Revised version, March 97

Some Remarks on Massey Products, tied cohomology classes and Lusternik–Shnirelman Category *

Claude Viterbo Equipe de Topologie et Dynamique, URA 1169 Département de Mathématiques, Bâtiment 425 Université de Paris–Sud, 91405 Orsay, France email: viterbo@dmi.ens.fr

Abstract:

Let X be a Hilbert manifold, F (X) the minimal number of critical points for a function on X, and cℓ(X) the cup length of X, that is the maximal integer k such that there are classes β_{1}, …, β_{k−1} in H^{∗}(X)−H^{0}(X) with nonzero cup product. Then it is classical that cℓ(X) ≤ F (X).

In section 2, we introduce the notion of tied cohomology classes, and use it to improve the cup length into a ”tie length” T ℓ(X), a lower bound for F (X). We then estimate T ℓ(X) using in particular Massey products.

In section 4 we deal with the ”stable tie length” tℓ(X). and again give some estimates. Applications to symplectic geometry and examples where tℓ(X) > cℓ(X) are given in the last section.

1. INTRODUCTION

Let f be a C^{1} function on a Hilbert manifold X, satisfying the Palais–Smale condition: (PS) Any sequence (x_{n}) such that f′(x_{n}) → 0 and f(x_{n}) is bounded has a converging subsequence.

Set for a ∈ , X^{a} = {x ∈ X|f(x) ≤ a}. Note that for −∞ < a ≤ b < +∞, (PS) implies that the set of critical points in the closure of X^{b} − X^{a} is compact, and the set of critical values is closed.

Let F (X) be the minimal number of critical points for a function bounded from below and satisfying (PS) on X (If X has a boundary ∂X, we assume lim_{x→∂X} f(x) = +∞). More generally for A submanifold (with boundary) of X, F (X, A) is defined as the minimal number of critical

* This is a corrected version of a previous preprint, circulated in August 92 under the title ”Some Remarks on Massey Products and Lusternik–Shnirelman Category” it contained a mistake in the definition of product length, one in the statement of theorem 2.1, and our definition of c lacked precision. points in X − A for a non-negative function X, satisfying (PS) and vanishing on A. Changing f to 1/f, we see that it is also the number of critical points for a function f on X − A such that lim_{x→A} f(x) = +∞.

A lower bound for F (X) was given by Lusternik and Shnirelman ([Lu-S]), denoted by cat(X), it is the minimal number of open subsets, contractible in X, needed to cover X. Note that the definition of cat(X) is valid on any topological space, not just a manifold.

A well-known consequence of Lusternik and Shnirelman’s work is that, denoting by cℓ(X) (= cup length of X) the number

cℓ(X) = max{k|∃β_{1}, …, β_{k−1} ∈ H^{∗}(X) − H^{0}(X), β_{1} · β_{2} . . . · β_{k−1} ≠ 0}

then cat(X) ≥ cℓ(X). Note that cℓ(X) depends on the choice of the coefficient ring.

In all sections but section 3, cohomology will be Alexander-Spanier cohomology with arbitrary coefficient ring.

In fact this may be restated more precisely. Let α ∈ H^{∗}(X^{b}, X^{a}) − {0}, and set

c(α, f) = inf{λ|α ≠ 0 in H^{∗}(X^{λ}, X^{a})} .

We then have

Theorem 1.1. The number c(α, f) is a critical value of f. If β ∈ H^{∗}(X), and α · β ≠ 0 in H^{∗}(X^{b}, X^{a}), then

(∗)  c(α · β, f) ≥ c(α, f) .

Moreover, in case (∗) is an equality, set K_{c} to be the set of critical points of f at level c = c(α, f) = c(α · β, f), then β is nonzero on H^{∗}(K_{c}). As a result dim K_{c} ≥ degβ, and in particular if degβ ≠ 0, K_{c} is uncountable (hence infinite).

The proof is a standard variation (cf. [Kl]) over the theorem of Lusternik and Shnirelman (cf. [Lu-S]). From the theorem it immediately follows that F (X) ≥ cℓ(X).

Remarks:

1.2 Usually the theorem is stated as follows : if (*) is an equality, then K_{c} is infinite. This stronger form turns out to be useful (see [V 3]).

1.3. If K_{c} is a manifold, the statement clearly implies in case of equality that dim(K_{c}) ≥ deg(β).

1.4 We may extend the definition of c(α, f) to a set of cohomology classes as follows: let E be a subset of H^{∗}(X^{b}, X^{a}), then

c(E, f) = sup{c(u, f) | u ∈ E}

Our theorem may then be generalized to c(E · F, f) ≥ c(E, f) with equality only if for all β ∈ F β is nonzero on K_{c}. Here E · F is the set of all products of an element in E and one in F.

The proof of this and subsequent theorems is based on the following

Main Lemma: Let K_{c} be the set of critical points at level c. Then for any neighbourhood V of K_{c}, there exist ϵ > 0 and an isotopy ϕ_{t} : X^{c+ϵ} → X^{c+ϵ} such that ϕ_{0} = Id and ϕ_{1} : X^{c+ϵ} → X^{c−ϵ} ∪ V.

Proof: Let V be a neighbourhood of K_{c}. According to the Palais-Smale condition, there exist η > 0 such that on X^{c+ε_{0}} − X^{c−ε_{0}} ∩ V, we have |df(x)| ≥ η.

Let φ_{s} be the Ωow of Y = −∇f(x), we want to compute the length of a trajectory of φ_{s} contained in (X^{c+ε} − X^{c−ε}) ∩ V. It is given by

\begin{aligned}∫_{s_{0}}^{s_{1}} |Y (φ_{s}(x))| ds & ≤ \frac{−1}{η} ∫_{s_{0}}^{s_{1}} df(φ_{s}(x))Y (φ_{s}(x))ds \\ & = \frac{−1}{η}[f(φ_{s_{1}}(x)) − f(φ_{s_{0}}(x))] ≤ \frac{2ε}{η} . \end{aligned}

Thus if we assume ε < \frac{η}{2} · δ this length is less than δ. Now let U be a neighbourhood of K_{c}, and V an other neighbourhood with K_{c} ⊂ V ⊂ \overline{V} ⊂ U. Since K_{c} is compact, \overline{V} may be assumed to be compact, and then δ = d(V, U) > 0.

If we add to V the portion of the trajectories of Y which go through V and are contained in X^{c+ε} − X^{c−ε} we get a neighbourhood W of K_{c}, such that W ⊂ U, and W ∩ (X^{c+ε} − X^{c−ε}) is “saturated” for Y (i.e. if z ∈ W ∩ (X^{c+ε} − X^{c−ε}) and φ_{s}(z) ∈ X^{c+ε} − X^{c−ε} then φ_{s}(z) ∈ W).

In other words, since \frac{d}{ds}f(φ_{s}(x)) = − |∇f(φ_{s}(x))|^{2} ≤ −η^{2} outside W, we have for z

in X^{c+ε} − X^{c−ε} and s_{0} = \frac{2ε}{η^{2}} that either:

(i) ∀s ≥ s_{0} φ_{s}(z) ∈ X^{c−ε} if the trajectory of z does not meet W,

or

(ii) ∀s ≥ s_{0} φ_{s}(z) ∈ W ∪ X^{c−ε} if the trajectory of z meet W

Note: Since we used the integrability of the gradient vector field, we should have assumed that f is at least C^{1,1}. But this is not really needed, as we may replace the gradient vector field by a pseudo-gradient vector field.

In practice one is interested in good lower bounds for F (X), as this number is very hard to actually compute. As far as we know, the only general lower bound for F (X) or cat(X) was given by cℓ(X) (at least if X is simply connected), until the paper by Faddell and Husseini [FH1] (see however [Fe-H], [J], [S], [T] for other estimates). They proved that if M is a closed manifold, ΛM its free loop space, i.e. ΛM = C^{0}(S^{1}, M), then F (ΛM) = +∞, while cℓ(ΛM) may be finite (this is the case if M = S^{2} or P^{n}, see the discussion on minimal models further on).

Their argument is based on a clever use of the fibration ΛM → M given by u → u(1) using the fact that fiber ΩM = {u ∈ ΛM|u(1) = x_{0}} has infinite cup length (cf. [S-VP]).

Our aim was then to see whether, at least for simple spaces, the cohomology of ΛM could explain this phenomenon. We were thus led to use Massey’s triple product to find more critical levels for the Energy function on ΛM. Eventually, we were led to define a new invariant, the tie length that gives a lower bound for the F-category. It is not clear whether this new invariant is always infinite for a free loop space, but it is at least so for complex projective spaces and even dimensional spheres (and many other examples are easy to construct in view of section 3.B). On the other hand this invariant has certain stability properties explained in section 2, that make it more widely applicable than the category. In particular this is usually the case for applications to symplectic topology (see [V 3], [V 4]). We now first recall the definition of Massey product.

Let α, Ω ∈ H^{∗}(X) β ∈ H^{∗}(X, A) be such that α · β = Ω · β = 0 (this is just what happens when the cup product will not give anymore critical points). Let a, b, c be cochain representatives of α, β, Ω, we may thus write a · b = dλ, b · c = dμ for some cochains λ, μ vanishing on A. Then a · μ − (−1)^{deg(a)}λ · c is closed (its differential is -up to sign- a(bc) − (ab)c), and its cohomology class is well defined in H^{∗}(X, A)/[α · H^{∗}(X, A) + Ω · H^{∗}(X, A)].

We denote by ⟨α, β, Ω⟩ the class we defined in the quotient of H^{∗}(X, A), it is called Massey’s triple product (of α, β, Ω).

Similarly, we may define ⟨α, β, Ω⟩ for α, Ω ∈ H^{∗}(X, A) and β ∈ H^{∗}(X).

Remarks:

1.4 Massey products are ”functorial” in the sense that if f : X → Y is a map such that f(A) ⊂ B, then the induced map f^{∗} : H^{∗}(Y, B) → H^{∗}(X, A) sends ⟨α, β, Ω⟩, an element of H^{∗}(Y, B)/[α·H^{∗}(Y, B)+Ω·H^{∗}(Y, B)] to ⟨f^{∗}α, f^{∗}β, f^{∗}Ω⟩, an element of H^{∗}(X, A)/[f^{∗}α·H^{∗}(X, A)+ f^{∗}Ω · H^{∗}(X, A)]. This means that f^{∗}(< α, β, Ω >) ⊂< f^{∗}α, f^{∗}β, f^{∗}Ω >.

1.5 There are also “higher order Massey products” for which we refer to the paper by Kraines and the book by McCleary ([K], [Mc-C] p. 290–297). The product ⟨u_{1}, …, u_{k}⟩ is defined only if ⟨u_{1}, …, u_{k−1}⟩ = ⟨u_{2}, …, u_{k}⟩ = 0 (but this condition is not sufficient). We shall not deal with them here, but our results and proofs extend to these higher order products.

1.6 It is known that the following relation holds (cf [McC] p. 293)

For p odd prime u ∈ H^{2m+1}(X; /p ) then βP^{m}(u) ∈ ⟨u, …, u⟩

Here β is the mod p Bockstein, and P^{m} the mod p Steenrod m-th power. It is then natural to ask whether the following hold

(1) if βP^{m}(u) ≠ 0 is it true that the critical levels associated to u and βP^{m}(u) are distinct unless the common critical level contains infinitely many critical points. In other words if 0 ∈ ⟨u, …, u⟩ then our method (see section 2) does not apply, however βP^{m}(u) may still be nonzero. Is that enough to get distinct critical points?*

(2) Same question with βP^{j}(u) for j ≠ m. However this cannot be expected for all j, since the Steenrod operations are stable operations (in a different sense than Massey products!!). As a result we have for instance that the Bockstein operation is non-zero on the lens spaces L(1, p). Hence it is also non-zero on its suspension. But the category of a suspension is at most two, while this would give us that it is at least three. Thus if there is any hope of a positive answer to our question, it seems that j has to be a function of m.

1.7 Of course the cup length may be defined for any cohomology theory (provided it is endowed with multiplication) like K-theory. It is not clear how one could extend the “product length”that we define in the next section.

1.8 In [F-H2], it is proved that if k is a field with suitable (nonzero) characteristic, H^{∗}(ΛX, k) has infinite cup length for X = S^{2n}, P^{n}, etc. However cohomology computations in nonzero characteristic are usually very hard, since one cannot make use of Sullivan’s minimal model. We will see that on the contrary, Massey products in rational cohomology are easily detected on the minimal model.

The article by Fadell and Husseini, [FH1], was brought to our attention by P. Majer, whom we wish to thank. This paper was written during a visit at the Forschungsinstitut Mathematik of the ETH in Zürich. We are grateful to Professors Moser and Zehnder for their invitation and hospitality, as well as the members and staff of the Institute for making our stay pleasant and productive, especially Mrs Rahel Boller for her patience in typing a hard to read manuscript. I

* In January 93, we received a preprint by Fadell and Husseini, which positively solves this question also thank Eric Séré for discussions related to 4.2. Finally, I am very grateful to an anonymous referee his extremely accurate and patient work. In particular for pointing out references [S] and [T], as welll as many inaccuracies, misprints, wrong references, and suggesting many useful improvements.

2. TIED COHOMOLOGY CLASSES

In this section we shall try to partially answer the following two questions:

(i) given a function f on the compact manifold X, and a class x in H^{∗}(X), which classes y satisfy c(y, f) ≥ c(x, f) for all f? We shall then say that y is weakly tied to x.

(ii) what are the classes y, weakly tied to x, such that equality cannot hold unless H̄^{j}(K_{c}) ≠ 0 for some j > 0. We then shall say that y is strongly tied to x.

Finally there is a stable version of this. If x, y are cohomology classes on M, we may consider an orientable vector bundle E on M, and the image T ∪x, T ∪y of x, y by the Thom isomorphism in H^{∗}(D(E), S(E)) where D(E), S(E) are the canonical disk and sphere bundle associated to E.Then, if for any orientable bundle E, T ∪ y is strongly (resp. weakly) tied to T ∪ x, we shall say that x and y are stably tied (resp. weakly stably tied).

Note that x is weakly tied to y in particular if the following property holds: for any map h : P → M we have h^{∗}(x) = 0 ⟹ h^{∗}(y) = 0.

We may similarly define the notion of tied sets of cohomology classes. We may thus set

Definition 2.1 We define T ℓ(M) to be the maximal integer k such that there is a sequence of cohomology classes α_{1}, β_{1}, α_{2}, β_{2}, … β_{k−1}, α_{k} such that β_{j} is weakly tied to α_{j} and α_{j+1} is strongly tied to β_{j} . The number tℓ(M) is defined similarly, with strongly tied replaced by stably tied and weakly tied by stably weakly tied.

Remark: Most of the time, we only need the case β_{j} = α_{j}.

The simplest example is of course given by Corollary 2.2 The classes x and y = x ∪ z are stably tied, provided of course y ≠ 0, and

deg(z) > 0.

This is nothing else than a rephrasing of theorem 1.1. Clearly cℓ(X) ≤ tℓ(X) ≤ T ℓ(X). Another example is given through cohomology operations. A cohomology operation is a ”natural map”, ψ : H^{∗}(X, R_{1}) → H^{∗}(X, R_{2}) where R_{1}, R_{2} are

rings, satisfying certain naturality assumptions. The most famous operations are the Steenrod squares. We refer to Spanier’s book for details on these questions. The only fact we need is that if h^{∗}(x) = 0 we have h^{∗}(ψ(x))) = ψ(h^{∗}(x)) = 0 (this is the naturality assumption). Applying this to the inclusion i_{λ} : X^{λ} → X, we easily get c(ψ(x), f) ≥ c(x, f). This is

Proposition 2.3 Given a cohomology operation, ψ, ψ(x) is always weakly stably tied to x. Remark: In general these two classes are not strongly tied. Even if in some instances the

classes are strongly tied they will not in general be stably and strongly tied.

Let us define the stable F-category of M, SF (M) to be the minimal number of critical points for a function f defined on a vector bundle over M, coinciding at infinity with a nondegenerate quadratic form in the fibers (see [V 3]).

Returning to the notion of tie length, an obvious result is Proposition 2.4 We have the inequality F (M) ≥ T ℓ(M) and SF (M) ≥ tℓ(M).

The rest of the section is devoted to giving examples of tied cohomology classes. Proposition 2.5 Let x, y ∈ H^{∗}(M) with deg(y) odd and x · y = dη. Then y · η is closed and

defines a cohomology class strongly tied to y. To be precise, we have if c(y, f) = c(y · η, f) = c that H̄^{j}(K_{c}) is non zero for some j ∈ {deg(x), deg(x) + deg(y) − 1}.

Proof: We set c = c(y, f) and shall prove, assuming H̄^{j}(K_{c}) vanishes for j in the above set, that c(y · η, f) > c + ϵ. According to our Main Lemma, we identify M^{c+ϵ} with M^{c−ϵ} ∪ V, where V is a neighbourhood of K_{c}. From the definition of c, we get a cochain ψ that we may assume to be defined on all of M, such that dψ = y on M^{c−ϵ}. Now change y to y′ = y − dψ, and η to η′ = η − (−1)^{deg(x)}x · ψ. Then we have that y · η and y′ · η′ are cohomologous. Indeed (y − dψ) · (η − (−1)^{deg(x)}x · ψ) = y · η − dψ · η − (−1)^{deg(x)}y · x · ψ + (−1)^{deg(x)}dψ · x · ψ =

y · η − d(ψ · η) + (−1)^{deg(ψ)}ψ · dη − (−1)^{deg(x)}y · x · ψ + (−1)^{deg(x)}dψ · x · ψ =

y · η − d(ψ · η) + ψ · x · y − x · y · ψ + (−1)^{deg(x)}dψ · x · ψ =

y · η − d(ψ · η) + \frac{1}{2}(−1)^{deg(x)}d(x · ψ · ψ).

Similarly, if we change x to x′ = x + dα and η to η′ = η + α · y the cohomology class of y · η and y · η′ are unchanged (note that have x′ · y = x · y + d(α · y) = d(η + α · y)). The same is true again if, without changing x or y, we add to η an exact cochain.

Now we may choose y′ to be zero in M^{c−ϵ} and x′ to be zero in V, we have dη′ = x′y′ = 0 in M^{c−ϵ} ∪ V. Now η′ is closed in V. Then, because H̄^{deg(η)}(K_{c}) = 0, we have for V′ a subset of V containing K_{c} but small enough, η′ vanishes in H^{deg(η)}(V′) = 0 (), so we may write η′ = dρ in V′. Then y′(η′ − dρ) is cohomologous to y′η′. But clearly y′(η′ − dρ) vanishes on M^{c−ϵ} ∪ V′, thus y · η is cohomologous to zero in H^{∗}(M^{c+ϵ}) , and we finally get c(y · η, f) ≥ c + ϵ.

Remark: The class of y · η is nothing else than the class of < y, x, y >. The only difference, is that in principle < y, x, y > is a family of cohomology class (differing by an element in y · H^{∗}(M)) while we proved that the class of y · η is independent of any choice. In fact of more careful study of Massey products shows that in many cases where some class is repeated in the product, the indeterminacy on the resulting Massey product may be reduced. In particular this is the case for < x, x, …, x > (see [Kr]).

Proposition 2.6 Let α, Ω in H^{∗}(M^{b}, M^{a}) and β ∈ H^{∗}(M). Then
c(< α, β, Ω >, f) ≥ min(c(α, f), c(Ω, f)).

Equality implies that H̄^{j}(K_{c}) ≠ 0 for j ∈ {deg(β), deg(α) + deg(β) − 1, deg(β) + deg(Ω) − 1}

Proof: Assume a, c are representatives of α, Ω, vanishing on M^{c−ϵ} ( with c = min(c(α, f), c(β, f))) and assume b = 0 on V, then

ab = dλ vanishes on M^{c−ϵ} ∪ V bc = dμ vanishes on M^{c−ϵ} ∪ V. Now assume the cohomology classes of λ and μ are zero on V. We may then assume that

these cochains actually vanish on V, hence aμ ± cλ = 0 on M^{c−ϵ} ∪ V. We thus get an element in < α, β, Ω > that vanishes in H^{∗}(M^{c+ϵ}) so

c(< α, β, Ω >, f) = sup{c(u, f) | u ∈< α, β, Ω >} > c.

Otherwise one of the cohomology classes of b, λμ is nonzero in H^{∗}(V), and since we may choose for V an arbitrarily small neighbourhood of K_{c}, our proof is complete.

Corollary 2.7 Assume in the above proposition that moreover α is weakly tied to Ω. Then the right hand side of the above inequality is equal to c(Ω, f) thus < α, β, Ω > is strongly tied to Ω.

Note that in particular, we allow the case α = Ω. We now compare c(< α, β, Ω >, f) to c(β, f) where β ∈ H^{∗}(M^{b}, M^{a}) and α, Ω ∈ H^{∗}(M).

Proposition 2.8 Assume < α, β, Ω > to be nonzero. Then, for c = c(β, f) , there is a critical level strictly above c, unless H̄^{j}(K_{c}) ≠ 0 for some j ∈ {deg(α), deg(Ω)}.

Remark: We cannot claim in general that < α, β, Ω > is even weakly tied to β.

Proof: Again, we may assume b = 0 on M^{c−ϵ} and a = c = 0 on V. Then ab = dλ and bc = dμ vanish on M^{c−ϵ} ∪ V, this last set being identified to M^{c+ϵ}. We may consider β̃the class of b in H^{∗}(M, M^{c−ϵ}). We shall distinguish two cases. Either we may choose λ and μ such that their cohomology classes on M^{c+ϵ} vanishes. Then we have α · β̃ = β̃· Ω = 0 in H^{∗}(M, M^{c+ϵ}). Thus < α, β̃, Ω > is well defined in H^{∗}(M, M^{c+ϵ}) and its image in H^{∗}(M) is < α, β, Ω > which does not contain zero by assumption. In this case c(< α, β̃, Ω >, f) ≥ c + ϵ. Otherwise, either α · β̃ or β̃ · Ω is non zero in H^{(}M, M^{c+ϵ}) and c(α · β̃, f) > c + ϵ.

Since we do not know in which case we end up, we may not infer that c(< α, β, Ω >, f) ≥ c + ϵ.

Remark: One of the unpleasant facts about this result is that we do not know exactly how the new critical level has been obtained. In particular this is unfit for iterating the procedure (this was one of the mistakes in the preprint version of Aug. 92) .

We now wish to prove that in the above propositions, we may replace strongly tied by stably tied (provided in corollary 2.7 we replace weakly tied by stably weakly tied).

Our main tool will be:

Lemma 2.9 (Stability of Massey products). Let p : E → M be an orientable vector bundle. Let T : H^{∗}(M) ⟶ H^{∗}(D(E), S(E)) be the Thom isomorphism. Then for α, Ω ∈ H^{∗}(M), β ∈ H^{∗}(M), we have (up to sign)

⟨p^{∗}α, T β, p^{∗}Ω⟩ = ⟨T α, p^{∗}β, p^{∗}Ω⟩ = ⟨p^{∗}α, p^{∗}β, T Ω⟩ = T · p^{∗}⟨α, β, Ω⟩ .

Proof of lemma: The Thom isomorphism is given by taking the product with the Thom class U ∈ H^{d}(D(E), S(E)) (d = dimension of the fibre). It is easy to check from the definition of Massey product that

⟨p^{∗}α, U · β, p^{∗}Ω⟩ = U · p^{∗}⟨α, β, Ω⟩ .

Proposition 2.10 Let E be an orientable vector bundle over M, f a function on D(E) going to infinity on S(E). Then if T is the Thom class of E, we have

c(T · ⟨α, β, Ω⟩, f) ≥ min(c(T · α, f), c(T · Ω, f)).

Similarly, we have for x, y in H^{∗}(M) such that deg(y) is odd and x · y = dη T · y and T · y · η are tied. Thus the classes which were found to be tied according to Prop. 2.5 and 2.6 are in fact stably tied.

One of the motivations for introducing the stability property comes from symplectic topology (see for example [L-S] or [V 3]). There we deal with functions on a vector bundle which are quadratic at infinity. This means that there is a function Q(x, ξ) (x is the base variable, ξ the fiber variable) which for fixed x is a non degenerate quadratic form, such that f(x, ξ) − Q(x, ξ) has compact support.

We then have the following easy consequence of the above results:

Corollary 2.11 Let E be an orientable vector bundle over M, f a function on E, quadratic at infinity. Then if T is the Thom isomorphism associated to the negative bundle of the quadratic form. Then

c(T · ⟨α, β, Ω⟩, f) ≥ min(c(T · α, f), c(T · Ω, f)).

Similarly, we have for x, y in H^{∗}(M) such that deg(y) is odd and x · y = dη, that T · y and T · y · η are tied.

Remarks:

2.12 We could define a third invariant, the product length, denoted by pℓ(X), defined in the same way as the tie length, except that we now assume that α_{j+1} is tied to β_{j} through propositions 3.1, 3.2, 3.4 or 2.7 that is through a cohomology operation, a cup product or Massey product (i.e. α_{j+1} is related to β_{j} either because α_{j+1} = ψ(β_{j}) where ψ is a cohomology operation, or α_{j+1} = β_{j} ∪ Ω_{j}, or α_{j+1} = ⟨β_{j}, Ω_{j}, α_{i}⟩ with i ≤ j, or....). Note that the analogue of our results for higher order Massey products may also be used. Let us point out that Massey products are sets of cohomology classes, so that the ”indeterminacy” increases at each step. It is thus important to specify in which order the operations are done, and this order actually has an effect on the length of the chain (because even though ⟨a, bd, c⟩ ⊂ ⟨a, b, c⟩d, equality does not hold in general). Clearly pℓ is a stable invariant, and yields a lower bound for tℓ.

2.13 It has been proved by Laudenbach and Sikorav ( [L-S], as a result of their proof of H. Hofer’s theorem [H 1]) that if φ_{t} is a Hamiltonian Ωow on T^{∗}M, then #(φ_{1}(O_{M})∩O_{M}) ≥ SF (M).

Since by our corollary SF (M) ≥ pℓ(M), we thus improve the previous bound cℓ(M).

2.14 A similar phenomenon occurs when T^{∗}M is replaced by a general symplectic manifold, and L is some lagrange submanifold as in [Fl]. Then Floer constructs a Ωow on a space X (which is not in general a manifold) which is decreasing for a function f, and such that p : X → L induces an injection in cohomology. Then it is clear that pℓ(X) ≥ pℓ(M) hence f|_{X} has at least pℓ(M) stationary points. One may then conclude that if φ_{t} is a Hamiltonian isotopy on a symplectic manifold (M, ω) such that [ω]π_{2}(M, L) = 0, then

#(φ_{1}(L) ∩ L) ≥ pℓ(L) .

2.15 A useful feature of critical points obtained as c(α, f) is that as we proved in [V2], if α ∈ H^{d}(X^{b}, X^{a}) then the level c(α, f) contains a critical point x, of index i(x), nullity ν(x) such that

i(x) ≤ d ≤ i(x) + ν(x) .

This may be an important information, since in many instances bounds on i(x) yield bounds on x ( see also 4.2).

3. EXAMPLES

A. Circle bundles over manifolds

Let N be a principal S^{1} bundle over the two torus T^{2} with nonzero Chern class. According to the Chern-Weil theory, there is a connection on this bundle, denoted by η, such that dη = x ∧ y, where x, y are generators of the 1 dimensional De Rham cohomology of T^{2}(for a cochain on T^{2} we denote by the same letter its pull-back on N). As a result, we see that in the sequence of cochains 1, x, x ∧ η, x ∧ y ∧ η each cochain is strongly tied to the previous one. Therefore the stable F-category of N is 4. We may generalize this as:

Proposition 3.A Let P be a circle bundle over a manifold M. Then if c_{1}(P) is the Chern class of the bundle, and there are classes in H^{∗}(M)−H^{0}(M) with x_{1}·x_{2} ⋯ x_{q} ≠ 0 and x_{1}·x_{2} ⋯ x_{q}·c_{1} = 0, we have that T ℓ(P) ≥ q. If moreover, p^{∗}(x_{1}) ≠ 0, we have T ℓ(P) ≥ q + 1

Proof: Using the Gysin exact sequence associated to the circle fibration, we see that in the cohomology of P the classes p^{∗}(u) · η ( where p is the projection p : P → N and η the connection cochain of P) when closed are always nonzero. The closedness is equivalent to u · c_{1} = 0. On the other hand, p^{∗}(u) is automatically close, and exact if and only if u = v · c_{1}. Thus the cochains p^{∗}(x_{1} ·x_{2} ⋯ x_{j})·η are closed by assumption, and nonzero in cohomology. This shows that T ℓ(P) ≥ q. If moreover p^{∗}(x_{1}) ≠ 0, we may consider the sequence p^{∗}(x_{1}), p^{∗}(x_{1}) · η, p^{∗}(x_{1} · x_{2} ⋯ x_{j}) · η, to show the second inequality.

Note in particular that if x_{1} has degree 1, p^{∗}(x_{1}) ≠ 0, since x_{1} cannot be of the cochain c_{1} · u, and if x_{1} has degree 2 and is not proportional to c_{1}, p^{∗}(x_{1}) ≠ 0 for the same reason.

B. The complement of the Borromean rings

Let B ⊂ S^{3} be the Borromean rings of figure 1

Figure 1

Note that B is the union of three pairwise unlinked circles which however may not be taken apart by an isotopy of ^{3}.

Massey proved this that ^{3} − B is not homeomorphic to the complement of three standard circles using Massey products as follows.

Let u, v, w ∈ H^{1}( ^{3} −B) be cohomology classes dual to the disk spanned by each of the circles in ^{3}. In other terms, if Ω is a loop in ^{3} − B, (u, Ω) is the algebraic intersection number of Ω with the disk bounded by the first circle.

Then u · v = v · w = u · w = 0, as this is the algebraic translation of the pairwise unlinkedness of the circles.

Now we claim that ⟨u, v, w⟩ ≠ 0 in H^{2}( ^{3} − B). Indeed, u · v = dα, v · w = dβ, where α and β may be visualized as follows on figure 2.

Since u · v may be represented (as a cochain) by the intersection of the disks D_{u} and D_{v}, α should be represented by the disk D_{α} with boundary D_{u} ∩ D_{v}. Since D_{v} ∩ D_{w} = ∅, ⟨u, v, w⟩ is equal to α · w, which is represented by D_{α} ∩ D_{w}. On figure 2, we see that this is a path from ∂D_{w} to ∂D_{u} which represents a nontrivial cochain as its intersection with a tubular neighbourhood of ∂D_{u} is ±1.

We may thus conclude

Proposition 3.B.1. F ( ^{3} − B) = 3.

Proof: The inequality F ( ^{3} − B) ≥ 3 follows from the following argument. Consider c(u, f) and c(w, f) and the two possibilities: either c(u, f) ≠ c(w, f), and since c(1, f) < c(u, f), c(1, f) < c(w, f), we get in this case three distinct critical levels, unless there are infinitely many critical points or c(u, f) = c(w, f). Then since < u, v, w > is nonzero, we have c(⟨u, v, w⟩, f) > min(c(u, f), c(w, f)) = c(u, f) = c(w, f) unless H̄^{1}(K_{c}) is non zero. So we have again that if f has only finitely many critical points, there are three critical levels: c(1, f), c(u, f), c((⟨u, v, w⟩, f) > c(u, f).

Because ^{3} −B is open, we have F ( ^{3} −B) ≤ dim( ^{3} −B) = 3 (see [T], p.206). This concludes our proof.

Let now U_{B} be a tubular neighbourhood of the Borromean rings, and define M_{B} to be the manifold obtained by gluing two copies of S^{3} − U_{B} along their common boundary (it is called the doubling of S^{3} − U_{B}) . We then have

Figure 2

Corollary 3.B.2. F (M_{B}) = 4.^{1}

Proof: Since M_{B} is a closed three-manifold, F (M_{B}) ≤ dim(M_{B}) + 1 = 4. Now the classes u, v, w ∈ H^{1}( ^{3} − B) extend to classes in H^{1}(M_{B}), that we denote by the same letters. Indeed doubling the disks D_{u}, D_{v}, D_{w} defined three closed surfaces in M_{B} which define by intersection three nonzero classes, since a meridian on the connected component of ∂U_{B} will have intersection ±1 with one of the surfaces. It is then easy to see that the Massey product of u, v, w is a nonzero class in H^{2}(M_{B}), hence by Poincaré duality, there is a class z in H^{1}(M_{B}) such that its product with ⟨u, v, w⟩ is the generator of H^{3}(M_{B}). We may again distinguish the cases c(u, f) ≠ c(w, f) from c(u, f) = c(w, f). In the first case we may for instance assume that c(u, f) < c(w, f). Denoting by w^{∗} a class Poincaré dual to w, we have, always assuming there are only finitely many critical points, that c(1, f) < c(u, f) < c(w, f) < c(w · w^{∗}, f). On the other hand, if c(u, f) = c(w, f), we may write as in the proof of 3.B.1, c(1, f) < c(u, f) = c(w, f) = min(c(u, f), c(w, f)) < c(⟨u, v, w⟩, f) < c(⟨u, v, w⟩ · z, f) and again we have four critical levels, and this concludes our proof.

1 as the referee pointed out, this follows from [T], theorem 3.3, once we admit that M_{B} is not a connected sum of a number of copies of S^{2} × S^{1} and S^{2} ˜×S^{1} (this last space denoting the total space of the non orientable S^{2} bundle over S^{1}). This could of course be deduced from the fact that Massey product do not all vanish on M_{B} !!

Thus this example is not the most striking one. We decided however to keep it as it illustrates a method that may be applied in higher dimensions.

C. Tied classes, Massey products and minimal models In this section all cohomologies will be with rational coefficients.

Massey products are then easily computed from the minimal model of a space.

Let (A, d) be a differential graded algebra, that is A = ⊕_{p≥0}A_{p}, A_{p} · A_{q} ⊂ A_{p+q}, and d : A_{p} → A_{p+1} satisfies d∘d = 0. Moreover, we assume xy = (−1)^{pq}yx, x ∈ A_{p}, y ∈ A_{q} (the algebra is thhen said to be commutative), and d(xy) = dx · y + (−1)^{p}x · dy.

We shall say that A is free if it is the tensor product of a symmetric algebra generated by elements of even degree and an exterior algebra generated by elements of odd degree. Then (A, d) is a minimal D.G.A. if it is free, and d(A_{n}) ⊂ ∑_{p=1}^{n} A_{p} · A_{n+1−p}.

The cohomology of a D.G.A. is given by H^{p}(A, d) = ker d|_{A_{p}}/ d(A_{p−1}).

Now if M is a manifold, Ω^{∗}M the D.G.A. of exterior forms on M, and d the exterior differential, a theorem of Sullivan ([S]) tells us that there is a minimal D.G.A. (M_{M}, δ) and a D.G.A. morphism M_{M} → Ω^{∗}M inducing an isomorphism in cohomology.

Results of Sullivan, Vigue–Poirrier (see [Su], [S-VP]) tell us how to compute the minimal model of ΛM, the free loop space of M, starting from M_{M}.

To compute Massey products in H^{∗}(M, R) from the knowledge of M_{M}, one proceeds as if (M_{M}, d) was the De Rham complex of M.

To illustrate this, let us give some examples. In what follows, subscripts will always indicate the degree of an element. We consider the space Λ( P^{n}). Its minimal model is S(x_{2}, ȳ_{2n}) ⊗ E(x̄_{1}, y_{2n+1}), where

S(x_{2}, ȳ_{2n}) is the symmetric algebra with one generator of degree 2, x_{2}, one of degree 2n, ȳ_{2n}; E(x̄_{1}, y_{2n+1}) is the exterior algebra on the generators x̄_{1}, y_{2n+1} of dimensions 1 and 2n + 1

respectively.

Moreover dx_{2} = 0, dx̄_{1} = 0, dȳ_{2n} = x_{2}^{n}x̄_{1}, dy_{2n+1} = x_{2}^{n+1}.

We claim that ⟨ȳ_{2n}^{q}x̄_{1}, x_{2}^{n}, ȳ_{2n}^{q}x̄_{1}⟩ = ȳ_{2n}^{q+1}x̄_{1} (up to some non zero constant factor).

Indeed, ȳ_{2n}^{q}x̄_{1} · x_{2}^{n} = dȳ_{2n}^{q+1}, ȳ_{2n}^{q}x̄_{1} has odd degree, so that ⟨ȳ_{2n}^{q}x̄_{1}, x_{2n}, ȳ_{2n}^{q}x̄_{1}⟩ is represented by ȳ_{2n}^{q+1} · ȳ_{2n}^{q}x̄_{1} + ȳ_{2n}^{q+1}ȳ_{2n}^{q} · x̄_{1}.

As a result,

c(ȳ_{2n}^{2q+1}x̄_{1}, f) ≥ c(ȳ_{2n}^{q}x̄_{1}, f))

so that ȳ_{2n}^{2q+1}x̄_{1} is strongly tied to ȳ_{2n}^{q}x̄_{1}.

Thus if the equality c(ȳ_{2n}^{2q+1}x̄_{1}, f) = c(ȳ_{2n}^{q}x̄_{1}, f) occurs,then K_{c} must have cohomological dimension at least 2n. (If f is the energy ∫_{S^{1}} |q̇|^{2}dt associated to the standard metric on P^{n}, then this is indeed what happens).

In particular tℓ(Λ P^{n}) = F (Λ P^{n}) = +∞, while cℓ(Λ P^{n}) = n + 2.

This is still true for any space X with the rational homotopy type of P^{n}.

Similar computations shows that cℓ(ΛS^{2n}) = 3 and T ℓ(ΛS^{2n}) = +∞ (in the odd dimensional case cℓ(ΛS^{2n+1}) = +∞). Indeed we have again generators x_{2n}, x̄_{2n}, y_{4n−1}, ȳ_{4n−2} with relations dx_{2n} = dx̄_{2n} = 0, dy_{4n−1} = x_{2n}^{2}, dȳ_{4n−2} = x_{2n} · x̄_{2n}. Again, x̄_{2n−1} · ȳ_{4n−2}^{2q+1} = ⟨ȳ_{4n−2}^{q}x̄_{2n−1}, x_{2n}, ȳ_{4n−2}^{q}x̄_{2n−1}⟩ is strongly tied to x̄_{2n−1} · ȳ_{4n−2}^{q}.

We now deal with an example by Felix and Halperin, concerning the category of the space CP^{2} ∨ S^{2} ∪_{ω} e^{7}. Note that as this is not a smooth manifold, and we are dealing with the standard category of the space. That our methods adapt to this setting is explained in remark 4.3.

In [Fe-H] it is proved that this space has category 4 (three with their conventions). Now a minimal model for this space is given by: x_{2}, x′_{2}, y_{3}, y′_{3}, y_{5}, v_{4}, v_{6}, x_{7}, ... satisfying dy_{3} = x_{2}x′_{2}; dy′_{3} = (x′_{2})^{2}; dy_{5} = (x_{2})^{3}; dv_{6} = y_{5}x′_{2} − y_{3}x_{2}^{2} − x_{7}; dv_{4} = y_{3}x′_{2} − y′_{3}x_{2}; ... (there are more generators and relations, but they do not occur in our computations). Now we see that using the sequence 1, x_{2}, x_{2}^{2}, y_{5}x′_{2} − y_{3}x_{2}^{2} we only need to prove that, if a function has finitely many critical points, then c(x_{2}^{2}, f) < c(y_{5}x′_{2} −y_{3}x_{2}^{2}). Here we are not able to prove that for instance tℓ(X) or even T ℓ(X) ≥ 4, but will instead be able to use corollary 2.7 to get the last ”extra” critical point. Indeed y_{5}x′_{2} − y_{3}x_{2}^{2} is simply < x_{2}, x_{2}^{2}, x′_{2} >, using the equalities d(y_{3} · x_{2}) = x′_{2} · x_{2}^{2}, d(y_{5}) = x_{2} · x_{2}^{2}. We may then apply corollary 2.7 to infer the above strict inequality. Note that due to the stability properties of the Massey product, this would still hold for a space with same the stable homotopy type as CP^{2} ∨ S^{2} ∪_{ω} e^{7}.

Question 3.C.1: If X is simply connected, is T ℓ(ΛX) = +∞?

The answer should be affirmative at least for ”formal spaces” (these are spaces X such that the minimal model of the space of differential forms coincides with the minimal model of the ring H^{∗}(X) with d = 0, examples are symmetric spaces, Kähler manifolds and their products).

4. Concluding remarks

4.1 After this paper was written, the author discovered the paper by James ([J]) where it is remarked that in a space with category at most 2, not only all cup products, but also Massey products should vanish. Moreover, the referee pointed out the reference [S], where secondary operations are used to give lower bounds of F (X).

4.2 Note also that the above method does not show the existence of infinitely many critical levels, for a function on a space with infinite tie length. Let us point out that on a space of infinite cup length, even if

c(x_{1} · x_{2} · . . . · x_{p}, f) = c(x_{1} · x_{2} · . . . · x_{p} · x_{p+1}, f) = … = c(x_{1} · x_{2} · . . . · x_{p} · x_{p+1} . . . · x_{p+r}, f)

then x_{p+1} . . . · x_{p+r} is nonzero on K_{c}, but by compactness of K_{c}, this can only occur for finitely many r. This argument breaks down for the tie length. However, an alternative approach to this problem is often possible.

(i)If the subsets X^{c} have the homotopy type of a finite dimensional space, then since H^{∗}(X^{c}) is finite dimensional for each value of c, the degree of a cohomology class with non zero image in H^{∗}(X^{c}) is bounded by a constant depending on c. Thus it is enough to have a sequence α_{k} of cohomology classes in H^{∗}(X) of unbounded degree to conclude that f has an unbounded sequence of critical levels. Note that the above assumptions are fulfilled for the Energy function on the loop space (i.e. E(Ω) = ∫_{S^{1}} |Ω̇(t)|^{2}dt)

(ii) The above argument still holds provided X^{b} − X^{a} has the homotopy type of a finite dimensional space and H^{∗}(X^{b}, X^{a}) contains cohomology classes of unbounded degree. (see [C-S] for related ideas). Note that whenever one is looking for solutions of a geometric, or reasonable analytic problem, the set of critical points usually must have finite (cohomological) dimension (e.g. the set of geodesics on a manifold has dimension at most twice the dimension of the manifold). It is enough to show that also the set of connecting trajectories has finite dimension to satisfy the above hypothesis.

Moreover, as X is a manifold and K a compact subset of X the set K may be covered by

finitely many ”convex sets”, that is K ⊂ ⋃_{j=1}^{p}V_{i} where the V_{i} are contractible as well as all the

intersections V_{i_{1}} ∩ V_{i_{2}} … V_{i_{q}}. Therefore, given a cohomology class of degree larger than p + 1 on

X it induces an exact class in the neighbourhood ⋃_{j=1}^{p}V_{i}. (this is obvious in Cech cohomology).

This implies that if the set of critical points and connecting trajectories in X^{b} − X^{a} is compact, for each pair (a, b), a sequence α_{k} of cohomology classes in H^{∗}(X) of unbounded degree will yield an unbounded sequence of critical levels. (see [C], [C-S]).

(iii) The Palais -Smale condition tells us that if the set of critical points is non compact, the set of critical levels must also be unbounded. Now assuming that f is C^{2}, we have that if d is the degree of the class α, for c = c(α, f) contains a critical point of index i, nullity ν such that d − ν ≤ i ≤ d (see 2.14). Thus if a sequence of c(α, f) is bounded, we have a compact set of critical points with either its index or its nullity unbounded. If we know that the index is finite at each critical point the index can not be unbounded, since it is lower semi continuous and finite at each point. If moreover we know that the nullity is bounded (this is for instance the case for the energy functional on the loop space, the bound being twice the dimension) we see that a bound on the degree implies a bound on the critical level. Notice also that this is redundant with (ii) since a bound on the nullity implies a bound on the dimension of the critical set, and a bound on the indices, a bound on the dimension of the connecting trajectories.

4.3 Some reader may wonder how much of our results carry over from F (X) to cat(X). The answer is “everything”: all results from section 1 to 3