L.S.-categorical Sections

Sections L.S.-catégoricales

Octavian Cornea University of Toronto Toronto, ON M5S 1A1

September 23, 1994

Abstract

Let X be a finite type, simply connected CW -complex. If the Lusternik-Schnirelmann category of the localizations of X at each prime is bounded from above by n, then the category of X is bounded from above by 2n+1; if X is finite, this upper bound can be improved to 2n.

1 Introduction

We will work here inside the pointed category of finite type, simply connected spaces with the homotopy type of a CW -complex.

The starting point of this paper is a question of Charles McGibbon [11]: Is catX≤ n a generic property in the sense of the Mislin genus ? Recall that the Mislin genus of a space is the set of all homotopy types of spaces which, when localized at any prime, coincide with the localization at that prime of the given space [13]. In particular, McGibbon conjectured that if the Lusternik-Schnirelmann category of the localization of a space at any prime is less or equal to n, then the category of the space is, at least, finite. It is this conjecture that we prove here.

If P is a set of primes we denote by P′ the complementary set of primes; X_{P} is the localization of X at P; X_{(0)} is the rationalization of X.

The results of this paper imply that:

1. For a finite space X:

a. There is a finite set of primes P such that cat(X_{P′})=cat(X_{(0)})

b. If cat(X_{(p)})≤ n for each prime p, then cat(X)≤ 2n.

2. For X of finite type if cat(X_{(p)})≤ n for each prime p, then cat(X)≤ 2n + 1.

The second point above is the main aim of the paper; it is deduced from the point 1 b with the help of the next result:

3. Let X be a CW -complex and let X^{(n)} be its n-th dimensional skeleton. We have cat(X^{(n)})≤cat(X)+1.

The tools for proving 1 b, which is clearly central to our approach, are provided by a close look at one of the basic constructions in the Lusternik-Schnirelmann theory. Recall first the definition of the L.S.- category of a space, X, [9] : cat(X)≤ n iff there is a covering of X by n + 1 open subsets each contractible in X, equivalently, iff the diagonal X ⟶ X^{(n+1)} factors, up to homotopy, through the inclusion of the fat wedge T^{n+1}X ⟶ X^{(n+1)}, or once more, iff the n-th fibration in the Milnor classifying construction [12], applied to ΩX: E_{n}ΩX ⟶ B_{n}ΩX ⟶^{p_{n}} X has a homotopy section. This type of section will be called a categorical section. The Milnor fibrations above appear as a particular case of the following construction due to Ganea [6]: given a fibration F ⟶ E ⟶ B define inductively the fibrations E_{n}(E, F) ⟶ G_{n}(E, F) ⟶ B by taking E_{0}(E, F) = F, G_{0}(E, F) = E, G_{n+1}(E, F) the homotopy cofibre of the inclusion E_{n}(E, F) ↪ G_{n}(E, F) and E_{n+1}(E, F) the homotopy fibre of the obvious projection p_{n+1} : G_{n+1}(E, F) ⟶ B. A formula of Ganea shows that E_{n+1}(E, F) ≃ E_{n}(E, F) ∗ ΩB. The classical case is obtained for the path-loop fibration. Notice that we denote E_{n}(∗, ΩB) = E_{n}ΩB and G_{n}(∗, ΩB) = B_{n}ΩB.

Our study of this construction was initiated by a natural question suggested to the author by John Moore: given two maps s_{1}, s_{2} : X ⟶ E such that when composed with the projection on B they become homotopic by a homotopy h : X × I ⟶ B, what is the smallest k such that after composing s_{1} and s_{2} with the inclusion i^{k} : E ⟶ G_{k}(E, F) they become homotopic by a homotopy that covers h ? As consequences of the results presented in the following we get:

4. The number k as above is at most cat(X).

5. There are examples where the least k is exactly cat(X) even when the initial fibration is the path-loop fibration.

Remark. For the path-loop fibration and h constant to the identity, point 4 above follows also from a result of Ganea [6]. The line of proof that we use here is different from that of Ganea.

The natural context for analyzing this problematic is that suggested by some ideas of Israel Berstein (as they were communicated to the author by Peter Hilton, see also [1]): for our space X, instead of considering the L.S.-category simply as a numerical invariant, focus on the finer structure consisting of a categorical section for X.

In the second section we will formalize these ideas by introducing the notion of n-B(erstein)-structures and prove, in this setting, the results relevant for 1, 2, 3 and 4. In the third section we discuss some other properties of the n-B-structures with a special emphasis on their behaviour with respect to push out squares. This is applied to prove 5.

The original question of McGibbon remains open. We believe that 1 b is, in fact, valid for infinite complexes too.

It turns out that the notions of cone-length and cone-decomposition are very useful in this study. A space X is an n-cone if there is a sequence of spaces X_{i} , 0 ≤ i ≤ n such that X_{0} = ∗, X_{i+1} = X_{i} ⋃ CZ_{i} for 0 ≤ i < n and some spaces Z_{i} (here CA is the cone over A). An n-cone decomposition of (the homotopy type of) X is a sequence of cofibration sequences Z_{i} ⟶ X_{i} ⟶ X_{i+1} with 0 ≤ i < n, X_{0} ≃ ∗, X_{n} ≃ X. The cone-length of X, Cl(X), is the least n such that there is an n-cone homotopy equivalent to X. We will use here the techniques developed for handling cone decompositions in [3] and [5]. In particular, it was shown there that, by adding in the definition of the cone-length the requirement that the spaces Z_{i} are i-th order suspensions, the invariant does not change. We will see in the following that there is a very strong relation between n-B-structures and n-cone-decompositions.

Remark. The notion of cone-length first appeared in [6] under the name of strong L.S.-category of X where it is shown to be also equal to the minimal number of self-contractible subcomplexes needed to cover some CW -complex of the homotopy type of X. It is proved by Ganea and Takens [14], that the strong L.S.-category is bigger than the usual one by at most one.

We will now discuss a rather surprising consequence of 2. We can think about the cone-length as a measure of how much a given homotopy type can be compressed.

Let X be a finite type CW -complex and let {p_{i}}_{1≤i≤∞} be a strictly increasing sequence of primes. Construct, starting from X, spaces X_{i} by means of cofibration sequences:

S^{u_{i}} ⟶^{t_{i}} X_{i−1} ⟶ X_{i}

with t_{i} being a p_{i}^{k_{i}}-torsion homotopy class and such that the sequence u_{i} is strictly increasing. Let X′ be the limit space. The points 1 b and 2 together with the inequality cat(S)≤Cl(S)≤cat(S)+1 for any space S, imply the curious fact that, independently of the choices involved in the construction, Cl(X_{i})≤ 2Cl(X)+3 for all i. and Cl(X′)≤ 2Cl(X)+4. We do not expect, apriori, to be able to compress in this fashion each such space.

Acknowledgements. My thanks go to Charles McGibbon for suggesting the original problem, to Steve Halperin for pointing out the last example in Section 3 and for numerous useful suggestions and to John Harper for a useful discussion.

2 Localization and n-B-structures

We start by defining the notion of Berstein structure which provides the natural context for our study of the relation between localization and the L.S.-category.

Definition 1 An n-B-structure (n Berstein strucure) for X is the fibre homotopy class of a section s : X ⟶ B_{n}ΩX . The (n+k)-extension of an n-B-structure represented by s is the (n+k)-B-structure represented by i_{n}^{k} ∘ s where i_{n}^{k} : B_{n}ΩX ⟶ B_{n+k}ΩX is the inclusion.

A map of two n-B-structures is defined by the existence of representatives s : X ⟶ B_{n}ΩX and r : X′ ⟶ B_{n}ΩX′ that are related by a map v : X ⟶ X′ making the following diagram commutative:

[figure: X; s; X′; r; v; B_{n}ΩX; p_{n}; B_{n}ΩX′; p_{n}; B_{n}(v); X; X′; v]

If the map v above is a homotopy equivalence the two n-B-structures are equivalent.

Remark. 1.For a fibration F ⟶ E ⟶ B, two sections that are homotopic as maps in E, are also fiber homotopic, [8].

2. In [1] a factorization of the diagonal through the fat wedge, as in the definition of cat(−), is called a (categorical) strucure map. It is clear that structure maps and categorical sections are equivalent notions. Also, a map v as above, is n-primitive in the sense of [1].

It is clear that two different n-B-structures should coincide when extended enough. We will see below how soon that happens.

Proposition 2.1 Let X be a space as above. Let F ⟶ E ⟶^{p} B be a fibration. Let s_{1,2} : X ⟶ E be two maps such that there is a homotopy h : X × I ⟶ B between p ∘ s_{1} and p ∘ s_{2}. If X is an n-cone, then there is a homotopy H : X × I ⟶ G_{n}(E, F) between i_{n} ∘ s_{1} and i_{n} ∘ s_{2} and such that p_{n} ∘ H = h. Here i_{n} : E ⟶ G_{n}(E, F) is the inclusion and p_{n} : G_{n}(E, F) ⟶ B is the projection.

In particular all n-B-structures on X extend to the same 2n-B-structure.

Proof. Notice the following facts:

Let Z ⟶ A ⟶^{j} D be a cofibration sequence and let U ⟶ V ⟶^{p} W be a fibration.

a. Given a map t : D ⟶ W and one k : A ⟶ V such that p ∘ k = t ∘ j, the obstruction to extending k to a map k′ : D ⟶ V such that p ∘ k′ = t lies in [Z, U].

b. Given two maps g_{1,2} : D ⟶ V together with homotopies r : A×I ⟶ V between g_{1} ∘ j and g_{2} ∘ j and r′ : D × I ⟶ W between p ∘ g_{1} and p ∘ g_{2} such that p ∘ r = r′ ∘ (j × 1_{I}), the obstruction to extending r to a homotopy b : D × I ⟶ V between g_{1} and g_{2}, such that p ∘ b = r′, lies in [ΣZ, U].

Point a. is trivial; b. is an easy exercise. Recall that CZ is the cone over Z. We asume D = A ⋃ CZ. The key point is to lift r′ over CZ ×I taking into account the fact that r gives us allready a lift over CZ × {0, 1} ⋃ I × Z = T. If we were working with unreduced cones and suspensions, then T would be homeomorphic with Σ′Z and it is simple to verify that CZ × I would be homeomorphic to CΣ′Z where Σ′Z is the unreduced suspension. Our result follows by recalling that we use pointed maps and homotopies.

Lemma 2.2 In the context of the point a. above there is a map t′ : D ⟶ G_{1}(V, U) such that p_{1} ∘ t′ = t and t′ ∘ j = i_{1} ∘ k. In the context of the point b. there is a homotopy b′ : D × I ⟶ G_{1}(V, U) such that p_{1} ∘ b′ = r′ and b′ ∘ (j × I) = i_{1} ∘ r. Here p_{1} : V/U ≃ G_{1}(V, U) ⟶ W is the projection and i_{1} : V ⟶ G_{1}(V, U) the collapsing map.

Proof of the lemma. Let α ∈ [Z, U] and β ∈ [ΣZ, U] be the two obstructions given respectively by the points a. and b. There is an obvious map of fibrations:

[figure: U; i′_{1}; V; i_{1}; W; id; p; E_{1}(V, U); G_{1}(V, U); W; v; p_{1}]

For the lemma it is enough to show that the images α′ of α and β′ of β in [Z, E_{1}(V, U)] and, respectively, in [ΣZ, E_{1}(V, U)] by the maps induced by i′_{1} are null. Notice that, because G_{1}(V, U) ≃ V/U, the composites v ∘ α′ and v ∘ β′ are null. The proof ends by remarking [6], that the connectant ΩW ⟶ E_{1}(V, U) is null.

We turn now to the proposition. The proof is completed by applying inductively the lemma. Indeed as X is an n-cone there is a sequence of cofibration sequences Z_{i} ⟶ X_{i} ⟶ X_{i+1} for 0 ≤ i < n with X_{0} = ∗ and X_{n} = X. We restrict all the maps and homotopies involved to the i-th level. Recall that G_{i+1}(E, F) ≃ G_{i}(E, F)/E_{i}(E, F) ≃ G_{1}(G_{i}(E, F), E_{i}(E, F)). The lemma shows how to extend a homotopy H_{i} : X_{i} × I ⟶ G_{i}(E, F) to H_{i+1} : X_{i+1} × I ⟶ G_{i+1}(E, F).

Remarks. 1. It’s clear that, under the circumstances above, knowing that Z_{i} is an (iterated) suspension can be of interest. This points out one reason why the the possibility to assume Z_{i} = Σ^{i}P_{i} in the definition of the cone-length is meaningful. Indeed, if we use above this type of cone decomposition, at the i-th step the obstruction will lie in π_{i+1}(P_{i}, U).

2. If H′ is another homotopy constructed the same way as H in the proposition, a similar argument as above can be used to show that, after extension to G_{n+1}(E, F), H and H′ can be connected by a homotopy K defined on (X × I) × I and which is constant to s_{1} on X × {0} × I and to s_{2} on X × {1} × I and which also covers h. We construct inductively K_{i} : X_{i−1}×I×I ⟶ G_{i}(E, F), a homotopy between H_{i−1}∘j_{i−1}^{1} and H′_{i−1}∘j_{i−1}^{1} where j_{i−1}^{1} is the inclusion of G_{i−1}(E, F) into G_{i}(E, F).

Corollary 2.3 If cat(X)≤ n replaces the condition of X being an n-cone in the above proposition, then the conclusions hold.

Proof. If cat(X)≤ n, by the results in [3], it follows that the space Y = X ∨ ΣΩE_{n}ΩX is of cone-length bounded by n. Of course X is a retract of Y. Hence, by composing with the collapsing map X ∨ΣΩE_{n}ΩX ⟶ X, we reduce the problem for X to the same problem for Y. Here it can be solved after possibly changing Y, inside the same homotopy type, as to become an n-cone. Finally we compose again with the inclusion X ⟶ Y. However, we start with s_{1}, s_{2} and we constructed a homotopy between s_{1} ∘ u and s_{2} ∘ u covering h ∘ u where u : X ⟶ X is homotopic to the identity and appears because of the change of Y into an n-cone. The result follows by recalling that X has the homotopy type of a CW -complex and, hence, u can be inverted.

Before going further it is useful to note that, when applied to simply connected spaces, the functors B_{n}Ω(−) commute with localizations.

We start now the comparison between geometric and localized n-B-structures. The first result shows that, for finite CW -complexes, rational n-B-structures can be pulled back to geometric ones when inverting some finite number of primes.

Proposition 2.4 Let X be a finite, simply-connected CW -complex. Given an n-B-structure s_{0} on the rationalization X_{(0)} of X, there is a finite set of primes P and an n-B-structure on X_{P′} which, when rationalized, coincides with s_{0}. In particular, cat(X_{P′})=cat(X_{(0)}).

Proof. Consider the following commutative diagram:

[figure: A; j; B; f; k; C; D; g]

Let Z ⟶ N ⟶^{h} M be a cofibration sequence and consider maps f′ : M ⟶ B, g′ : M ⟶ C, l : N ⟶ A such that f ∘ f′ ≃ g ∘ g′ and k ∘ l ≃ f′ ∘ h, j ∘ l ≃ g′ ∘ h. The obstruction to the construction of a map s : M ⟶ A such that k ∘ s ≃ f′ and j ∘ s ≃ g′ lies in [Z, L] where L is the homotopy fibre of the obvious map of A into the homotopy pull back of f and g. A similar statement holds in the case of homotopies.

We are going to apply this remark to the following situation: let s′ be a representative of s_{0}. Let X^{(i)} be the i-th skeleton of X and let l_{o} : X ⟶ X_{(0)} be the rationalization. Suppose that we have constructed: s_{i} : (X^{(i)})_{P′_{i}} ⟶ B_{n}ΩX_{P′_{i}}, P_{i} a finite set of primes such that the rationalization of s_{i} is the restriction of s′ to (X^{(i)})_{(0)}. We have the square:

[figure: B_{n}ΩX_{P′_{i}}; B_{n}ΩX_{(0)}; p_{0}; X_{P′_{i}}; X_{(0)}; l_{i}]

Here the horizontal arrows are rationalizations and the vertical ones are the usual projections. Notice that the homotopy fibre of the obvious map from B_{n}ΩX_{P′_{i}} to the pull back of l_{i} and p_{0} has only torsion homotopy goups. Recalling that X is finite it follows that, by inverting some other (finite number of) primes, we can kill all the obstructions to the construction of s_{i+1} satisfying the induction hypothesis. At the end we are left with a section s′′ : X_{P′} ⟶ B_{n}ΩX_{P′} whose rationalization is s′; P being the union of the P_{i}’s.

Remark.1. The same method as before can be used to ”derationalize” homotopies.

2. The finiteness condition is necessary. Indeed, if p_{i} is an infinite sequence of primes and n_{i} is an increasing sequence of integers ≥ 3, let T be the wedge of the Moore spaces P^{n_{i}}(p_{i}). Clearly, the category of T is 1 and inverting any finite number of primes does not reduce it. However, rationally T is contractible.

Theorem 2.5 Let X be simply connected and of finite type. Let s_{0} be a rational n-B-structure on X_{(0)} and let s_{p} be n-B-structures on X_{(p)} for each prime p.

a. If X is finite there is a 2n-B-structure on X which, when localized at p, for any prime p, is equivalent with the one extending s_{p} and when rationalized gives the extension of s_{0}. In particular, cat(X)≤ 2n.

b. For X not necessarily finite, there is a (2n+1)-B-structure with the analogous properties. In particular, cat(X)≤ 2n + 1.

Proof. We begin with the finite case.

Let s′ represent s_{0}. By Proposition 2.2 , there is a finite set of primes P such that there is a section s′_{P′} : X_{P′} ⟶ B_{n}ΩX_{P′} whose rationalization is s′ Let u_{P} be the 2n-extension of s′_{P′} and let r be the 2n-extension of s′. From now on all the primes involved belong to P.

Denote by l^{p} : X_{(p)} ⟶ X_{(0)} a rationalization. We denote the rationalizations at the level of the spaces B_{n}ΩX_{(p)} by Bl^{p}. We may assume Bl^{p} to be a fibration for all the p’s. Let s′_{p} : X_{(p)} ⟶ B_{n}ΩX_{(p)} be a section that represents s_{p}. Denote by s′_{p,0} the localization of this section. Let r_{p} be the composite i_{n}^{n} ∘ s′_{p,0} where i_{n}^{n} : B_{n}ΩX_{(0)} ⟶ B_{2n}ΩX_{(0)} is the inclusion. Also, let u_{p} be the 2n-extension of s′_{p} (r_{p} is the rationalization of u_{p}). By applying Proposition 2.1 and its corollary it follows that there are homotopies H_{p} : X_{p} × I ⟶ B_{2n}ΩX_{0} such that H_{p}(0) = r ∘ l^{p}, H_{p}(1) = l^{p} ∘ r_{p} and p_{2n,0} ∘ H_{p}(t) = l^{p} for any fixed t ∈ I; p_{2n,0} : B_{2n}ΩX_{(0)} ⟶ X_{(0)} being the projection. We have the square:

[figure: B_{2n}ΩX_{(p)}; p_{2n}; B_{2n}ΩX_{0}; p_{2n,0}; Bl^{p}; X_{(p)}; X_{0}; l^{p}]

The same type of argument as that at the beginning of the proof of Proposition 2.2 can be used to infer the existence of a homotopy H′_{p} : X_{(p)} × I ⟶ B_{2n}ΩX_{(p)} which is a lift of H_{p} in the sense that Bl^{p} ∘ H′_{p} = H_{p} and with the additional properties : H′_{p}(1) = u_{p}, p_{2n} ∘ H′_{p}(t) = id_{X} for each fixed t. Let u′_{p} = H′_{p}(0). It is clear that u′_{p} is fiber homotopic with u_{p}. Moreover the rationalization of u′_{p} is r.

Let us now consider the following commutative double cube diagram:

[figure: Y; k; X_{P′}; w; ∏_{p∈P} X_{(p)}; ∏_{p∈P} X_{(0)}; ∏ r; G; B_{2n}ΩX_{P′}; ∏ B_{2n}ΩX_{(p)}; ∏ B_{2n}ΩX_{(0)}; Y; X_{P′}; w; ∏ X_{(p)}; ∏ X_{(0)}; ∏ l^{p}]

Here w is the rationalization composed with the diagonal; in the bottom cube the vertical arrows are projections; in the top cube the vertical, non marked arrows are ∏ u′_{(p)} and u_{P′}.

We may assume the rationalizations Bl^{p}, l^{p} to be fibrations. All the horizontal squares are pull backs. In this case, by standard fracture lemmas [2], there is a weak homotopy equivalence X ⟶ Y^{v}. Also, the space G comes with a weak homotopy equivalence with domain B_{2n}ΩX. We obtain the map k which provides the 2n-B-structure that we are looking for. Indeed, to conclude the point a. of the theorem notice that, the localization of the n-B-structure represented by k coicides, trivially, for the primes in P with the extension of s_{p} and for the primes in P′ it is the n-th extension of an n-B-structure (induced by the derationalization of s′) hence, by Proposition 2.1 it again coincides with the extension of s_{p}.

Remark. Another variant of the fracture lemmas [2] can be used to avoid the ”derationalization” argument.

We will now pass to the infinite case. The reduction to the finite one is provided by the next result.

Proposition 2.6 Let f : Y ⟶ X be a map of simply connected spaces with Y being of dimension m and let s be a n-B-structure on X. If H_{∗}(f) is an isomorphism for ∗ < m and an epimorphism for ∗ = m, then there is an (n+1)-B-structure on Y, s′, such that, f maps the (n+2)-extension of s′ into the (n+2)-extension of s.

In particular cat(Y)≤cat(X)+1.

Remark. 1. A simple example when cat(Y)=cat(X)+1 is provided by the trivial map S^{m} ⟶ ∗ with m ≥ 2.

2. It would be interesting to know if we can find a CW -decomposition for X such that for each m cat(X^{(m)})≤cat(X). Rationally it is true that there is a CW -decomposition for X such that the category of each skeleton is smaller or equal than that of X.

Proof. Let r be a representative of s. We have the following commutative square:

[figure: B_{n}ΩY; p_{n}^{Y}; B_{n}ΩX; p_{n}^{X}; B_{n}(f); Y; X; f]

Let H be the pull back of f and p_{n}^{X}. Let g : B_{n}ΩY ⟶ H be the map induced by B_{n}(f) and p_{n}^{Y}. Let k : Y ⟶ B_{n}ΩX be the map induced by id_{X} and r ∘ f. If k could be lifted over g, then that lift would provide a section with all the required properties (it would give, in fact, even an n-B-structure !). Hence, it is natural to see what are the obstructions to such a lift. Let F be the homotopy fibre of g.

Lemma 2.7 With the notations above, F is (m-2)-connected.

Proof of the lemma. It is easy to see that the map H_{∗}(B_{n}(f)) is an isomorphism in dimensions less or equal to m − 1 and an epimorphism in dimension m. Indeed, this follows from the coresponding property for f and the fact that the fibre of E_{n}ΩY = (ΩY)^{∗(n+1)} ⟶ (ΩX)^{∗(n+1)} = E_{n}ΩX is 2n + m − 2 connected.

Obviously, the map u : H ⟶ B_{n}ΩX has the same connectivity as f. As u ∘ g = B_{n}(f), it follows that H_{∗}(g) is an iomorphism for ∗ ≤ m − 1. However, it might not be an epimorphism when ∗ = m. This means that F is (m − 2)-connected and, in general, it is not more connected than that.

We return now to the proof of the proposition. The lemma implies that when trying to construct a lift of k over g we do not encounter any obstacles to the construction of a map v : Y^{(m−1)} ⟶ B_{n}ΩY which is a lift of the inclusion j : Y^{(m−1)} ⟶ Y and such that r ∘ f ∘ j = B_{n}(f) ∘ v. Of course, Y can be otained from Y^{(m−1)} by attaching a number of m-cells. By applying now Lemma 2.2 we get a section r′ : Y ⟶ B_{n+1}ΩY which extends i_{m}^{1} ∘ v. The fibre homotopy of B_{n+2}(f) ∘ i_{n+1}^{1} ∘ r′ and i_{n}^{2} ∘ r ∘ f follows by noticing that the needed commutativity already exists on Y^{(m−1)} and applying again the Lemma 2.1.

We are now in good shape for proving the point b of the theorem. We will construct the needed section by constructing coherent lifts of the inclusions j^{m} : X^{(m)} ↪ X into B_{2n+1}ΩX.

The first step is to notice that we can build a double cube similar to that above only that the top square will be replaced by its restriction to the m-th skeleton X^{(m)}. More precisely, if we represent s_{p} by s′_{p} we can restrict these sections to (X^{(m)})_{(p)} and similarly for s_{0}. Now the derationalization argument works. Keeping in mind that, by the proposition above, the category of (X^{(m)})_{(p)} and (X^{(m)})_{0} is bounded by n + 1, the exact same argument as that used at the point a implies the existence of some lifts k^{m} : X^{(m)} ⟶ B_{2n+1}ΩX of j^{m} with the properties that, when localized at any prime p, they are fiber homotopic with the restrictions of the (2n+1)-extensions of s′_{p}, s_{p}^{m}. The proof of the theorem is concluded by noticing that for each p, s_{p}^{m} form a coherent system of lifts in the sense that s_{p}^{m+1} composed with the localization of the inclusion j_{1}^{m} : X^{(m)} ↪ X^{(m+1)} gives s_{p}^{m}. This implies that k^{m} is homotopic (by a fracture lemma [2] and even fiberwise by [8]) to k_{m+1} ∘ j_{1}^{m}.

An infinite telescope argument shows this to be enough for the construction of the needed section.

Remark. A more direct but inefficient way to approach the point b. of the theorem is to use a result of Hardie [7, 9] which shows that the category of a direct limit of spaces of category less than n is bounded from above by 2n. Applying this together with the point a. of the theorem and the last proposition we would get an upper bound for the category of X in point b. of the order of 4n.

In fact Hardie’s result can be restated in a somewhat stronger form in terms of n-B-structures: given n-B-structures on a direct system of spaces there is a 2n-B-structure on the direct limit whose restriction to the spaces in the system coincides with the extension of the original B-structures. This is an immediate consequence of Proposition 2.1 .

3 Cone-length and n-B-structures

It is clear from the previous section that there is a strong relation between n-B-strucures and cone-decompositions. We will try to make it now more explicit. In [4] a map of cone-decompositions (level-preserving map), f :

X ⟶ X′, is defined by the diagrams:

[figure: Z_{i}; X_{i}; f_{n}; X_{i+1}; f_{n+1}; Z′_{i}; X′_{i}; X′_{i+1}]

Here 0 ≤ i < n, both horizontal sequences are cofibration sequences defining cone-decompositions for X_{n} = X and, respectively, X′_{n} = X′; f = f_{n}. Two cone-decompositions are equivalent if they are related by a map of cone-decompositions which is also a homotopy equivalence. Any n conedecomposition can be trivialy extended to one of length n + 1 by defining Z_{n} = ∗, X_{n+1} = X_{n}.

Lemma 3.1 Each n-B-structure on a space X induces an equivalence class of cone-decompositions of length n on X ∨ΣΩE_{n}ΩX and an equivalence class of cone-decompositions of length n + 1 on X.

Conversely, any cone-decomposition of lenght n on X induces a n-B-structure on X. The (n + 1)-extensions of the n-B-structures induced by equivalent cone-decompositions coincide.

Moreover, the two constructions are inverse at the level n + 1.

As we will not use this lemma in the following, we just mention the two basic constructions. First, how to get a cone-decomposition from a section s of p_{n} : B_{n}ΩX ⟶ X. The idea, descibed in detail in [3], is to pull back the standard cone-decomposition of B_{n}ΩX to X thus getting a certain sequence of push out squares. Then we collapse the homotopy fibre of s into each of the spaces appearing in the push out squares. What we get in this way is an n-cone-decomposition of X ∨ ΣΩE_{n}ΩX. As E_{n}ΩX is an n-suspension this immediately provides an (n+1)-decomposition for X. On the other hand, given a cone-decomposition Proposition 2.1 shows how to construct a section associated with it. The whole point is to show that these constructions are well-behaved with respect to the different equivalence relations involved. This follows by simple checking and by applying again Proposition 2.1. Also, it is not difficult to see that, given an initial n-cone-decomposition which induces a section s, the (n+1)-cone-decomposition constructed out of s is equivalent to the (n+1)-cone-decomposition that trivially extends the initial one.

Remark. 1.There is a result similar to the one above concerning the relation between maps of cone-decompositions and maps of n-B-structures.

2.There is some assymetry in the statement of the lemma. The reason is immediate: n-B-structures behave well with respect to retracts. In particular an n-B-structure on X ∨ΣΩE_{n}ΩX induces one on X. Also any n-B-structure on X can be extended to one on X ∨ΣΩE_{n}ΩX. The same type of arguments do not work for cone-decompositions.

In some circumstances it is easier to work with cone-decompositions in others with n-B-structures. The results above show that from many points of view it does not make too much difference which variant we choose. In particular the following lemma will be stated in terms of cone-decompositions. It immediately iimplies an obvious analogue in terms of n-B-structures (compare with Theorem 3.4 in [1]).

Lemma 3.2 A map of n-cone-decompositions, f : X ⟶ Y induces on the cofibre Y/X an n-cone-decomposition such that the collapsing map is a level preserving map.

Proof. Let us take a look at the diagrams given by f:

[figure: Z_{i}; X_{i}; f_{i}; X_{i+1}; f_{i+1}; Z′_{i}; Y_{i}; Y_{i+1}]

with f = f_{n}. By collapsing the top row into the bottom one we get a new sequence of cofibration sequences: Z′_{i}/Z_{i} ⟶ Y_{i}/X_{i} ⟶ Y_{i+1}/X_{i+1}.

Remark. Obviously, in general (that is, when there are no restrictions on f) there is a map of (n+1)-cone-decompositions between the trivial extension of the decomposition on Y and the (n+1)-cone-decomposition on Y/X obtained by first building up Y and then adding the cone over X.

Another interesting fact that is worth mentioning here is the following:

Lemma 3.3 If F ⟶ E ⟶ B is a fibration then for any n-B-structure on B there is an n-B-structure on E/F such that the projection E/F ⟶ B is a map of n-B-structures.

This follows from arguments described in [3] where the cone-decomposition version is presented in detail. We will say that the corresponding n-B-structure on E/F is inherited from that on B.

Having the above results at our disposal we intend to discuss (and prove) the proposition:

Proposition 3.4 There is a space X that supports two t-B-structures with different (t+k)-extensions for all k < t.

Proof. To start consider the following construction: assume we are given the homotopy push out square:

[figure: A; g; B; f; D; C]

Assume also that we have n-B-strucures on B and D represented respectively by sections s and r. Let F be the homotopy fibre of f. It is not hard to see that we also have a homotopy push out quare:

[figure: A/F; g′; B; f′; D/F; C]

Here f′ and g′ are induced by f and g; the map F ⟶ D used in the collapsing is the composite of g and the inclusion of F. Now on A/F there is an n-B-structure represented by a section s′ inherited from s. On D/F there is an (n+1)-B-structure represented by r′ that is induced by r. Let F′ be the homotopy fibre of g′. We have a homotopy push out square:

[figure: (A/F)/F′; g′′; B/F′; f′′; D/F; C]

It is constructed in a similar fashion with the last one. We also get two (n+1)-B-structures on (A/F)/F′ and one on B/F represented respectively by sections r′′ inherited from r′, s′′ induced by s′ and s_{1} induced by s. Moreover, f′′ is a map of (n+1)-B-structures between s′′ and s_{1} and g′′ is a map of (n+1)- B-structures between r′′ and r′. Suppose now that the (n+1+k)-extensions of s′′ and r′′ coincide. It is easy to see that we can assemble the (n+1+k)- extensions of s′′, s_{1} and r′ to obtain a (n+1+k)-B-structure on C. This provides a method to show that, for k small, (n+1+k)-extensions of (n+1)- B-strucures do not generally coincide. Indeed the above argument shows, in particular, that cat(C)≤ n + 1 + k. However, it is well-known that the result that is valid in general is cat(C)≤ 2n + 1. An example when cat(C)= 2n + 1 would show that k ≥ n in general. However, as cat(A/F)/F′≤ n + 1 this type of result is not enough for the proposition (we would need k ≥ n + 1).

In the following we will present a modification of the construction and a relevant example in order to prove the proposition.

Lemma 3.5 Consider a map f : X ⟶ Y. Let F be the homotopy fibre of f. If Cl(Y)≤ n, then there is map of (n+1)-cone-decompositions, f′ : X/F ⟶ Y ∨ ΣF, which, up to homotopy is induced by f after collapsing F in both X and Y.

Remark. This result allows us to replace any map by a level preserving one (in n+1-stages, though) such that the cofibre of the replacement is the same as that of the original.

Proof of the lemma. We assume f to be a fibration. Let Z_{i} ⟶ Y_{i} ⟶ Y_{i+1} be a sequence of cofibration sequences with 0 ≤ i < n, X_{0} = ∗, Y_{n} ≃ Y.

We pull back these cofibration sequences, thus getting the cubes:

[figure: Z_{i} × F; F; X_{i}; X_{i+1}; Z_{i}; ∗; Y_{i}; Y_{i+1}]

Here the vertical faces are pull backs and, hence, the top square is a push out, [9]. By collapsing F in each of the spaces appearing in the cube we get:

[figure: (F × Z)/F; ∗; X_{i}/F; X_{i+1}/F; Z_{i} ∨ ΣF; ΣF; Y_{i} ∨ ΣF; Y_{i+1} ∨ ΣF]

We remark that for i ≥ 1 we may assume Z_{i} to be a suspension. Also for i = 0 in the bottom square the two arrows with domain Z_{0} ∨ ΣF are equal. This makes it possible to use a technique discussed in [5] to transform the bottom push out square into a cofibration sequence: Z_{i} ∨ ΣF ⟶ Y_{i} ∨ ΣF ∨ ΣF ⟶ Y_{i+1} ∨ΣF. It is easy to see that this cofibration sequence can be transformed into : Z_{i} ⟶ Y_{i} ∨ ΣF ⟶ Y_{i+1} ∨ ΣF into which maps the top cofibration sequence appearing in the cube: (F × Z)/F ⟶ X_{i}/F ⟶ X_{i+1}/F.

Now let f : X ⟶ Y, g : X ⟶ K be two maps that are related by homotopy equivalences in the sense that there is a commutative square:

[figure: X; g; X; f; id; K; Y]

with all horizontal arrows homotopy equivalences. Given a n-cone-decomposition on Y the lemma above shows how we can replace f by f′ : X/F ⟶ Y ∨ ΣF (F is the homotopy fibre of f and also of g) with f′ a map of (n+1)-cone decompositions. Given another n-cone-decomposition on K there is an induced cone-decomposition on X/F and an induced map of n-cone-decompositions: g′ : X/F ⟶ K. Clearly we can trivially assume g′ to be a map of (n+1)- cone-decompositions.

The corresponding statement for n-B-structures is that, after fixing a n-B-structure on Y and one on K, we may construct (n+1)-B-structures such that f′ and g′ become maps of (n+1)-B-structures.

It is easy to see that the push out of f′ and g′ is, up to homotopy, the same as that of f and g. Let us denote by T this push out.

Suppose that the two different (n+1)-B-structures on X/F have the same n + 1 + k extension, then we can construct on T an (n+1+k)-B-structure by pasting the (n+1+k)-structures constructed on K and Y ∨ΣF. In particular, the category of T is bounded from above by n + 1 + k.

Notice that the category of X/F is less or equal to n (by lemma 3.3). The proposition is proved by constructing an example such that cat(T)= 2n + 1.

Example. In the (2n+2)-dimensional complex space C^{2n+2} consider two orthogonal (n+1)-dimensional complex subspaces C_{1} = C^{n+1}, C_{2} = C^{n+1} such that C_{1} corresponds to points with all the last (n+1) coordinates zero and C_{2} to points with all the first (n+1) coordinates zero. Let U_{j} be the set of all lines in C^{2n+2} which are not contained in C_{j}, j = 1, 2. It is easy to see that U_{j} ≃ CP^{n} and, of course U_{1} ⋃ U_{2} = CP^{2n+1}. Recall that cat(CP^{k})= k. The only thing that remains to be shown is that l_{j} : U_{1} ⋂ U_{2} ↪ U_{j}, j = 1, 2 are related by homotopy equivalences. This follows from the fact that the following square is commutative:

[figure: U_{1} ⋂ U_{2}; j_{1}; U_{1} ⋂ U_{2}; j_{2}; w; U_{1}; U_{2}; w]

Where w is a homeomorphism that corresponds to the restriction of the map on C^{2n+1} which permutes the i-th cordinate and the i + n + 1 one for all 1 ≤ i ≤ n + 1.

Remark. The behaviour of the L.S.-category with respect to push outs is prescribed by the following inequality: let T be the homotopy push out of f : X ⟶ Y and g : X ⟶ K. We have [7, 9] :

cat(T)≤max{cat(Y)+cat(K)+1, cat(Y)+cat(X), cat(K)+cat(X)}

The cone-length variant of this inequality is also true. To underline the usefullness of cone-decompositions we will present here a short proof for:

Cl(T)≤max{Cl(Y)+cat(X), Cl(K)+Cl(X)}.

Indeed if X is a point this is obvious. The general case follows by induction on Cl(X). Let Z ⟶ X′ ⟶ X be a cofibration sequence with Cl(X′)<Cl(X). It is enough to inspect the following double cube where all horizontal squares are push outs and all columns are cofibration sequences:

[figure: Z; ∗; ∗; ΣZ; X′; K; Y; T′; X; K; Y; T]

Here the maps from X′ into Y and K are the composites of f and, respectively g with the inclusion X′ ↪ X.

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