Generating Functions, Symplectic Geometry and Applications

Claude Viterbo

Département de Mathématiques, Bâtiment 425 Université de Paris-Sud, F-91405 Orsay Cedex, FRANCE. email: viterbo@dmi.ens.fr

1. Symplectic manifolds, their Lagrange submanifolds and generating functions 2. Existence and uniqueness theorems for generating functions

3. Symplectic invariants, solutions of Hamilton-Jacobi equations and applications.

4. Generalized generating functions and applications to Floer homology computations.

5. Applications to Hamiltonian dynamics and obstructions to Lagrange embeddings

1. Symplectic manifolds, their Lagrange submanifolds and generating functions

A symplectic form on a manifold is a closed two form ω, nondegenerate as a skew-symmetric bilinear form on the tangent space at each point. Integration of the form on a two dimensional submanifold S with boundary ∂S in M associates to S a real number (positive or negative) the “area of S” which due to Stoke’s formula only depends on the curves ∂S, and the homology class of S rel ∂S. If moreover the form is exact, that is ω = dλ, the area of S is obtained by integrating λ over ∂S. In this case it is also possible to integrate λ on loops non homologous to zero and we get the notion of “area enclosed by a loop ”. However this area depends on the choice of λ. If this choice is fixed once for all, we shall then talk about an exact manifold. One should be careful about the fact that this notion is slightly different from that of a symplectic manifold with exact symplectic form (because in the latter case we have not chosen the primitive of ω).

It is a theorem of G. Darboux, that the simplest example, R^{2n} with the constant symplectic form σ = ∑_{j=1}^{n} dx_{j} ∧ dy^{j}, is also the universal local model (i.e. any symplectic manifold is locally symplectomorphic to (R^{2n}, σ)

The main example for us will be the exact manifold, T^{∗}N, the phase space of N, where the exact form is the “contraction tensor” given in local coordinates by ∑_{j=1}^{n} p_{j}dq^{j}, where q_{j} are local coordinates on N, and the p_{j} the dual coordinates.

Submanifolds of a symplectic manifold inherit naturally the two form induced by ω. This is naturally a closed form, but only exceptionally nondegenerate. In this case the submanifold is called symplectic. Other remarkable cases occur if the form vanishes on the submanifold, which is then called isotropic (if 2·dim(V) < dim(M)) or Lagrangian (if 2 · dim(V) = dim(M)). It is then maximal among isotropic submanifolds. In terms of area, any contractible curve on a Lagrange submanifold has zero area. In an exact manifold, any two homologous curves on L have the same area. If moreover any two curves have the same area (which is then necessarily zero), the Lagrange submanifold is called exact. This is equivalent to the exactness of the pull-back of λ on L.

In T^{∗}N there is a particularly simple family of Lagrange submanifolds. To any closed one form α we may associate L_{α} = {(q, α(q)) | q ∈ N}. The form induced by λ on L_{α} is just α, thus ω = dλ induces dα = 0. In particular, if α = df is exact we get an exact Lagrange submanifold.

A remarkable property of a Lagrange submanifold in T^{∗}N, is that it intersects the zero section more often than differential topologist would expect. For L_{f}, we see that points in L_{f} ∩ 0_{N} are in one to one correspondence with critical points of f. In any case, as N is compact, there are at least two such points. One of the Arnold conjectures, partially solved by H. Hofer in 1983 (see [H]), claims that for L exact and obtained from the zero section by a Hamiltonian isotopy, the number of points in L ∩ 0_{N} is bounded from below by the Lusternik-Shnirelman category of N (i.e. the minimal number of critical points for a function on N). For L_{f} this conjecture is obvious. However, the Lagrange submanifolds that may be written as L_{f} are exactly those for which the projection p : T^{∗}N → N restricts to a diffeomorphism.

Our main interest will be on Lagrange manifolds, and we shall represent them through their generating functions, an idea first introduced by L. Hörmander for different purposes (see [Hö]).

A generating function for the Lagrange submanifold , L, is a function S : E → R, defined on a vector bundle E over N, and such that

L = \{(x, \frac{∂S}{∂x}) | \frac{∂S}{∂ξ} = 0\}

(assuming that 0 is a regular value for \frac{∂S}{∂ξ}) S is Generating functions have the double advantage of preserving the following interesting property: the points in L_{S}∩0_{N} are in one to one correspondence with the critical points of S. The apparent drawback is that there are of course functions on E with no critical points (since E is non compact) but this may be restored if we restrict ourselves to Generating Functions Quadratic at Infinity (abbreviated as G.F.Q.I. ):

Definition (G.F.Q.I. ). A generating function S is a G.F.Q.I. if and only if there exist a fibrewise quadratic nondegenerate form Q(x, ξ) such that S(x, ξ) − Q(x, ξ) has compact support.

The main example is associated to a symplectic diffeomorphism, ϕ, that we assume to be the time one flow of a compact supported Hamiltonian. Then Γ_{ϕ} = {(z, ϕ(z)) | z ∈ R^{2n}} is a Lagrange submanifold of R^{2n} × R̄^{2n}(R̄^{2n} is simply R^{2n} with the symplectic form −ω). We shall identify R^{2n} ×R̄^{2n} with T^{∗}(Δ) = T^{∗}(Γ_{Id}) (where Δ is the diagonal). Note that if ϕ_{t} has compact support, Γ_{ϕ} coincides with the zero section outside a compact set. We may thus compactify the base of T^{∗}(Δ) to T^{∗}(S^{2n}), and simultaneously compactify Γ_{ϕ} to Γ̄_{ϕ}. Then Γ̄_{ϕ} coincides with the zero section near the point at infinity on S^{2n}, and is obtained from 0_{S^{2n}} by a Hamiltonian isotopy. We shall usually work in this last setting, since it is more pleasant to work with compact bases.

2. Existence and uniqueness theorems for generating functions

From now on, except in section 4, we shall only consider generating functions quadratic at infinity. Let L be the space of Lagrange submanifolds hamiltonianly isotopic to the zero section, G the set of G.F.Q.I. of elements of L. The obvious projection G → L is denoted by π., and G_{L} will be π^{−1}(L).

For S, T two G.F.Q.I. on E and F, we denote by S ⊕T the G.F.Q.I. on E ⊕F defined by (S ⊕ T)(x, ξ, η) = S(x, ξ) + T (x, η)

Let us introduce the following two equivalence relations on G:

(a) S_{1} ≃ S_{2} if and only if there are nondegenerate quadratic forms Q_{1}, Q_{2} on F_{1}, F_{2} and a fibre preserving diffeomorphism Φ : E_{1} ⊕ F_{1} → E_{2} ⊕ F_{2} such that (S_{1} ⊕ Q_{1}) = (S_{2} ⊕ Q_{2}) ∘ Φ.

(b) S_{1} ≃ S_{2} if and only if there are G.F.Q.I. for the zero section Σ_{1}, Σ_{2} defined on F_{1}, F_{2}, and a vector bundle isomorphism Ψ from E_{1} ⊕ F_{1} → E_{2} ⊕ F_{2} such that (S_{1} ⊕ Σ_{1}) = (S_{2} ⊕ Σ_{2}) ∘ Ψ.

Note that in (a) there are more permissible “stabilizations” but fewer isomorphisms than in (b). However the equivalence relation(b) seems weaker than (a). The point of using (b) is that it makes certain proofs easier, without weakening too much the conclusions. Making the quotient of G by one of these equivalence relation a topological space, and a CW complex is needlessly complicated; it is better to notice that the concept of continuous map from a cube in G/ ≃ (where ≃ is one of the equivalence relations (a) or (b)) is clear to everyone: it is the projection of a continuous map from the cube to G. It is then clear what a continuous map from a polyhedron into G/ ≃ will be. Smooth maps from a finite dimensional manifold to G/ ≃ are similarly defined .

The existence and uniqueness results may now be stated as follows.

Theorem (existence and uniqueness of G.F.Q.I. ). ([L-S], [V 1], [Th], [V 2])

Let G̃ be one of the quotient spaces corresponding to (a) or (b). Then the map π̃ : G̃ → L induced by π is a Serre fibration. The fibre is reduced to a line described by the set of S + C with C ∈ R, and S any element of G.

As an application, it is easy using the equivalence relation(b) to see that

Corollary (GF Symplectic homology). ([El],[Tr])

Given any two G.F.Q.I. for L there is a unique integer m and a unique real number l, such that H^{k}(S^{b}, S^{a}) = H^{k−m}(S^{b−l}, S^{a−l}).

In particular, if we normalize in any reasonable way the constant, we have a well defined ring H^{∗}(S^{b}, S^{a}), for any pair of real numbers (a, b). Note moreover, that there is an action by the cup product of H^{∗}(E) = H^{∗}(N) on this ring (so it is actually an H^{∗}(N)-module). This is in particular the case if we consider the Lagrange submanifolds Γ̄_{ϕ} associated to a compact supported Hamiltonian isotopy ϕ. Then since Γ̄_{ϕ} coincides with the zero section of T^{∗}(S^{2n}) outside a compact set, the generating function has a unique critical point over the point at infinity. We shall normalize S so that this critical level is zero.

Remark Note that, as was pointed out by Eliashberg and Gromov, G.F.Q.I. yield more precise information than just their homology. For instance, if the base space N has nontrivial fundamental group, for instance the Reidmeister torsion may be recovered from S, a more refined information than just the homology of the space. Also, for families of Lagrange (or Legendre) submanifolds, invariants coming from pseudo-isotopy may be obtained (see [E-G]). It also implies a priori more critical points than one may expect from the Morse inequalities, or the ring structure of the cohomology (using Lusternik-Shnirelman theory). In fact, because we are dealing with real topological spaces, we get for free the secondary operations, which yield also more critical points (see [V6]).

As an example of this last fact, we may consider the manifold N, total space of a nontrivial circle bundle over the two torus. Even though its cup length is 3, any function on N has 4 critical points. Moreover, for any Hamiltonian isotopy ϕ_{t} on T^{∗}N, ϕ_{1}(0_{N}) ∩ 0_{N} has at least 4 critical points (and not three as one may expect from Floer’s theory).

Remark: There is no reason to limit oneself to the cohomology functor. In fact any strongly stable homotopy functor is invariant through the equivalence relation(a) (for (b) it is more delicate). We shall see below how this is connected to the idea that there exists a stable Floer homotopy (see [CJS]).

3. Symplectic invariants, solutions of Hamilton-Jacobi equations and applications.

We shall briefly sketch a particularly simple construction of a symplectic invariants, related to the “capacity” defined by Ekeland and Hofer in [EH1] and [EH2], and also a new class of solutions for first order Hamilton-Jacobi equations.

Let S(x, ξ) be a G.F.Q.I. for the Lagrange submanifold L. Note that for c large enough, E^{c}, E^{−c} has the homotopy type of D(E^{−}), S(E^{−}), where E^{−} is the sum of the negative eigenspaces of the quadratic form defined by S at infinity.

Given a cohomology class α in H^{∗}(N), Thom isomorphism T : H^{∗}(N) → H^{∗+d}(D(E^{−}), S(E^{−})) is given associates the class T ∪ p^{∗}(α). Now to S, we may associate the critical level

c(α, S) = inf{λ | T ∪ α is nonzero inH^{∗}(E^{λ}, E^{−c})}

By strongly stable homotopy functor, F, we mean that given an orientable vector bundle, E over X, and D(E), S(E) its unit disc and sphere bundle respectively, then F (X) is isomorphic to F (D(E)/S(E)). K-theory for instance, would be such a functor.

To simplify notations, we shall again denote by α the class T ∪ p^{∗}(α).

The number c(α, S) is a critical value for S, which implies that the corresponding level of S contains at least one critical point. In turn, this yields an intersection point of L_{S} and 0_{N}.

Let μ_{L} be the generator of H^{n}(N), and 1 the generator of H^{0}(N). The number γ(L) = c(μ, S) − c(1, S) is a sort of ”norm” for L: γ(L) = 0 if and only if L = 0_{N}.

Remark: this holds only for embedded Lagrange submanifolds. There are immersed Lagrange submanifolds admitting a G.F.Q.I. , such that γ(L) = 0 but L ≠ 0_{N} (however it is always true that L contains 0_{N}).

The most interesting case is when L = Γ_{ϕ}. Then using the compactification explained at the end of section 1, we get two critical values c(1, S) and c(μ, S). that we denote as c_{−}(ϕ) and c_{+}(ϕ) respectively. The main properties of c_{+} and c_{−} are summarized in the following proposition:

Theorem 2. (Properties of c_{−}(ϕ) and c_{+}(ϕ))

i c_{(}ϕ) = −c_{+}(ϕ^{−1}).

ii c_{(}ϕ) ≤ 0 ≤ c_{+}(ϕ) and c_{(}ϕ) = 0 = c_{+}(ϕ) holds only if ϕ = Id.

iii c_{+}(ϕψϕ^{−1}) = c_{+}(ψ) and the same holds for c_{−}.

iv c_{+}(ϕψ) ≤ c_{+}(ϕ) + c_{+}(ψ), and the equality is reversed for c_{+} replaced by c_{−}.

The only nontrivial result, is (iv).

Most of the basic results of symplectic homology can be proved very simply using this approach. One defines a symplectic invariant, the capacity, by setting

c(U) = sup{c_{+}(ϕ) | ϕ is the time one map of a Hamiltonian isotopy generated by

a Hamiltonian supported in U }

Clearly, c(ψ(U)) = c(U) for any symplectic diffeomorphism ψ, and if U ⊂ V, we clearly have c(U) ≤ c(V). Since we may also prove that c(B^{2n}(r)) = c(B^{2}(r) × R^{2n−2}) = πr^{2}, we get immediately Gromov’s theorem

Theorem (Gromov). If there is a symplectic embedding from B^{2n}(r) into B^{2}(r′) × R^{2n−2}, then we must have r ≤ r′

This implies for instance that the group of symplectic diffeomorphisms is closed for the C^{0} topology in the group for all volume preserving diffeomorphisms.

A particular feature of the invariant we defined, is to be suited for comparing the capacity of a set with the capacity of a symplectic reduction. This follows for instance from the fact that we have the following inequalities: Let L be a Lagrange submanifold in T^{∗}(N × P), and for each p ∈ P, let L_{p} = (L ∩ T^{∗}N × T_{p}^{∗}P)/T_{p}^{∗}P that we consider as a submanifold of T^{∗}N.Then, if S is a G.F.Q.I. for L, we have that L_{p} has S_{p} as a G.F.Q.I. , where S_{p}(n, ξ) = S(n, p, ξ).

Let us now see how generating functions may be used to find some remarkable solutions of Hamilton-Jacobi equations (this is partially due to M. Chaperon and J.C. Sikorav, [Ch2]).

The idea is that if L is a Lagrange submanifold in T^{∗}(N × [0, T]) contained in the hypersurface τ + H(t, x, p) = 0, and if L has a G.F.Q.I. S, then the function defined by

u(x) = inf{λ | the generator of H^{∗}(E_{x}^{c}, E_{x}^{−c}) in H^{∗}(E_{x}^{λ}, E_{x}^{−c})is nonzero}

“satisfies the Hamilton-Jacobi equation”

\frac{∂u}{∂t} + H(t, x, \frac{∂u}{∂x}) = 0

Moreover this is a C^{0} solution, satisfying the equation almost everywhere, and having a certain number of additional properties. Our main result is that these solutions extend to the case where H and u_{0}(x) = u(x, 0) are only continuous.

Theorem 3. Let J be the map from C^{k}(N) × C^{k}([0, T] × T^{∗}N) into C^{Lip}([0, T] × N), which associates to (u_{0}, H) the above constructed solution of the equation

\frac{∂u}{∂t} + H(t, x, \frac{∂u}{∂x}) = 0 ; u(0, x) = u_{0}(x).

This map is continuous for the natural topology on each space. Then J extends to a map J̄ from C^{0}(N) × C_{q}^{Lip}([0, T] × T^{∗}N) to C^{0}([0, T] × N) (which also sends C^{Lip}(N) × C^{Lip}([0, T] × T^{∗}N) to C^{Lip}([0, T] × N)). Moreover u = J̄(u_{0}, H) solves almost everywhere the above Hamilton-Jacobi equation.

Here C_{q}^{Lip}([0, T] × T^{∗}N) means continuous in the p variable, and Lipschitz in the q variable.

The remarkable class of solutions thus defined is different from the one defined by P.L. Lions under the name of “viscosity solutions”, as was shown by A. Ottolenghi (personnal communication).

4. Generalized generating functions and applications to Floer homology computations

The first idea in this section, is to give two infinite dimensional extensions of the notion of generating function, and to prove a suitably adapted uniqueness theorem.

The first adaptation of the definition of G.F.Q.I. to the infinite dimensional setting is easy under some natural assumptions. First we may replace the finite dimensional vector bundle over N, by a Banach bundle over N. Then S : E → R is a smooth function satisfying the Palais-Smale condition. The smoothness and transversality assumptions clearly extend to the Banach space setting. And finally, our proof of uniqueness can be also extended without any new ingredient to this case. Note however that G.F.Q.I. be such that the quadratic function at infinity should have finite index. A typical example is as folllows: let L(t, q, q̇) be a Lagrangian on T N such that \frac{∂^{2}}{∂q̇^{2}} L is invertible.

S(q) = ∫_{0}^{1} L(t, q, q̇)dt

defined on the Banach bundle P = {q : [0, 1] → N | q̇(0) = 0} and π : P → N is given by π(q) = q(1). We claim that S is a generating function for ϕ_{1}(0_{N}) where ϕ_{t} is the flow associated to the Hamiltonian H obtained from L by Legendre duality. The computation is omitted, and we refer to the sequel of this section for a very similar computation. Even though, S will never be quadratic at infinity, but if L(t, q, q̇) = ∥q̇∥^{2} outside a compact set, it is easy to show that H^{∗}(E^{c}, E^{−c}) = H^{∗}(N) for c large enough (it is easy to prove this by comparison with L = ∥q̇∥^{2})

Now we turn to the more subtle version of infinite dimensional generating functions, that we shall call Floer generating functions.

This relies on Floer’s idea to deal with the variational theory of the action functional, as there was not, prior to Floer’s work, any reasonable approach to the variational study of this functional on a general manifold (see [R] for a finite dimensional approach in the case of R^{2n}).

One way to understand the introduction of generating functions is to consider the action functional as such a function. Let E be the set of paths, E = {γ = (q, p) : [0, 1] → T^{∗}N | p(0) = 0} (we do not specify the regularity of the path, as it is of no interest for the moment) and π : E → N be the map γ → q(1). Then consider the function A_{H} defined as

A_{H}(γ) = ∫_{0}^{1}[pq̇ − H(t, q, p)]dt

.

We have that

DA_{H}(γ)δγ = ∫_{0}^{1}[(q̇ − \frac{∂H}{∂p})δp − (ṗ − \frac{∂H}{∂q})δq](t, q, p)dt + p(1)δq(1).

Thus, the set of γ such that the derivative of A_{H} in the direction of the fibres of π vanishes correspond to solutions of

q̇ − \frac{∂H}{∂p} = 0 , ṗ − \frac{∂H}{∂q} = 0

that is γ(t) = ϕ_{t}(γ(0)) = ϕ_{t}(q(0), 0). Now since for such a γ, \frac{∂A_{H}}{∂q(1)} = p(1) so the set of points (q(1), \frac{∂A_{H}}{∂q(1)}) is ϕ_{1}(0_{N}).

The function A_{H}, is particularly difficult to study from a variational point of view, because all its critical points are of infinite index and coindex. As a result, we have for example that H^{∗}(A_{H}^{b}, A_{H}^{a}) vanishes. In a famous series of papers, A. Floer explained, using some ideas of Thom, Smale, Conley and Witten, how to define the groups F H^{∗}(A_{H}^{b}, A_{H}^{a}) that are a sort of `‘middle dimensional cohomology´’ of (A_{H}^{b}, A_{H}^{a}) (see [F1], [F2], and also [F-H-W]).

In particular these groups only depend on L, so we denote them by F H^{∗}(L; a, b). Now we have:

Theorem (Uniqueness of symplectic homology).

F H^{∗}(L; a, b) = GH^{∗}(L; a, b)

This circle of ideas may be adapted to the periodic orbit problem. In general one studies the number of fixed points for a Hamiltonian flow on M. The periodic orbits are critical points of the action functional ∫_{D^{2}} u^{∗}ω − ∫_{S^{1}} H(t, z)dt, where u is any map such that u_{|S^{1}} = z. Of course the value of ∫_{D^{2}} u^{∗}ω may depend on the choice of u, not only on z, but this is not the case provided ω vanishes on π_{2}(M), that we shall henceforth assume. Then, one encounters the same difficulties and cure (through Floer methods) as in the Lagrangian case. With the above assumptions, for compact M, one gets the isomorphism

F H^{∗}(M) = H^{∗}(M)

Of course such an equality is not perfectly honest, unless we say which structures are transported from the right hand side to the left hand side. For instance, the original statement of Floer only deals with the additive group structures. In [V3], we proved that also the multiplicative structure extends to F H^{∗}(M) (and the isomorphism between F H^{∗}(M) and H^{∗}(M) preserves this structure. This implies a weak version of the Arnold conjecture for these manifolds, and has been extended to more general manifolds by several authors (Floer, Hofer-Salamon, ...). In [V3], we proved that also the multiplicative structure extends to F H^{∗}(M) (and the isomorphism between F H^{∗}(M) and H^{∗}(M) preserves this structure. Work by Cohen-Jones-Segal and Fukaya, seems to indicate that much more structure is defined on F H^{∗}(M) (for example cohomology operations ...)

On noncompact manifolds, however the situation is more interesting. For example in T^{∗}N, let us look for periodic orbits of a Hamiltonian H, such that H(q, p) = ∥p∥^{2} at infinity.

We shall denote the Floer homology associated to H as F H^{∗}(T^{∗}N). It is clear as in the compact case, that the Floer homology only depends on the behavior of H at infinity. Now for H(q, p) = ∥p∥^{2}, the Hamiltonian flow is the geodesic flow, and we know perfectly well its periodic orbits: they are in one to one correspondence with closed geodesics. There is however another functional with the same critical points: the energy functional E(q) = ∫_{S^{1}} q̇^{2} defined on the loop space of N , ΛN. This is not enough to prove that F H^{∗}(T^{∗}N) = H^{∗}(ΛN), for this one would have to show that the connecting trajectories for the gradient flow of E are in one to one correspondence with the Floer trajectories, solutions of ∂_{J}u = −∇H.

As this is not easy to prove, we shall use an approach derived from the ideas of section 2. Indeed, we introduce the notion of Floer generating function as a generating function of the form:

F_{H,S}(γ, ξ) = A_{H}(γ) ⊕ S(γ(1), ξ)

where S is a G.F.Q.I. for L_{S}. It is not hard to see, that formally, F is a generating function for ϕ^{−1}(L_{S}). But beside this, we want to define F H^{∗}(F^{b}, F^{a}): this is done as in Floer homology, by considering critical points and connecting trajectories.

It is easy to see that for H ≡ 0, F H^{∗}(F^{b}, F^{a}) = H^{∗}(S^{b}, S^{a}), while for S = 0, F H^{∗}(F^{b}, F^{a}) = F H^{∗}(L; a, b). These are the main ingredients of the theorem’s proof.

Remark: Let us notice that the isomorphism between F and G homology has not been proved to be natural. If this would be true, as it seems likely to be, it would have quite interesting consequences, as all cohomology operations that are easily defined for G-homology would automatically be well defined on Floer cohomology, a much less trivial fact. Of course, if one could prove that the spaces S^{b}/S^{a} may be canonically identified up to suspension, we would have realized the program of Cohen Jones Segal of defining the “stable Floer homotopy type ” in this setting.

5. Hamiltonian dynamics, Obstructions to Lagrange embeddings

The previous theorem has a number of important applications. The first one will deal with the following question:

given L, N is there a Lagrange embedding from L to T^{∗}N.

The answer is of course yes for L = N, and no other example is known if we moreover require the embedding to be exact. On the other hand, without the exactness assumption, it is not even known whether there is a Lagrange embedding of the Klein bottle into R^{4}. In the sequel, all Lagrange embeddings will be assumed to be exact.

A conjecture by Arnold, claims that the only such embeddings are obtained by applying a symplectic diffeomorphism to the zero section. At least, since proving this statement seems to be out of reach, we may try to show

Conjecture Let L be an exact Lagrange submanifold in T^{∗}N. Then the projection of L onto N has nonzero degree.

Note that this implies Gromov’s theorem, claiming, in Sikorav’s formulation, that there is no exact embedding into T^{∗}M for open M (remember that L is always compact).

The connection with the former section is as follows. A Lagrange embedding j of L into T^{∗}N extends to an embedding of a neighborhood of the zero section of T^{∗}L into T^{∗}N. This induces a map from the Floer cohomologies, from Φ(j) : F H^{∗}(T^{∗}L) = H^{∗}(ΛL) → F H^{∗}(T^{∗}N) = H^{∗}(ΛN), satisfying the following algebraic property:

Φ(j)(x ∪ Λ(j)^{∗}(y)) = Φ(j)(x) ∪ y

It is easy to construct, for many pairs of manifolds (L, N) obstructions to the existence of such maps. We refer to [V 6] for many examples. Let us just quote the solution to a previously open question

Theorem . There is no exact embedding from T^{2} to T^{∗}S^{2}.

Remark The method of proof in [V7] (see also [V5])is more complicated, because the relation between Floer cohomology and generating functions had not been established yet. Thus the whole proof is based on generating functions methods.

We also mention another application of the above computation to the Weinstein conjecture in cotangent bundle. A compact hypersurface in a symplectic manifold is said to be of contact type, if there is a conformal vector field (i.e. L_{ξ}ω = ω) defined in a neighborhood of the hypersurface, and transverse to it. The characteristic flow on a hypersurface is the flow of a time independent Hamiltonian having the hypersurface as a regular energy level. The special feature of a contact type hypersurface, is that a neighborhood of it is foliated by hypersurfaces having diffeomorphic characteristic flow.

The conjecture of Weinstein, claims that such a hypersurface always has a periodic characteristic. This was first proved in R^{2n} by the author ([V1]), later extended in joint work with Hofer and Floer ([HV1], [FHV], [HV2]). In particular in [HV1], it is proved that the conjecture holds for a contact hypersurface in T^{∗}N surrounding the zero section. But, strangely enough, it was left open for a general contact hypersurface in , even though, as pointed out by M. Chaperon, if N has a Lagrange embedding in R^{2n}, then the Weinstein conjecture in T^{∗}N holds as a consequence of it holding in R^{2n}.

The above computation, together with the informations on the structure of the ring H ∗(ΛN) in the simply connected case, due to Goodwillie (see [Go]), may be exploited to prove (see [V7]):

Theorem (Weinstein conjecture in s.c. cotangent bundles). The Weinstein conjecture holds in T^{∗}N for N a simply connected compact manifold.

References.

[C1] M., Chaperon,Une idée du type géodésiques brisées pour les systèmes hamiltoniens, Comptes Rendus Acad. Sci. Paris, 298(1984), pp.293-296.

[C2] M. Chaperon, Lois de co