Some properties of the relative Lusternik-Schnirelmann category

Octavian Cornea Université de Lille 1

U.F.R. de Mathématiques & U.R.A. 751 au CNRS 59655 Villeneuve D’Ascq, France

cornea@gat.univ-lille1.fr

December 10, 1996

Abstract

We study some properties of the relative Lusternik-Schnirelmann category of Fadell. In particular, it is shown that it admits a Ganea type description. We also show that, in a compact setting, it bounds the minimal number of critical points of a certain type of functions that have a more complicated boundary behaviour than that usually allowed. This is used to relate some forms of the Arnold conjecture in symplectic geometry to the Ganea conjecture in homotopy theory.

1 Homotopical remarks.

We work in the pointed category of the spaces with the homotopy type of a CW -complex. Let f : A ⟶ X be a fixed inclusion. Following Fadell the relative Lusternik-Schnirelmann category of X, cat(X, A), is the minimal n for which there is a covering of X with n+1 open subsets U_{i} of X, 0 ≤ i ≤ n, such that there is a homotopy of pairs (U_{0}×I, A×I) ⟶ (X, A) that contracts U_{0} to A and, for i ≥ 1, U_{i} is contractible in X. The absolute, classical case corresponds to A = ∗.

This invariant has been used in many variational problems (originally, in the absolute case, by Lusternik and Schnirelmann [13]) and has been generalized to the equivariant setting by Clapp and Puppe [1].

The properties of the relative invariant parallel those of the absolute case (compare with [12]). In particular [7] one immediately obtains a Whitehead type definition by adapting the proof from the absolute setting. We first define the fat wedge relative to A: T^{1}(X, A) = A, T^{n+1}(X, A) = (A × X^{n}) ⋃(X × T^{n}(X)) ⊂ X^{n+1}. Where T^{n}(X) is the standard fat wedge. For further use note that T^{n}(X, A) = (T^{n−1}(X, A) × X) ⋃(X^{n} × ∗). We suppose that X is connected. Then cat(X, A)≤ n iff the diagonal X ⟶ X^{n+1} can be deformed in the fat wedge T^{n+1}(X, A) by a homotopy H : (X, A) × I ⟶ (X^{n+1}, T^{n+1}(X, A)) which verifies p_{1} ∘ H_{t} ∘ f = f for all 0 ≤ t ≤ 1 where p_{1} is the projection on the first factor.

We want now to give another description of the relative category. We consider the following sequence of fibrations: E_{i}(X, A) ⟶^{j_{i}} B_{i}(X, A) ⟶^{p_{i}} X, 0 ≤ i with A ≃ B_{0}(X, A); p_{0} ≃ f; E_{i}(X, A) is the fibre of p_{i}; j_{i} is the inclusion of the fibre; B_{i+1}(X, A) is of the homotopy type of the cofibre of j_{i} and p_{i+1} is, up to homotopy, the map canonically induced by p_{i}. Notice that if f is ∗ ⟶ X this sequence of fibrations is equivalent to that given by the succesive filtrations in Milnor’s classifying construction for ΩX. We also denote by q_{i} the obvious map A ⟶ B_{i}(X, A) (q_{0} ≃ id). We have that q_{i+1} factors as t_{i} ∘ q_{i} where t_{i} : B_{i}(X, A) ⟶ B_{i}(X, A)/E_{i}(X, A) is the collapsing map.

This construction is commonly called the fibre-cofibre construction and is due to Ganea [9] (see also [10]).

Proposition 1.1 If X is connected cat(X, A) equals the least n such that, with the notations above, p_{n} admits a section s that verifies s ∘ f ≃ q_{n}.

Proof. The key point is to show that:

Lemma 1.2 Let p′_{n} be the fibration obtained by transforming the inclusion T^{n+1}(X, A) ⊂ X^{n+1} into a fibration. The fibration p_{n} is the pull back of p′_{n} over the diagonal.

Indeed, assuming the lemma one easily sees that the existence of a deformation like the one in the Whitehead type definition is equivalent to the existence of a section of p_{n} that respects the maps out of A.

Proof of the Lemma. We proceed by induction. This is clear for n = 0. We suppose it holds for n ≤ k. Recall that T^{k+1}(X, A) is the push out of the inclusions: i : T^{k}(X, A)×∗ ⟶ X^{k}×∗ and j : T^{k}(X, A)×∗ ⟶ T^{k}(X, A)×X. We transform all inclusions in X^{k+1} into fibrations and then take the pull backs over the diagonal. We obtain in this way spaces U, V, W, K being respectively the pull backs of T_{k}(X, A)×∗, X^{k}×∗, T^{k}(X, A)×X, T^{k+1}(X, A). There are also maps i′ : U ⟶ V, j′ : U ⟶ W and V ⟶ K, W ⟶ K. Moreover, one sees that by applying the cube theorem (on the pull back of a push out square) we find that the square given by these maps is a homotopy push out square. By induction, the fibration W ⟶ X coincides with p_{k−1}. The space V is contractible and j′ is identified with the inclusion of the fibre E_{k−1}(X, A) ⟶ B_{k−1}(X, A).

We will assume in the following that the biggest set in the pairs of spaces that we use is connected.

Remarks. We recall a few immediate consequences of the three descriptions above.

1. If the pair (X, A) is a pair retract of (Y, B), then cat(X, A)≤cat(Y, B).

2. If f factors as A ⟶ X′ ⟶ X, then we have cat(X, A)≤ cat(X, X′)+cat(X′, A).

3. We have cat(X, A)≥ min\{n : H^{∗}(X, A)(\overline{H}^{∗}(X))^{n} = 0\}.

4. If cat(X, A)≠ 0, then cat(X, A)=cat(X ∨ΣT, A) where the map A ⟶ X ∨ ΣT is the obvious composition.

Another invariant that is useful in relation with the category is the conelength (see [4,14] for more details).

Definition 1 The cone-length of X is the least n (or ∞) such that there are cofibration sequences Σ^{i}Z_{i} ⟶ X_{i} ⟶ X_{i+1} with 0 ≤ i < n, X_{0} ≃ ∗, X_{n} ≃ X.

The relation between category and cone-length is described next [4]:

Proposition 1.3 Assume cat(X)= n. There is a space Z such that Cl(X ∨ Σ^{n}Z)= n.

The covering description of the relative category implies cat(X, A)≤cat(X)+1. A special case is of interest:

Corollary 1.4 If f admits a homotopy section, then cat(X, A)≤cat(A).

Proof. We use the interpretation of Cl− in terms of obstruction theory (see [5] for details). Let g : Y ⟶ X be a map and suppose that g_{1} and g_{2} are two lifts of g up to A. We can certainly obtain maps g_{1}^{i} = q_{i} ∘ g_{1}, g_{2}^{i} = q_{i} ∘ g_{1}. They are liftings of g through p_{i}. The key point is that if i ≥cat(Y) we have g_{1}^{i} ≃ g_{2}^{i}. This is seen in the following way. First, after possibly wedging Y with a suspension we may assume Cl(Y)=cat(Y). Consider now a minimal cone decomposition of Y, T_{n} ⟶ Y_{n} ⟶ Y_{n+1}, (Y_{0} ≃ ∗) and suppose inductively that we have constructed a homotopy between the restrictions to Y_{k} of g_{1}^{k} and g_{2}^{k}. The obstruction to extend this homotopy to Y_{k+1} lies in [ΣT_{k}, E_{k}(X, A)] and is killed when passing to B_{k+1}(X, A). So we get the next step. The process starts for k = 0 because all our maps are pointed.

We turn now to our context and apply the above argument to the identity A ⟶ A and to the composition of the provided section with f.

2 Critical points.

We show here how one can use the relative category to bound the minimal number of critical points in cases when the boundary is present and when the behaviour of the function on the boundary does not allow a direct application of the standard proof of the Lusternik-Schnirelmann inequality. In fact, we obtain the inequality as a consequence of a somewhat stronger result.

We consider a smooth, riemannian, compact manifold (possibly with boundary) N and a smooth function h : N ⟶ R. Inside N we consider a topological compact manifold with boundary, M, of the same dimension as that of N. We also assume that M is smooth everywhere except that its boundary might have some corners. Precisely, ∂M = E ⋃ I where E and I are closed, smooth and the union is taken over ∂E = ∂I (which could be void). Moreover, we assume that there is an ϵ > 0 such that (∇h(x)/||∇h(x)||) · ν_{E} ≤ −ϵ for x ∈ Int(E) and (∇h(x)/||∇h(x)||) · ν_{I} ≥ ϵ for x ∈ Int(I). Here ν_{E}, ν_{I} are respectively normal, unit vectors to E and I, pointing out of M. We will refer to this as condition T.

These conditions are not too restrictive: for example if N is a cobordism such that ∂N is the disjoint union of V_{1} and V_{2} and h is maximal, constant and regular on V_{2} and minimal, constant and regular on V_{1}, then M can be obtained from N by just deleting a small colar of the boundary.

We fix a notation: γ_{t}(x) is the flow induced by −∇h. It points out on E which will be thus called the exit set of M with respect to γ and in on I which will be called the entrance set.

We are also going to assume that E is connected (which implies that M is connected too).

Assume that S is the maximal invariant set of γ in the interior of M. Let M^{−}(S) be the set of points x of M that verify lim_{t→−∞}γ_{t}(x) ∈ S.

Proposition 2.1 If U is a closed neighbourhood of M^{−}(S), then the pair (M, E) is a homotopy retract of pairs of (U ⋃ E, E).

Proof. The idea is the same as that of Proposition 2.1 in [6] applied here with some caution needed to control the boundary behaviour.

Define the function: t : M ⟶ R ⋃{∞}, t(x) = sup{τ ∈ R : γ_{τ}(x) ∈ M}. Call this function the departure-time function relative to E. It is obvious that t(x) = ∞ iff lim_{τ→∞}γ_{τ}(x) ∈ S. Also, if t(x) ≠ ∞, then γ_{t(x)}(x) ∈ E. Moreover, each point x such that t(x) ≠ ∞ has a neigborhood on which the function t is continuous. Indeed, the continuity of γ implies that there is a neighborhood of x containing only points for which the departure time is finite. As ∇h is never tangent to E it follows that for a given ϵ > 0 there is a possibly smaller neighborhood of x, U, such that for all y ∈ U we have that t(x) − ϵ < t(y) < t(x) + ϵ. This implies the continuity we are looking for.

For further use notice also that all these arguments do not actually depend on the smootheness of E. It is enough for E to be a topological manifold admitting a dense set of points where the normal is defined and for condition T to be satisfied with respect to these points.

Notice that the pair (K, E) is an index pair in the sense of the Conley index theory [2] for the flow γ and the pair (K, I) is an index pair for the inverse flow that we denote by γ′ (it is induced by ∇h). We use Lemma 5.3 of [16] and apply it to γ′. It provides a continuous function g : M ⟶ [0, 1] that has the following properties: g(x) = 1 iff x ∈ M^{−}(S), g(x) = 0 iff x ∈ I; for t < 0 and x satisfying g(x) ∈ (0, 1), γ_{τ}x ∈ M for τ ∈ [t, 0], we have g(γ_{t}(x)) < g(x). Put L = E ⋃ g^{−1}([δ, 1]). We choose δ close enough to 1 such that L ⊂ E ⋃ U.

We end the proof by showing that L is a deformation retract of M relative to E.

For this define the following function: j : M ⟶ R, j(x) = min(t(x), u(x)) with u(x) = sup{t ∈ R : g(γ_{t}(x)) ≤ δ} if g(x) ≤ δ and u(x) = 0 in the opposite case. We see that u is continuous around all points where it is finite. This implies that j is continuous and by compacity there is a T = max{j(x) : x ∈ M}. In turn, this allows us to define the deformation d(t, x) = γ_{t}(x) if t ≤ j(x) and d(t, x) = γ_{j(x)}(x) otherwise.

Corollary 2.2 cat(M, E)≤cat(U ⋃ E, E).

Corollary 2.3 The number of critical points of h is at least equal to cat(M, E).

Proof. Assume that the number of critical points of h is finite. We may also assume by modifying a bit the function in small neighbourhoods of the critical points that they are situated on different level sets. Consider two regular values of h, a < b. Take h^{−1}(a) ⋂ M and push it a little bit such that it will cut transversely E and I as well as their common boundary. We denote the resulting hypersurface by H_{a}. We clearly may assume that ∇h is never thangent to H_{a}. We proceed similarly for b thus obtaining H_{b}. Let K be the portion of M comprised between H_{b} and H_{a}. We put E′ = H_{a} ⋃(E ⋂ K) and I′ = H_{b} ⋃(I ⋂ K). Notice that the proof of the proposition above applies to K (relative to E′ and I′). In particular, if there is no critical point in K we have cat(K, H_{a})= 0. If there is just one critical point and a is sufficiently close to b then we may take as neighbourhood of K^{−}(S) a disk that intersects transversely H_{a} and is disjoint from ∂(M). Hence cat(K, H_{a})≤ 1. This implies the statement.

Here is the relation with the Arnold conjecture.

Corollary 2.4 If (M, ω) is a compact symplectic manifold with Euclidean cover, then any smooth 1-periodic Hamiltonian system has at least cat(M, M × S^{n}) contractible 1-periodic orbits for any n ∈ N^{∗} sufficiently large.

Proof. We use the function g constructed in the proof of Theorem 3.2 in [15]. We see that this function is defined on M × D^{2m}, it points in on M × ∂D^{m} × D^{m} and out on M × D^{m} × ∂D^{m} and the estimates at the end of the proof of Theorem 3.2 show that it satisfies condition T. We can multiply M × D^{2m} by a disk D^{q} and add to g a quadratic form q defined on D^{q} and which is negative definite. This sum g ⊕ q satisfies again condition T and has the same number of critical points as g, the exit set in this case is M × D^{m} × S^{m+q−1}. We apply our theorem to g ⊕ q.

Final comments. A form of Arnold’s conjecture claims that even in the absence of the Euclidean cover condition the number of 1-periodic orbits is at least equal to the least number of critical points of a smooth function on M (crit(M)). Results in this direction started with the work of Conley and Zehnder [3] and were then pursued by Floer [8] and Hofer [11]. They show that if [ω]|π_{2}(M) = 0, then the number of 1-periodic orbits is strictly bigger than the cup-length of the cohomology of M (with arbitrary coeficients). Clearly, this cup-length is smaller or equal than cat(M) and cat(M)+1 ≤ crit(M). It was proven by McCord and Oprea [15] that the cup-length can be replaced in the Euclidean-cover case by a homotopy invariant: the rational Toomer invariant.

The relative category cat(X, X×S^{n}) is bigger or equal than the cup-length of X+1 (with any coefficients). Indeed, take a class of maximal length in H^{∗}(X) and multiply it by δ(i^{n}) with i^{n} the fundamental class of S^{n}; δ being the connectant H^{n}(X × S^{n}) ⟶ H^{n+1}(X, X × S^{n})). It can also be shown that for all simply connected spaces X and in all the cases covered by the results of McCord and Oprea cat(X, X × S^{n}) is also bigger or equal than the rational Toomer invariant +1 of X (for all n > 0).

At the other end of the inequality conjectured by Arnold, the methods in [6] show that in general one might expect cat(M, M × S^{n}) to be a close estimate of the minimal number of critical points of a function g like the one in the proof of the corollary above (however, possibly not coming from a Hamiltonian).

This immediately suggests the following statement (that one could call the ”homotopical Arnold conjecture”):

Conjecture: cat(X, X ×S^{n})=cat(X)+1 for all connected, finite type CW - complexes X (n ∈ N^{∗}).

By Corollary 1.4 if this equality is true then cat(X × S^{n})=cat(X)+1 so the homotopical Arnold conjecture implies the Ganea conjecture.

Clearly, the following question is also open:

Problem: Is the statement of Corollary 2.4 valid in the absence of the Euclidean cover condition ?

References

1. M.Clapp, D.Puppe -Critical point theory with symmetries. J. reine angew. Math. 418 (1991), 1-29.

2. Ch.Conley -Isolated invariant sets and the Morse index. CBMS Regional Conf.Ser. in Math 38, Amer. Math. Soc. , Providence, R. I. , 1976.

3. Ch.Conley, E.Zehnder -The Birkhoff-Lewis Fixed Point Theorem and a conjecture of V.I. Arnold. Invent. Math. 73 (1983), 33-49.

4. O.Cornea -Cone-length and Lusternik Schnirelmann category. Topology 33 (1994), 95-111.

5. O.Cornea -Lusternik-Schnirelmann categorical sections. Ann. Scient. Ec. Norm. Sup. 28 (1995), 689-704.

6. O.Cornea -Cone-decompositions and degenerate critical points. Preprint Fall 1995.

7. E.Fadell, S.Y.Husseini -Relative category, products and coproducts. Preprint 1994.

8. A.Floer -Cuplength estimates on Lagrangian intersections. Comm. Pure Appl. Math. 42 (1989), 335-356.

9. T.Ganea -Lusternik-Schnirelmann category and strong category. Ill. J. of Math. , 11 (1967), 417-427.

10. W.Gilbert -Some examples for weak category and conilpotency. Ill. J. of Math. 12 (1968), 421-432.

11. H.Hofer -Ljusternik-Schnirelmann theory for Lagrangian intersections. Ann.Inst. H. Poincaré 5 (1988) 465-499.

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

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

14. H.Marcum -Cone-length of the exterior join. preprint 1996.

15. Ch.McCord, J.Oprea -Rational Ljusternik-Schnirelmann category and the Arnold conjecture. Topology 32 (1993) 701-717.

16. D.Salamon -Connected simple systems and the Conley index of isolated invariant sets. Trans of the A.M.S. 291 (1985) 1-41.