Strong L.S.-category equals cone-length ^{∗}

Octavian Cornea University of Toronto Toronto, ON M5S 1A1

cornea@math.toronto.edu

January 25, 1994

Abstract

If a path-connected CW -complex can be covered with n + 1 selfcontractible, subcomplexes, then there is a homotopy equivalent space that can be constructed from a point by iteratively attaching cones, in n steps, such that, at the step k, the respective cone is attached over a k-th order suspension.

1 Introduction

We work in the pointed category of spaces with the homotopy type of a CW -complex. The strong Lusternik-Schnirelmann category of X, CatX, was defined by Ganea, [6], as the least n such that there is a CW -complex of the same homotopy type as X that can be covered by n + 1 self-contractible, subcomplexes. It approximates the Lusternik-Schnirelmann category of X, catX. This last invariant is of significance in analysis because cat+1 gives, [13], a lower bound for the minimal number of critical points of a smooth function on a smooth manifold (here cat(∗)= 0). In topology it represents an important finitness condition [11],[5]. Recently it was also applied to problems concerning the complexity of algorithms,[16]. The inequality catX ≤CatX ≤catX + 1, proven by Takens and Ganea, [17], shows that the two notions are indeed close.

∗1991 Mathematics Subject Classification: Primary 55P50; Secondary 55P40

For X path-connected define, as in [1], the cone-length of X, Cl(X), to be the least n such that there is a sequence of cofibration sequences:

Z_{i} ⟶ X_{i} ⟶ X_{i+1}  (1)

with 0 ≤ i < n, X_{0} contractible, X_{n} of the same homotopy type as X and Z_{i} = Σ^{i}U_{i} for some spaces U_{i}. If no such n exists put ClX= ∞.

It was shown in [1] that catX≤ClX≤catX+1. Clearly this suggests the result that we prove in this note:

Theorem 1.1 For X path-connected we have : CatX=ClX

Other notions of cone-length can be obtained by imposing different restrictions on the spaces Z_{i}. It was proved in [6] that the invariant obtained when we do not impose any conditions on the spaces Z_{i} in (1) equals CatX. This is the description of the strong L.S.-category that we use, in the second section, for the proof of the theorem; it also implies the inequality CatX≤ClX. At the opposite extreme, take the Z_{i}’s to be wedges of spheres. Denote the least n in this case by cl_{S}(X). We may consider ClX to be a compromise between CatX and cl_{S}(X). There are examples where the difference cl_{S}X− CatX is as large as we want: it suffices to remark that CatX = 1 iff X is a suspension and, if a composite of k Steenrod squares does not vanish in the cohomology of X, then cl_{S}(X) > k. From this point of view the theorem shows that (iterated) suspensions are as efficient building bricks as general spaces. However, they don’t touch at all the wide gap between Cat and cl_{S}.

Remarks. 1. Our notion of cone-length is less unnatural than it might appear because if F_{k} : E_{k}ΩX ⟶ B_{k}ΩX ⟶ X is the k-th Milnor fibration in the classifying construction applied to ΩX, then catX≤ n iff F_{n} admits a section (up to homotopy); on the other hand, there are homotopy cofibration sequences E_{k}ΩX ⟶ B_{k}ΩX ⟶ B_{k+1}ΩX and E_{k}ΩX is a k-th order suspension [8],[15].

2. The starting point for this work is a rational (still open) conjecture of Lemaire and Sigrist, [12]: catX_{o}=ClX_{o} ( (X)_{o} is the rationalization of X; we assume X to be simply-connected, of finite type, whenever we rationalize). Notice that Cl(X_{o})=cl_{S}(X_{o}).

Any co-H-space that is not a suspension is an example where catX= 1 and CatX= 2 hence it is clear that the geometric version of the conjecture is precisely the statement of the theorem.

3. The rational equality CatX_{o}=ClX_{o} was proven in [2] by comparing both invariants (and showing that they both coincide) with the homotopic nilpotency of the algebra of P L-forms on X (the relevant notion of homotopic nilpotency first appeared in [4]).

4. It is possible to define relative notions of cone-length. Given a map Y ⟶^{g} X construct Cl(X, Y) (Cat(X, Y)) by replacing, in the first cofibration sequence in (1), ∗ with Y and constructing the other cofibration sequences the same way as in (1) with the additional condition that the composite Y ↪ X_{i} ⟶ X should be homotopic to g. Then Cat(X, Y)=Cl(X, Y).

5. All the results discussed here hold in more general categories where homotopy theory is possible and where the cube axiom of Doeraene, [3], is valid (for spaces, this axiom coresponds to the second cube theorem of Mather, [14]).

2 A useful formula

The proof of Theorem 1.1 is based on the following fact:

Proposition 2.1 Suppose F : F ⟶ E ⟶^{p} B is a fibration and suppose there is a cofibration sequence C : Z ⟶ B′ ⟶^{j} B. Let E′ be the pull-back of p over j. Then, up to homotopy, there is a cofibration sequence :

Σ(F ∧ Z) ⟶ B′/E′ ⟶ B/E

Assuming the proposition let us see how the theorem follows. It is clearly enough to show that if Y ⟶^{f} X ⟶ C is a cofibration sequence, then Cl(C)≤Cl(X)+1. Let Z_{i} ⟶ X_{i} ⟶ X_{i+1}, 0 ≤ i < n, X_{0} ≃ ∗, X_{n} ≃ X, Z_{i} = Σ^{i}U_{i} be a cone-decomposition of X. Transform f into a fibration and pull it back over the inclusions X_{i} ⟶ X thus getting fibrations:

T_{i} : F ⟶ Y_{i} ⟶ X_{i}

F being the homotopy fibre of f. We apply our formula for the fibration T_{i+1} and the cofibration Z_{i} ⟶ X_{i} ⟶ X_{i+1}. Notice that the pull-back of T_{i+1} over the inclusion X_{i} ⟶ X_{i+1} is precisely T_{i}. Hence, we get cofibration sequences:

Σ(F ∧ Z_{i}) ⟶ X_{i}/Y_{i} ⟶ X_{i+1}/Y_{i+1}

To end we remark that Σ(F ∧Z_{i}) ≃ Σ^{i+1}(F ∧U_{i}), X_{0}/Y_{0} ≃ ΣF, X_{n}/Y_{n} ≃ C.

Proof of the proposition. First we pull-back the fibration F over the cofibration C . We obtain the following cube diagram with the vertical faces pull-backs and the bottom face a push-out.

[figure: F × Z; E′; F; E; Z; B′; ∗; B]

The top square is a homotopy push-out by the second cube theorem of Mather, [14]. Next, transform the vertical arrows into inclusions and collapse the top square into the bottom one. It is an easy exercise to see that we get a new homotopy push-out square:

[figure: Z/(F × Z); B′/E′; ΣF; B/E]

Notice that Z/(F × Z) ≃ Σ(F ∧ Z) ∨ ΣF. We need the following, general, fact:

Lemma 2.2 If the following square:

[figure: ΣT; v; A; w; B; D]

is a homotopy push-out, then there is a homotopy cofibration sequence:

ΣT ⟶^{w+v} A ∨ B ⟶ D

Proof. It is easy to see that we can construct a commutative, cube diagram which, up to homotopy, coincides with :

[figure: T; ∗; ∗; ΣT; w+v; ∗; A; B; A ∨ B]

Here, both horizontal squares are homotopy push-outs. Notice that, by collapsing the top square into the bottom one, we obtain a square which, up to homotopy, is the same with the one in the statement.

Remark. The simply-connected case can be easily proved without the use of cubes. The above argument avoids a special discussion for the nonsimply-connected context.

Applying the lemma we get a homotopy cofibration sequence :

Σ(F ∧ Z) ∨ ΣF ⟶^{u} B′/E′ ∨ ΣF ⟶ B/E

We attach now, simultaneously, a cone over the inclusion ΣF ↪^{i} Σ(F ∧ Z) ∨ ΣF and one over the composite u ∘ i. The first attachement gives Σ(F ∧ Z); denote by G the result of the second. We have a new homotopy cofibration sequence:

Σ(F ∧ Z) ⟶ G ⟶ B/E

The last step consists in verifying that G ≃ B′/E′. This follows from the fact that pr ∘ u ∘ i ≃ id where pr : B′/E′ ∨ ΣF ⟶ ΣF is the projection.

Remark. Our formula clearly implies the existence of the homotopy cofibration sequence : B′/E′ ⟶ B/E ⟶ Σ^{2}(F ∧ Z). This can also be obtained from a formula of Gray, [9]: E/E′ ≃ F/(F × Z) that extends one of Ganea, [7], and also James, [10], dealing with the case when E ≃ ∗; Gray’s formula is an immediate consequence of the fact that, in our first cube diagram, the top square is a homotopy push-out.

As E′ is a pull-back of p over j, the inclusion Z ↪ B′ together with the trivial map Z ⟶^{∗} E give a map Z ⟶ E′. Making use of this map we can construct a new homotopy cofibration sequence :

E′/Z ⟶ B ⟶^{t} B′/E′

In conjunction with Gray’s formula this cofibration sequence can be used to produce an alternative proof of the proposition.

It is easy to check that the collapsing map B ⟶ B/E factors : B ⟶^{t} B′/E′ ⟶ B/E (up to homotopy). An interesting consequence: if X ⟶^{f} Y is a map and if there exists a homotopy cofibration sequence Z ⟶ X ⟶^{f} Y, then cat(X/F)=catY (where F is the homotopy fibre of f). Indeed we just have to apply the previous discussion to the cofibration sequence Z ⟶ X ⟶ Y and the path-loop fibration over Y. It results that Y is a homotopy retract of X/F. Hence cat(X/F)≥catY. But the opposite inequality always holds ([1]).

Acknowledgements. Thanks are due to Paul Selick and Bill Richter who pointed out to me Gray’s formula and the relevant bibliography and to Steve Halperin for many helpful suggestions.

References

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