Homotopical dynamics of gradient flows

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

August 23, 1998

We review here some results focused on the following basic question: what are the homotopy properties that one can extract from a gradient flow?

We approach this question by decomposing it in three parts:

1. Identify what are the homotopical objects that are provided by the flow (e.g. critical points, Conley indexes).

2. Discover what are the relations that have to be satisfied by these objects (e.g. Morse inequalities, Lusternik-Schnirelmann type inequalities).

3. Realizability problem: given some homotopy objects that satisfy the relations from 2. is there a corresponding flow ? Is this flow unique up to some equivalence relation ?

We will consider only gradient flows induced by functions with isolated critical points, restrict our discussion to the finite dimensional, compact context and concentrate on the non-Morse case. One way to look at these questions in this case is to first treat numerical invariants, then local invariants, afterwards pairwise invariants (that concern pairs of consecutive critical points)...

At each such level we can look at the points 1.,2.,3. above.

Numerical and local data.

The information is grouped in a table.

We start with the well understood Morse case and proceed towards that of general critical points. In between, one has the ”reasonable” critical points [8], a class that contains all the isolated critical points of locally analytic functions.

Fix some notations first. Let W be a smooth, riemannian, compact, nmanifold with boundary and let f : W ⟶ R be smooth, maximal, regular and constant on ∂W. Consider W′ = f^{−1}[a, b] with a and b regular values, V_{0} = f^{−1}(a). Assume f has a single critical point in W′ which is denoted by P. If P is reasonable let A_{f}(P) ≃ f^{−1}(−∞, f(P)] ⋂ S(P, ϵ), where S(P, ϵ) is a sphere of radius ϵ around P. Its homotopy type does not depend on the different choices made if ϵ is small enough. The Conley index [5,31] of the critical point P with respect to the flow induced by −∇f is denoted C_{f}(P) it is the homotopy type of the quotient W′/V_{0}. We denote by CX the cone over the space X; Cat(X) is the strong Lusternik-Schnirelmann category of X (it is the minimal number of cone attachments needed to construct a space of the homotopy type of X starting from a point [22]) and cat(X) is the Lusternik-Schnirelmann category of X [23,25] .

Morse  Reasonable  General
Numerical inv. no. crit pts. of  no. crit. pts.  crit(f)
index k = crit_{k}(f) = crit(f)
Relations crit_{k}(f) ≥ b_{k}(W) crit(f) ≥ crit(f) ≥
Cat(W) + 1  cat(W) + 1
Realization π_{1}W = 0 π_{1}W = 0, π_{2}W = 0 same
n ≥ 6  π_{1}∂W = 0, W ≃ X
∃f perfect  dim(X) < n/2
∃g, crit(g) =  ∃g, crit(g) =
Cat(W) + 1  cat(W) + 2
Local inv.  index (P)  C_{f}(P)  C_{f}(P)
Relations  W′ = V_{0} ⋃ D^{k−1}  W′ = V_{0} ⋃ CA_{f}(P) C_{f}(P) =
C_{f}(P) = S^{k}  C_{f}(P) = ΣA_{f}(P) co-H-space
Realization ∀k ∀X Open
∃f, P  ∃f, P
index (P)= k  C_{f}(P) = ΣX

The information contained in the table is classical in the case of Morse functions. This case appears in the first vertical column (recall, that a Morse function is perfect if it has the minimal possible number of critical points allwowed by homological restrictions). The second column deals with the functions having only reasonbale critical points and the third with that of functions with general critical points. The results in these two last columns follow from [8]. The top inequality in the general case is the classical motivation for the introduction by Lusternik and Schnirelmann of the notion of category [25].

In the ”realization” lines we mention results providing functions and critical points realizing the restrictions indicated. For example, in the Morse case, the existence of a perfect Morse function on any simply connected manifold (this is a key, classical result, equivalent to the h-cobordism theorem ) or, in the last entry in the reasonable column, the ”realization” result claims that any finite suspension can be realized as the Conley index of some reasonable critical point.

The construction of the function g in the reasonable case implies the result in general because cat(X) ≤ Cat(X) ≤ cat(X) + 1 [33].

In fact g can be assumed to be self-indexed in the sense that its critical points P_{i}, −1 ≤ i ≤ Cat(X) can be indexed such that f(P_{i}) = i, the hessian of f has index i at P_{i} for i > −1 and P_{−1} is a minimum. Modulo the h-cobordism theorem this is the direct ”critical points” translation of the homotopy fact [6,7] that Cat(X) equals the cone-length Cl(X) = min{n : ∃ Σ^{i}Z_{i} ⟶ X_{i} ⟶ X_{i+1} cofibration sequences , 0 ≤ i < n, X_{0} ≃ ∗, X_{n} ≃ X}.

The relationship between the Lusternik-Schnirelmann category and the cone length plays an important role here. Its study was initiated by the papers [18] and [24]. In particular a conjecture of Lemaire and Sigrist claimed the equality of Cl and cat for rational spaces. The conjecture is now known to hold for rational Poincaré duality spaces [11]. In spite of the fact that strong category and cone length agree (even geometrically) the conjecture has been disproved by an example of Dupont [15] who constructed a rational space Y with Cl(Y) = 4 but cat(Y) = 3. Rationally, if cat(Y) = 2 then Cl(Y) = 2 also [19].

The fact that the Conley index of any isolated critical point is a co-H-space appears in non-explicit ways in the literature, for example in [14], it was made explicit, in a more general setting, in [8] and independently in [28].

There are no examples of critical points with a Conley index different from a suspension.

Pairwise data of consecutive critical points and stable homotopy.

Assume W′′ = f^{−1}[a, c] with a as above and c > b regular value such that f has a single critical point in W′′ − W′ that we denote by Q. We also assume that P and Q are both reasonable critical points.

We have two cofibration sequences A_{f}(P) ⟶ V_{0} ⟶ W′ and A_{f}(Q) ⟶ W′ ⟶ W′′. This provides a relative attaching map k(Q, P) : A_{f}(Q) ⟶ ΣA_{f}(P). We also have the same construction for the function −f. We get another relative attaching map k(P, Q) : A_{−f}(P) ⟶ ΣA_{−f}(Q).

The homotopy classes of these two maps are the invariants that we propose in the pairwise (consecutive) case.

The relation in this case is that the two maps are Spanier-Whitehead duals up to some twisting coming from the stable normal bundle of W [9]. In the stably parallelizable, Morse-Smale case this was proved by Franks [20]. The homological consequence of this duality result follows also from a result of McCord [26]. Again in the Morse-Smale case but for general manifolds the stable difference between k(P, Q) and k(Q, P) can be explicitely estimated in function of certain Hopf invariants and the stable normal bundle of the manifold. In particular, it belongs to the image of the obvious factor J^{q} : π_{∗}Ω^{q}Σ^{q}SO ⟶ π_{∗}^{S} of the classical J-homomorphism (q = min{index(P), index(Q)}). In turn, this can be viewed as an obstruction used to detect non-smoothable Poincaré duality spaces [13].

More stable ideas.

Let q : R^{m} ⟶ R a quadratic function of index k.

One of the key ingredients of both the duality result above and of the construction of the function g in the table is that, given a flow γ on some manifold M, the flow γ × −∇q on M × R^{m} suspends all the homotopical information k-times [9].

Understanding the behaviour of flows under suspension is also motivated by the ”stable” Morse /Lusternik-Schnirelmann theory of Eliashberg and Gromov [16] developed in the context of Lagrangian intersections.

Stable ideas are also relevant to this same subject in a different way. Consider the finite dimensional setting for the Arnold conjecture discussed in [16]. In this case, one needs to evaluate the minimal number of critical points of a function that is quadratic at infinity on M × R^{m}. As shown in [10] this number is estimated by cat(M, M × S^{k−1}) (if the index of the quadratic part is k). Here, cat(X, A) is the relative category introduced by Fadell [17].

The Arnold conjecture suggests that, in general, the inequality

cat(X, X × S^{k}) ≥ cat(X) + 1  (1)

can be expected to hold.

A stable version of this inequality has been recently proven by Moyaux [27]. He shows that cat(X, X ×S^{k−1}) is bigger than the least n for which the k-th suspension of the projection p_{n} : B_{n}ΩX ⟶ X admits a section (this number has been introduced and studied by Vanderbrouk [34] who called it the σ^{k}-category of X ; the existence of a section for p_{n} is equivalent to cat(X) ≤ n [22]; p_{n} is the projection at the n-th stage of Milnor’s classifying construction applied to ΩX).

As shown in [10] the inequality (1) implies the Ganea conjecture which claims cat(X × S^{k}) = cat(X) + 1. Iwase has recentely disproved the Ganea conjecture hence (1) can not be valid in full generality. Finding a universal, unstable lower bound for cat(X, X × S^{n}) is an interesting open question. Properties of the relative category in relation to counting critical points have also been investigated in [2] especially in the equivariant context. For other interesting applications see also [1].

Stable ideas have also been used efficiently by Rudyak [29] and Strom [32] in the homotopical study of the Lusternik-Schnirelmann category in relation to the Ganea conjecture. Rudyak has also applied his ideas to the Arnold conjecture context and, in particular, proved together with Oprea [30] that the category of each symplectic manifold (M, ω) with [ω]|_{π_{2}(M)} = 0 is equal to dim(M). It also follows from [30] and [27] that, in this case, cat(M, M × S^{k}) = cat(M) + 1.

Final Comments.

In what ways do the homotopical objects above - the Conley indexes of the critical points and pairwise data for critical points - caracterize the flow on W or, at least, the space W itself ?

In the Morse-Smale case this is quite well understood. Pairwise, consecutive information is insufficient to determine W even if, as indicated above, the stable difference between k(P, Q) and k(Q, P) is highly relevant for the stable normal structure. At the same time, the moduli spaces of connecting flow lines for all the pairs of critical points together with their compactifications provide the homeomorphism type of the manifold [3].

In the case of functions with isolated but general critical points much less is clear. Essentially, one only knows how to use the maps induced in homology by the relative attaching maps to determine the homology of W. This is the technique of connection matrices of Franzosa [21]. Homological results abund in Conley index theory and they apply to general flows, not only to gradient ones. However, at the geometric level, in this same setting of general flows, the only available general results are extensions of the suspension and duality statements mentioned before [12].

In what concerns going beyond homology in reconstructing the homotopy type of the manifold, the discussion above suggests that only stable global information can be expected to be recovered from pairwise flow data in the non-Morse context. For the reconstruction of the stable homotopy type it is likely that one would need to use techniques similar to those of [4].

References

1. T.Bartsch, Critical point theory and group actions, Springer 1993

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

3. R.Cohen, J.D.S.Jones, G.Segal, Homotopical Morse Theory, preprint 1992

4. R.Cohen, J.D.S.Jones, G.Segal, Floer’s infinite dimensional Morse theory and homotopy theory, The Floer memorial Volume, Birhauser (1995)

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

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

7. O.Cornea, Strong L.S.-category equals cone-length, Topology 34 (1995)

377-381

8. O.Cornea, Cone-decompositions and degenerate critical points, Proc. of the London Math. Soc. 77 (1998) 437-461

9. O.Cornea,Spanier-Whitehead duality and critical points, Homotopy Theory via Algebraic Geomtery and Group Representations, AMS Contemp. Math. 220 (1998)

10. O.Cornea, Some properties of relative L.S.-category, Fields Inst. Commun. 19, AMS (1998) 67-72

11. O.Cornea, Y.Felix and J.M.Lemaire, Rational Category and Cone-length of Poincaré Complexes, Topology 37 (1998) 743-748

12. O.Cornea, Homotopical Dynamics:Suspension and Duality, to appear in Journal of Erg. Theory & Dyn. Syst.

13. O.Cornea, Homotopical Dynamics II: Hopf invariants smoothings and the Morse complex, preprint August 1998

14. E.N.Dancer,Degenerate critical points, homotopy indices and Morse inequalities, J. Reine Ang. Math. 350 (1984), 1-22

15. N.Dupont, A counterexample to the Lemaire-Sigrist conjecture, to appear in Topology

16. Y.Eliashberg, M.Gromov, Lagrangian Intersection Theory, preprint December 1996

17. E.Fadell, S.Y.Husseini, Relative category, products and coproducts preprint 1994

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

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

20. J. Franks, Morse-Smale flows and homotopy theory, Topology 18 (1979), 199-215

21. R.Franzosa, Connection matrices for the Conley index, Trans. AMS

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

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

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

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

26. Ch. McCord, Poincare-Lefschetz duality for the homomlogy Conley index, Trans. of the A.M.S. 329 (1992) 233-252 27. P.M.Moyaux, Two lower bounds for the relative Lusternik Schnirelmann category preprint Spring 1998

28. J.Pears, Degenerate Critical Points and the Conley Index Thesis, University of Edinburgh, 1995

29. Y. Rudiak, On category weight and its applications, To appear in Topology

30. Yu.Rudyak, J.Oprea, The Lusternik Schnirelmann category of Symplectic Manifolds, preprint, Summer 1997

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

32. J.Strom, Two special cases of the Ganea conjecture, To appear in Trans. of the A.M.S.

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

34. L.Vanderbrouck, Adjunction spaces satisfying the Ganea conjecture, preprint 1997