}) ⊗_{d} A′; g”; (A_{1} ⊗ A) ⊗_{d} A′; g′; A_{1} ⊗_{d} A′; 0; j; p_{1}; 0; (A_{1} ⊗ \overline{B}) ⊗_{d} A′; (A_{1} ⊗ B) ⊗_{d} A′; A_{1} ⊗_{d} A′; 0; j′; p′_{1}]

where both short sequences are exact. It is easy to see that A_{1} ⊗_{d} A′ is acyclic, hence both j and j′ induce isomorphisns in cohomology in positive degrees. Let B_{1} = Ker(j′) ⊕ Q, R = Ker(j) ⊕ Q. As p_{1} ∘ i′_{2} = 0, p′_{1} ∘ i_{2}” = 0, i′_{2} and i_{2}” factor respectively through R and B_{1}.

Notice now that both B and B_{1} are algebras with trivial multiplication. The inclusion h_{1} : B ⟶ Q ⊕ ((A_{1} ⊗ \overline{B}) ⊗ A′) = B_{1} which is a quasiisomorphism gives an algebra splitting B_{1} = B ⊕ W. The elements in W are sums of monomials a ⊗ b ⊗ c with a ∈ A, b ∈ \overline{B}, c ∈ A′ at least one of a or c not being a constant. This shows that the splitting is also differential because for such a monomial we have d(a ⊗ b ⊗ c) ∈ W. It follows that the projection B ⊕ W ⟶ B is a dga morphism. Let u : B_{1} ⟶ B be this projection. Clearly, it will also be a quasi-isomorphism. Define g′′′ = u ∘ g”. We obtain a commutative diagram:

[figure: A; s; B; h_{1}; g; R; ψ_{2}; g′′′; B_{1}; u; g”; A_{1}; B]

Here ψ_{2} = ψ_{1} ∘ j, j ∘ s = i′_{2}, ψ_{2} ∘ s = f, s is a quasi-isomorphism and u ∘ h_{1} = id.

Lemma 3.10 There is an acyclic ideal J ⊂ R such that g′′′ and ψ_{2} vanish on J and (\overline{R/J})^{(n+2)} = 0.

This result implies immediately the factorization of both g′′′ and ψ_{2} through A_{2} = R/J. Proof of lemma. Consider the short exact sequence:

0 ⟶ (A_{1} ⊗ \overline{A}) ⊗_{d} A′ ⟶ (A_{1} ⊗ A) ⊗_{d} A′ ⟶ A_{1} ⊗_{d} A′ ⟶ 0

Let F_{p}(A_{1}) = \overline{A_{1}}^{p}, F_{q}(A) = \overline{A}^{q}, F_{n}((A_{1} ⊗ A) ⊗_{d} A′) = ∑_{p+q=n} F_{p}(A_{1}) ⊗ F_{q}(A)⊗A′, F_{n}(A_{1}⊗_{d}A′) = F_{n}(A_{1})⊗A′ and F_{n}(R) = F_{n}((A_{1}⊗A)⊗_{d}A′) ⋂ R. All of these are differential filtrations. They induce spectral sequences and at the E_{0} term we get a short exact sequence:

0 ⟶ E_{0}(R) ⟶ E_{0}(A_{1}) ⊗ E_{0}(A) ⊗ A′ ⟶^{h} E_{0}(A_{1}) ⊗ A′ ⟶ 0

The internal differential in A′ is null. It follows that, when we pass to E_{1}, the map H^{∗}(h) will remain surjective. Hence at the E_{1} term we get a short exact sequence:

0 ⟶ E_{1}(R) ⟶ E_{1}(A_{1}) ⊗ E_{1}(A) ⊗ A′ ⟶^{H^{∗}(h)} E_{1}(A_{1}) ⊗ A′ ⟶ 0

Now E_{1}(A) = E_{1}(A_{0}) ( recall A = A_{0} ⊗ F with A_{0} minimal and F acyclic). Notice that E_{2}^{>n,∗}(A_{1} ⊗_{d} A′) = 0. Also E_{2}^{>n,∗}((A_{1} ⊗ A) ⊗_{d} A′) = 0. This last fact follows because we have an obvious isomorphism: (E_{1}(A_{1}), d_{1}) ⊗ H ≃ ((E_{1}(A_{1}) ⊗ E_{1}(A_{0})) ⊗_{d} A′, d_{1}) with H free acyclic, H = (A_{0} ⊗_{d} A′, d_{1}) (as algebras E_{1}(A_{0}) = A_{0}; the isomorphism coresponds to just a change of generators).

The induced long exact sequence of the E_{2} terms shows E_{2}^{>n+1,∗}(R) = 0. Let I = F_{n+2}(R) and let {u_{i}} be a minimal set of elements of F_{n+1}(R) such that :

1. The u_{i}’s survive at the E_{1}^{n+1} term of the spectral sequence giving the classes [u_{i}].

2. {d_{1}([u_{i}])} spans the image of d_{1} in E_{1}^{n+2}.

In this case J = I+∑ Qu_{i} is an acyclic ideal of R. Moreover (\overline{R/J})^{n+2} = 0 and both ψ_{2} and g′′′ are null on J.

Remark. The spectral sequence argument used above is analoguous to one used by Felix and Halperin in their study of the Ganea spaces. By a similar method it is possible to show that if F ⟶ X ⟶ Y is a fibration of finite type, simply-connected spaces, then nil(X/F)≤nilX+1. This strongly suggests CatX=nilX.

Corollary 3.11 If ΣZ ⟶^{f} X ⟶ Y is a cofibration sequence of finite type, simply-connected, rational spaces, then nilY ≤nilX+1.

Obviously this implies nil≤Cl. Proof.Because ΣZ is a formal space we may represent f by a map f′ : M(X) ⟶ H = H^{∗}(ΣZ). In a standard way we may find a free dga M′, quasi-isomorphic to M(X) and a surjective map f′′ : M′ ⟶ H which represents also f. By applying the previous proposition we get a commutative diagram:

[figure: M′; h; H; id; f′′; A; H; g]

where A is simply connected of finite type, h is a quasi-isomorphism and \overline{A}^{(n+2)} = 0; Q ⊕ Ker(g) represents Y and it is nilpotent of the right order.

Remark. The ideal proof for the inequality nil≤Cl would consist in an extension of Lemma 3.2 to the case of Lie algebras with differential filtrations of arbitrary length. It is not clear if such a generalization is true.

References

[1] M.Clapp, D.Puppe -The generalized Lusternik-Schnirelmann category of a product space. Trans. of the A.M.S. , 321 (1990), 525-532

[2] J.P.Doeraene -LS-catégorie dans une catégorie à modèles. Thesis, Université Catholique de Louvain (1990)

[3] Y.Félix -La dichotomie elliptique-hyperbolique en homotopie rationelle. Astśrisque 176, Soc.Math.de France (1989)

[4] Y.Félix, S.Halperin -Rational L.S. category and its applications. Trans. of the A.M.S. , 273 (1982), 1-37

[5] Y.Félix, J.C.Thomas -Sur la structure des espaces de L.S. catégorie deux. Ill. J. of Math, 30 (1986), 574-593

[6] T.Ganea -Lusternik-Schnirelmann category and strong category. Ill. J. of math, 11 (1967), 417-427

[7] K.Hess, J.-M.Lemaire -On L.S. category in model categories. Preprint (1991)

[8] I.M.James -On category in the sense of Lusternik-Schnirelmann. Topology 17 (1978), 331-348

[9] J.-M.Lemaire, F.Sigrist -Sur les invariants d’homotopie rationelles liés à la L.S. catégorie. Comm. Math. Helv. 56 (1981), 103-122

[10] L.Ljusternik, L.Schnirelmann -Methodes topologiques dans les problèmes variationels. Hermann, Paris, (1934)

[11] F.Takens -The Lusternik-Schnirelmann categories of a product space. Comp.Math. 22 (1970), 175-180

[12] D.Tanré -Homotopie rationelle: Modéles de Chen, Quillen, Sullivan. Lecture Notes in Math. 1025 (1983)