The cup-product on the Thom-Smale-Witten

complex, and Floer cohomology

C. VITERBO

To the Memory of Andreas Floer

Introduction.

Let M be a finite dimensional smooth manifold, f a C^{2} Morse function on M satisfying the Palais-Smale condition

(PS) Any sequence (x_{n})_{n≥1} such that df(x_{n}) → 0 and f(x_{n}) is bounded has a converging subsequence.

In the sequel we shall considera “nice” pseudogradient vector field, X, that is

(i) df(x) · X(n) ≥ ε|df(x)|^{2} (ii) X is a Morse-Smale vector field, that is the stable manifold of a

critical point is transverse to the unstable manifold of another critical point. The (PS) assumption guarantees that the flow of X is well defined. Given two critical points x et y with f(x) < f(y), we denote by

P(x, y) the set of maps γ : R → M such that γ̇(s) = X(γ(s)) and

lim_{s→+∞} γ(s) = x  lim_{s→+∞} γ(s) = y .

As we have an obvious R-action on P(x, y) defined by (t ∗ γ)(s) = γ(s + t), it is natural to define

P̂(x, y) = P(x, y)/R .

Finally we denote by M(x, y) the image of P(x, y) by the evaluation map γ ↦ γ(0). In otherwords M(x, y) is the set of points on some curve of P(x, y).

Cauchy’s uniqueness theorem tells us that the evaluation map is a diffeomorphism from P(x, y) onto M(x, y).

Let us denote by i(x) the Morse index of the critical point x, we then have by our transversality assumptions

\begin{aligned}dim P(x, y) & = i(y) − i(x) \\ dim P̂(x, y) & = i(y) − i(x) − 1 . \end{aligned}

Now for any a < b regular values for f, we may construct the Thom-Smale-Witten complex by defining W^{k}(f; a, b) to be the free R-module with one generator for each critical point of f with index k and critical value in [a, b]

δ : W^{k}(f; a, b) → W^{k+1}(f; a, b)
δ(x) = Σn(x, y) · y

where n(x, y) is the number of points in P̂(x, y) counted with the proper sign (i.e. as an intersection point of W^{u}(x) and W^{s}(y)). As we are not particularly interested about this aspect, we shall just say that at each critical point, we may orient arbitrarily W^{u}(x).

This gives an orientation of W^{s}(x) if we require that W^{s}(x) ∩ W^{u}(x) = (+1) · x.

It has been proved by several authors that the cohomology of (W^{k}, δ^{k}) is isomorphic to H^{∗}(M^{b}, M^{q}). (cf. [T], [M], [Fl 4], etc...)

The above isomorphism is an isomorphism of R-modules, but there is more structure in H^{∗}(M) module in particular through the cup-product operation

\begin{aligned}H^{∗}(M) ⊗ H^{∗}(M^{b}, M^{a}) & → H^{α}(M^{b}, M^{a}) \\ ω ⊗ α & → ω ∪ α \end{aligned}

We now would like to read this structure on the complex W^{k}(f; a, b) directly.

In section 1, we define an H^{∗}(M)-module structure on the cohomology of W^{k}(f; a, b).

In section 3, we shall show that our argument extends to the more general framework of Floer cohomologies (in the symplectic or instanton case) with minor modifications, to yield an H^{∗}(ΛM) module structure (in the symplectic case) or an H^{∗}(BG)/(⨁_{k≥8}H^{k}(BG)) in the instanton

case (BG is the classifying space of the gauge group).

In section 2 we prove, returning to the case of a finite dimensional manifold, that the module structure defined in section 1 coincides with the cup-product. We then use this result to prove that the H^{∗}(ΛM), (ΛM is the free loop space of M) structure defined on the symplectic Floer cohomology reduces, in the case where this Floer cohomology may be identified with H^{∗}(M), to the composition of cup-product and i^{∗} : H^{∗}(ΛM) → H^{∗}(M) (i : M → ΛM) is the inclusion of constants).

The existence of the product is sketched in [Fl 3], where the associativity result of Lemma 4 is conjectured. If M has holomorphic spheres, things are slightly more complicated( see section 3). In particular we compute our product for complex projective spaces.

Let us point out that according to ideas of Cohen, Jones and Segal ([CJS]), one may define a kind of “stable Floer-Witten homotopy type” associated to the set of all bounded trajectories of X, such that its holomogy is just H^{∗}(M^{b}, M^{a}) (or the Floer cohomology).

However, on stable objects, there is no cup-product, thus there seems to be no way to reduce our cup-product to the above object, if we require a construction relying only on the spaces of bounded trajectories (this is the only thing we may hope to extend in the infinite dimensional case).

The author would like to thank Prof. J-M. Bismut, R. Cohen, F. Laudenbach and G. Segal, for useful discussions.

1. An H^{∗}(M)-module structure on the cohomology of (W, δ).

From now on, we assume that R = Let (W (f; a, b), δ) be the Thom-Smale-Witten chain complex defined in the introduction, taking real coefficients, and let [ω] ∈ H^{∗}(M) be a De Rham cohomology class.

Theorem 1. Let ω be a closed form on M, and let π(ω) be the map

\begin{aligned}π(ω) : W^{k}(f; a, b) & → W^{k+d}(f; a, b) \\ x & → ∑_{y} ( ∫_{M(x,y)} ω) · y \end{aligned}

Then π(ω) commutes with δ, inducing a map in cohomology

P (ω) : HW^{k}(f; a, b) → HW^{k+d}(f; a, b) .

This map itself only depends on the cohomology class of ω in H^{∗}(M), moreover we have

P (ω)P (ω′) = P (ω ∪ ω′) (associativity) .

As a result P defines an H^{∗}(M) module structure on HW^{∗}(f; a, b).

Remark.

In the above theorem, it is understood that P (ω) does not depend on the choice of the pseudo gradient vector fiels X.

The proof of the above theorem rests on several lemmas. We start by describing the closure of M(x, z) (or P(x, z)).

Lemma 1. (cf. [Fl 1], [Fl 3]). The closure of M(x, z) may be described as \overline{M}(x, z) = ⋃ M(x, y_{1}) ∪ ∪M(y_{q}, z) the union being over all sequences y_{1}, …, y_{q} of critical points such that M(x, y_{1}), M(y_{1}, y_{2}), …, M(y_{q}, z) are all non empty.

Moreover for any such sequence (y_{1}, …, y_{q}), there is a map G : P̂(x, y_{1}) × … × P̂(y_{q}, z) × Δ^{q+1} → \overline{M}(x, z)

where Δ^{q+1} = {(λ_{0}, …, λ_{q}) ∈ [ − ∞, +∞ ]^{q+1} | ∀j, 1 + λ_{j} ≤ λ_{j+1}}.

(1) The image of G is a neighbourhood of M(x, y_{1}) ∪ ∪M(y_{q}, z) in \overline{M}(x, z).

(2) The restriction of G to P̂(x_{1}, y_{1}) × ×P̂(y_{q}, z)× Δ^{∘}^{q+1} is a diffeomorphism onto its image.

(3) G(a_{0}, …, a_{q}, −∞, …, −∞, μ_{j}, …, μ_{j+p}, +∞, …, +∞) = G(a_{j}, …, a_{j+p},

μ_{j}, …, μ_{j+p}).

Proof:

We shall only sketch the construction of G (the first part of the theorem is standard). An essentially equivalent construction may be found in [Fl 3] or [C-J-S]. It is based on the glueing of flow trajectoires of X, that one may find in [Fl 3].

Finally, we point out that we only need the lemma for q = 1.

We may find a section of the natural projection M(a, b) → P̂(a, b) to identify P̂(a, b) to a submanifold of M(a, b).

Using a tubular neighbourhood of M(a, b), we may construct a submanifold Σ(a, b) such that

(i) Σ(a, b) ∩ M(a, b) = P̂(a, b) (ii) T_{u}Σ(a, b) ∩ T_{u}M(a, b) = T_{u}P̂(a, b) (iii) T_{u}Σ(a, b) + T_{u}M(a, b) = T_{u}M

(One usually says that Σ(a, b) and M(a, b) have “clean intersection” along P̂(a, b)). (iv) The projection x : Σ(a, b) → P̂(a, b) makes Σ(a, b) into a disk bun-

dle.

Now let γ be a trajectory in P(a, b) near M(x, y_{1}) ∪ ∪M(y_{q}, z). Then γ(R) intersects all the Σ(y_{j}, y_{j+1}) (we set y_{0} = x, y_{q+1} = z) at times λ_{j} (j = 0, …, q), and π(γ(λ_{j})) = a_{j} ∈ P̂(y_{j}, y_{j+1}).

Conversely, given a sequence (a_{j}, λ_{j}) with λ_{j+1} > 1 + λ_{j}, there is a unique γ in P(a, b) such that γ(λ_{j}) ∈ Σ(y_{j}, y_{j+1}) and π(γ(λ_{j})) = a_{j}. This follows from the transversality of stable and instable manifolds, and the implicit function theorem.

So our map G is defined on

P̂(x, y_{1}) × ×P̂(y_{q}, z)× Δ^{∘}^{q+1} .

and is a diffeomorphism onto its image.

We now have to extend G to the values of λ_{j} in {±∞}.

If λ_{0}, …, λ_{j} go to −∞, then for any t ∈ R, γ(t) goes to a point in \overline{M(y_{j}, z)}. Indeed, if γ is a trajectory, and γ(λ_{j−1}) ∈ Σ(y_{j−1}, y_{j}), γ(λ_{j}) ∈ Σ(y_{j}, y_{j+1}), then \frac{1}{|λ_{j+1}−λ_{j}|} measures how close γ passes y_{j} (because going near y_{j} slows down the trajectory), so that as λ_{j} → −∞ while λ_{j+1} > −C, γ converges to a trajectory starting from y_{j}.

The same argument shows that if λ_{k+1}, …, λ_{q} go to +∞, γ(0) converges to a point in \overline{M(x, y_{k})}.

Moreover as λ_{0}, …, λ_{j} got to −∞, λ_{k+1}, …, λ_{q} to +∞, we have that the definition of γ, only depending on (a_{j+1}, …, a_{k}, λ_{j+1}, …, λ_{k}) will eventually be given by G(a_{j+1}, …, a_{k}, λ_{j+1}, …, λ_{k}). This proves (3) and concludes our proof.

Lemma 2. Let x, z with i(x) − i(z) = k + 1. Define

∂M(x, z) = ∑_{i(y)=i(x)+1} n(x, y)M(y, z) + ∑_{i(y)=i(z)−1} n(y, z)M(x, y) .

Then we have for any k -form φ

∫_{∂M(x,z)} φ = ∫_{M(x,z)} dφ .

Remarks.

This is our version of Stoke’s formula. Hence the “sum” in the definition of ∂M(x, z) should be understood as a summation of integration currents.

Proof: (see also [L], prop. 2).

As in the proof of Stoke’s theorem, we may, using a partition of unity, restrict our attention to a neighbouhrood of M(x, y_{1}) ∪ ∪M(y_{q}, z).

Since G^{−1} is a diffeomorphism of such a neighbourhood into P̂(x, y_{1})× … × P̂(y_{q}, z)× Δ^{∘}_{q+1} , we may apply the change of variable formula (to carry back our integration on the latter set) the Stoke’s formula for manifolds with corners(see the appendix for details). We see that G^{∗}φ must be

integrated only on P̂(x, y_{1})×…×P̂(y_{q}, z)×{−∞}× Δ^{∘}_{q} ⋃ P̂(x, y_{1})×…×

P̂(y_{q}, z)× Δ^{∘}_{q} ×{+∞}, and since G(a_{0}, a_{1}, …, a_{q}, −∞, λ_{1}, …, λ_{q}) does

not depend on a_{0} , integration is really only P̂(y_{1}, y_{2})×…×P̂(y_{q}, z)× Δ^{∘}_{q} ∪P̂(x, y_{1}) × … × P̂(y_{q−1}, y_{q})× Δ^{∘}_{q}. Moreover this set has dimension > k unless i(x) − i(y_{1}) = 1 (or i(y_{q}) − i(z) = 1).

Thus the only integrals of φ that appear, are these on M(y, z) (resp. M(x, y)) with i(x) − i(y) = 1 (resp. i(y) − i(z) = 1) and the integral appears once for each element in P(x, y) counted with the propre sign.

This concludes the proof.

Consider now the map π(ω). It is a linear map, and depends linearly on ω.

Lemma 3.

π(dω) = δπ(ω) ± π(ω)δ

where δ is the coboundary on W^{∗}(f; a, b).

Proof:

π(dω)x = ∑_{t} ( ∫_{M(x,t)} dω)t

= ∑_{t} ( ∫_{∂M(x,t)} ω)t

= ∑_{t} ( ∑_{i(y)=i(x)+1} n(x, y) ∫_{M(y,t)} ω + ∑_{i(z)=i(t)−1} n(z, t) ∫_{M(x,z)} ω)t

= ∑_{i(y)=i(x)+1} n(x, y) ∑_{t} ( ∫_{M(y,t)} ω)t + ∑_{i(z)=i(t)−1} n(z, t)( ∫_{M(x,z)} ω)t

= (δπ(ω) + π(ω)δ)x

Remark.

This is the analogue of the formula d(φ∧α) = dφ∧α+(−1) deg(φ)φ∧ dα for φ ∈ Ω(M), α ∈ Ω(M^{b}, M^{a}).

Corollary. If ω is closed, π(ω) induces a map P (ω), HW^{k}(f; a, b) → HW^{k+d}(f; a, b) which depends only on the cohomology class of ω.

Proof:

If dω = 0, δπ(ω) = −π(ω)δ, hence (−1)^{deg} · π(ω) is a chain map, hence define a map in cohomology.

If ω = dφ, π(dφ) = δπ(φ) + π(φ)δ hence π(δφ) sends cycles to boundaries : it induces the zero map in cohomology.

We now prove

Lemma 4. Let ω_{1}, ω_{2} be closed forms. Then we have P (ω_{1} ∧ ω_{2}) = P (ω_{1}) · P (ω_{2}).

Remark. It is understood thatP (ω_{1}), P (ω_{2}) and P (ω_{1}∧ω_{2}) only depend on the cohomology classes of ω_{1} and ω_{2} .

Proof: We have

P (ω_{1} ∧ ω_{2}) · x = ∑_{z} ( ∫_{M(x,z)} ω_{1} ∧ ω_{2})z
P (ω_{1}) · P (ω_{2}) · x = P (ω_{1}) · ∑_{y} ( ∫_{M(x,y)} ω_{1})y

= ∑_{z} ( ∑_{y} ( ∫_{M(x,y)} ω_{1} ∫_{M(y,z)} ω_{2})z .

It will then be sufficient to prove that adding to ω_{1} and ω_{2} a suitable exact form, we have

(∗)  ∑_{y} ∫_{M(x,y)} ω_{1} ∫_{M(y,z)} ω_{2} = ∫_{M(x,z)} ω_{1} ∧ ω_{2} .

Let k_{1} = deg ω_{1} , k_{2} = deg ω_{2} , then if dim M(x, y) = i(x) − i(y) < k_{1} , we may chane ω_{1} by an exact form, so that it vanishes in a neighbourhood of M(x, y). The same holds for ω_{2} near M(y, z) whenever i(y)−i(z) < k_{2} .

Thus we only have in the left hand side of (*) terms with i(y) = i(x) + k_{1} = i(z) − k_{2} .

In other terms, if we also use the action (i.e. the flow) to enlarge the region where ω_{1} and ω_{2} vanish, we may assume that ω_{1} vanishes on any (given) compact subset of M(x, z) and M(y, z), and ω_{2} vanishes on any compact subset of M(x, z) and M(x, y).

Now, we see that ω_{1} vanishes outside a neighbourhood of the \overline{M(x, y)}, and ω_{2} outside a neighbourhood of the \overline{M(y, z)}.

Thus we may assume in proving (*) that ω_{1} and ω_{2} vanish outside neighbourhood of \overline{M(x, y)} and \overline{M(y, z)}, i.e. prove (*) for forms supported near \overline{M(x, y)} ∪ \overline{M(y, z)}.

Now we claim that this neighbourhood is the image CP̂(x, y) × CP̂(y, z), where CA is the cone over A.

Here, we identify CP̂(x, y) with P̂(x, y) × [−∞, +∞[/P̂(x, y) × {+∞}and CP̂(y, z) with P̂(y, z)×] − ∞, +∞]/P̂(y, z) × {+∞}.

According to lemma 1, a neighbourhood of M(x, y) ∪ M(y, z) is described as the image of P̂(y, y) × P̂(y, z) × Δ^{∘}_{2} by the glueing map.

We then associate to (a_{1}, t_{1}, a_{2}, t_{2}) ∈ CP̂(x, y) × CP̂(y, z) the image by the glueing map of (a_{1}, a_{2}, t_{1}, t_{1} + t_{2}), and (a_{1}, −∞, a_{2}, +∞) goes to {y}.

We now pull back ω_{1} and ω_{2} on CP(x, y) × CP̂(y, z), and we get two forms φ_{1}, φ_{2} .

From the fact that ω_{1} vanishes away from M(x, y), we get that in fact

ω_{1} ∈ H^{∗}(CP̂(x, y) × CP̂(y, z), D_{1})

ω_{2} ∈ H^{∗}(CP̂(x, y) × CP̂(y, z), D_{2}) .

where D_{i} = {(a_{1}, t_{1}, a_{2}, t_{2}) | t_{i} ≥ C} In other words if CA = A×[0, 1]/A× {0}, A is identified with A×{1}, we have with A = P̂(x, y), B = P̂(y, z)

\begin{matrix}φ_{1} ∈ & H^{∗}(CA × CB, A × CB) \\ φ_{2} ∈ & H^{∗}(CA × CB, CA × B) \\ φ_{1} ∧ φ_{2} ∈ & H^{∗}(CA × CB, A × CB ∪ CA × B) \end{matrix}

The fact that ∫_{CA×CB} φ_{1} ∧ φ_{2} = ∫_{CA} φ_{1} ∫_{CB} φ_{2} is given by the following

Sublemma. Let A, B be manifolds without boundary, of dimension a, b.

Let

∫_{CA} : H_{c}^{a+1}(CA, A) → R

be the isomorphism given by integration of forms (note that H^{i}(CA, A) ≃ H^{i−1}(A) for i ≠ 0). Then for α ∈ H_{c}^{a+1}(CA × CB, A × CB), β ∈ H_{c}^{b+1}(CA × CB, CA × B), we have

∫_{CA×CB} α ∧ β = ∫_{CA×{b_{0}}} α ∫_{{a_{0}}×CB} .

Proof: Note that α∧β ∈ H^{∗}(CA×CB, A×CB∪CA×B) ≃ H^{∗}(CA, A)^{⊗} H^{∗}(CB, B) (Künneth isomorphism). Now, because H_{c}^{∗}(CB) = 0, α is always cohomologous to \overline{α}^{⊗} 1, β to 1^{⊗} \overline{β}, with \overline{α} ∈ H_{c}^{∗}(CA, A), \overline{β} ∈ H_{c}^{∗}(CB, B).

Then

\begin{aligned}∫_{CA×\{b_{0}\}} α & = ∫_{CA} \overline{α} \\ ∫_{\{a_{0}\}×CB} β & = ∫_{CB} \overline{β} \end{aligned}

and α ∧ β is the image by Künneth of \overline{α}^{⊗} \overline{β}, hence

∫_{CA×CB} α ∧ β = ∫_{CA×CB} \overline{α}^{⊗} \overline{β} = ∫_{CA} \overline{α} ∫_{CB} \overline{β} = ∫_{CA×\{b_{0}\}} α ∫_{\{a_{0}\}×CB} .

This concludes our proof.

We still want to prove that

∫_{C_{P}(x,y)} φ_{1} = ∫_{M(x,y)} ω_{1} ,

but CP̂(x, y) × {(a_{1}, +∞)} goes to M(x, y), thus this follows from the change of variable formula.

We still want to prove that P (ω) does not depend on the choice of (X, f). In other words, let (X_{t}, f_{t}) be a family of data, such that M_{t}^{b} = M_{0}^{b}, M_{t}^{a} = M_{0}^{a}.

Then there is a chain map K : W^{∗}(f_{0}, X_{0}; a, b) → W^{∗}(f_{1}, X_{1}; a, b) inducing as isomorphism

\overline{K} : HW^{∗}(f_{0}, X_{0}; a, b) → HW^{∗}(f_{1}, X_{1}; a, b) .

Proposition.

\overline{K}P (ω) = P (ω)\overline{K} .

Proof: Assume

\begin{aligned}F_{t}(x) & = f_{0}(x) + \frac{ε}{2}t^{2} 0 ≤ t ≤ δ ε > 0 \\ & = f_{1}(x) − \frac{ε}{2}t^{2} 1 − δ ≤ t ≤ 1 \end{aligned}

and set F (t, x) = f_{t}(w).

Then F : M × [0, 1] has the following pseudogradient vector field

X(t, X) = X_{0}(x) + \frac{ε}{2}t\frac{∂}{∂t} 0 ≤ t ≤ δ

= X_{t}(x) + \frac{∂}{∂t}f_{t}(x) δ ≤ t ≤ 1 − δ

= X_{1}(x) − \frac{ε}{2}t\frac{∂}{∂t} 1 − δ ≤ t ≤ 1

Thus the flow preserves M ×{0} and M ×{1}. Moreover if x is a critical point of f_{0} (resp. f_{1} ) with index i, (x, 0) (resp. (x, 1)) is a critical point of F with index i (resp. i + 1).

For x a critical point of f_{0} , we set

K(x) = ∑_{y} n(x, y).y

where y is a critical point of f_{1} , and n(x, y) is the number of trajectories of X from (x, 0) to (y, 1).

Thus K defines a map W^{∗}(f_{0}, X_{0}, a, b) ↦ W^{∗}(f_{1}, X_{1}, a, b).

Note that if we consider W^{∗}(f_{0}, X_{0}, a, b) and W^{∗−1}(f_{1}, X_{0}, a, b) as subcomplexes of W^{∗}(F, X, a, b), the boundary map Δ on this complex may be written as a matrix

Δ = ( \begin{matrix}δ_{0} & 0 & ⋮ & 0 \\ K & δ_{1} & ⋮ & ? \\ … & … & ⋮ & … \\ ? & 0 & ⋮ & ? \end{matrix} )

where the first and second rows (resp. column) correspond to W^{∗}(f_{0}, X_{0}, a, b) and W^{∗}(f_{1}, X_{1}, a, b). Since Δ^{2} = 0, we get δ_{0}K + Kδ_{1} = 0 and K is (up to sign) the required chain map.

Now let ω̃ denote the obvious extension of ω to M × [0, 1]. Then, since we proved that Δ ∘ P (ω̃) = P (ω̃) ∘ Δ, this implies K ∘ P (ω̃) = P (ω̃) ∘ K, but P (ω̃) corresponds to P (ω) on W^{∗}(f_{0}, X_{0}, a, b) and W^{∗}(f_{1}, X_{1}, a, b).

This proves our proposition.

Remark. It would be interesting to understand what are the other cohomology operations that we may define using only trajectory spaces. In particular in view of our paper [V 2], one would expect to be able to define Massey products < ω, x, ω′ >, but we were unable to prove this, as the definition of Massey products makes crucial use of the existence of a product, defined on cochains, which is already associative on the cochain level.

Remark. It seems that all this may also be done with any coefficients ring. For this one would have to triangulate the spaces M(x, y), and define a “fundamental class”, (that should be the sum of all oriented simplexes in M(x, y) so that for any cocycle c in M, we may define < c, M(x, y) >, and the equivalent of lemma 2 is satisfied.

2. Identification with the “usual” cup-product structure.

We consider for example the case where f_{0} has a nondegenerate critical manifold N, assumed to be compact (if M is compact we could have M = N). Assume that all critical points in f^{−1}([a, b]) are on N.

Proposition 2.1. With the above assumptions, let j : N → M be the inclusion, i the index of N. Then H^{∗}(M^{b}, M^{a}) ≃ H^{∗−i}(N), and under this identification P (ω)α = j^{∗}(ω).α. In particular, when M is compact, a = −∞, b = +∞, P (ω) induces the usual cup-product on M.

Proof: We shall start with the case M = N. Even in this case we have to explain the meaning of our proposition, since the identification HW^{∗}(M) → H^{∗}(M) has not been explicited.

In this case, if we moreover assume f to be an ordered function, that is all critical values are distinct, and for x, y two critical points of f, f(x) < f(y) implies i(x) ≤ i(y).

Let (a_{i})_{i=0,...,n} be real numbers separating the critical values, so that f^{−1}([a_{i}, a_{i+1}]) contains all critical points of index i.

Since

H^{j}(M^{a_{i+1}}, M^{a_{i}}) = 0  j ≠ i
= R^{m_{i}} j = i if f has m_{i}

critical points with index i

we see that the map

d_{i} : H^{i}(M^{a_{i}}, M^{a_{i−1}}) → H^{i+1}(M^{a_{i+1}}, M^{a_{i}})

may be identified with the map

δ^{i} : W^{i}(f, X) → W^{i+1}(f, X)

as follows:

each class in H^{i}(M^{a_{i+1}}, M^{a_{i}}) may be identified to a sum of duals of W^{s}(x) for x critical point of index i, that is to x we associate α ↦ ∫_{W^{s}(x)} α.

In other words, we get a map

\begin{matrix}H^{i}(M^{a_{i+1}}, M^{a_{i}}) & → W^{i}(f, X) \\ ω & ↦ ∑_{x} (∫_{W^{s}(x)} ω) x \end{matrix}

Let us check that this identification is compatible with the differentials d_{i} and δ_{i}.

Now we know from [M], that d_{i} corresponds to counting the number of points of W^{s}(x) ∩ W^{u}(y) ∩ M^{a_{i}}. This is exactly how δ_{i} is computed.

Now let ω ∈ H^{d}(M), we may assume that ω vanishes on M^{a_{d}} since H^{d}(M^{a_{d}}) = 0.

Then P (ω) . 1 = ∑_{x} (∫_{W^{s}(x)} ω) x = μ(ω) where μ is the map H^{∗}(M) → HW^{∗}(f, X) that we defined above.

This proves our claim for N = M.

The general case follows from this, since N has a tubular neighbourhood ν, that may be decomposed as ν = ν^{+} ^{⊕} ν^{−}, positive and negative eigenspaces for d^{2}f.

Let g be a function on N, and set

f_{ε}(x, y) = f(y) + ε(|y|^{2})g(x)

(here y ∈ ν_{x} and f(y) ≃ |y_{+}|^{2} − |y_{−}|^{2} ).

Then df_{ε}(x, y) = (df(y) + ε′(|y|^{2}).yg(x), ε(|y|^{2})dg) provided |ε′(|y|^{2})| < 1 and |g(x)| < 1

df_{ε}(x, y) = 0 ⇔ y = 0dg(x) = 0

and we may choose X_{ε}(x, y) = (0, ∇g(x)) on {y = 0}.

Thus the trajectories of X_{ε} are in one to one correspondence with those of g. Choosing g to be well ordered, we are reduced to the case treated.

3. Generalizations to Floer cohomology.

Here we shall only deal with the case of symplectic Floer homology, even though there is no obstruction to applying the same methods for the instanton case.

Let (P, ω) be a symplectic manifold, H a C^{2} function on P × S^{1} . We choose an almost complex structure on P compatible with ω, that is

ω(Jξ, η) = g(ξ, η)

where g is a Riemannian metric. Note that for fixed ω, the set almost complex structures compatible with ω make a contractible space.

Let ΛP be the free loop space of P , i.e. the set of maps S^{1} → P . Floer associates to the action functional on ΛP that is defined as

A_{H}(x) = ∫_{D^{2}} u^{∗}ω − ∫_{S^{1}} H(θ, x)dθ

where u : D^{2} → P satisfies u|∂D = x. Note that

(i) A_{H} is only defined on the connected component Λ_{0}P of the set of constant maps (otherwise u does not exist)

(ii) u is not unique. In fact A_{H} is multivalued (i.e. defined on some covering space of Λ_{0}P ) unless [ω]π_{2}(P) = 0. We first deal with the classical case [ω]π_{2}(P) = 0. Then, the L^{2}

gradient flow of A_{H}, studied by Floer, is given by the equation

\frac{∂u}{∂t} = −J \frac{∂u}{∂θ} − ∇H(θ, u)

that is

\overline{∂}u = −∇H(θ, u) .  (∗)

Moreover, the critical points of A_{H} are the 1 periodic solutions of

(H) ẋ = J∇H(θ, x) = X_{H}(θ, x)

so that the connecting trajectories are solutions of

(E)  \{ \begin{matrix}\overline{∂}u = −∇H(u) \\ lim_{t→+∞} u(t, θ) = y(θ) , lim_{t→−∞} u(t, θ) = x(θ) . \end{matrix}

The set P(x, y) of solution for the above equation then behaves exactly as in the finite dimensional case. In particular we have both the compactness modulo break up, and the glueing of solutions ([Fl 3], p. 584, prop. 1b), as well as a homeomorphism P(x, y) → M(x, y) obtained from the evaluation map.

The only difference comes from the Morse index. Despite the fact that the Morse index is alway infinite, we may define a finite number, still called index that measures the size of the negative eigenspace of D^{2}A_{H}. For the definition we refer to [V 1], [Fl ], [S-Z]. It is now dependent on v, in fact if u, v : D^{2} → P are such that v = w on ∂D^{2} , i(v)−i(w) = 2c_{1}(P)(v−w) where (v − w) : S^{2} → P is the map obtained by glueing w and v along ∂D^{2} , c_{1}(P) is the first Chern class of (P, J) (cf. [V]).

Let Γ be the subgroup of Z generated by the values of 2c_{1}(P) over all spheres. Then we may define i(x) in Z/Γ for x a solution of (H).

We first assume Γ = {0}. Then one has a well defined index. The Floer complex is then constructed like the Thom-Smale-Witten complex. We denote its cohomology by HF^{∗}(P, H; a, b). Then we get a module structure of HF^{∗}(P, H; a, b) over the rign H^{∗}(ΛP).

Note that for P compact, a = −∞, b = +∞, HF^{∗} does not depend on H. If we take H to be independent of θ and C^{2} small, the solutions of (E) will be u(θ, t) = u(t) with u̇(t) = −∇H(u), that is u is a path of constant loops given by the flow trajectories of −∇H on P .

Thus HF^{∗}(P, H) ≃ H^{∗}(P), and according to proposition 1 of section 2, the H^{∗}(ΛP) structure is given by ω.α = c^{∗}(ω).α where c : P → ΛP is the inclusion of constant loops.

Now let us deal with the case [ω]π_{2}(P) ≠ 0 and/or c_{1}(P)π_{2}(P) ≠ 0. Let Λ̃P be the covering of Λ_{0}P defined as

Λ̃P = {v : D^{2} → P } / ∼ where

v ∼ w ⇔ v = w on ∂D^{2}

and

⟨ω, v − ω⟩ = ⟨c_{1}(P), v − ω⟩ = 0 .

Then, A_{H} is well defined on Λ̃P , and each critical point has a well defined index.

Now let u be a solution of (E), it will then be a trajectory from (x, v) to (y, w) where w is homotopic to v^{#}u, that is we glue v and u along ∂D^{2} to get a map from the disk obtained by glueing D^{2} and S^{1} × R (along ∂D^{2} ≃ S^{1} × {−∞}).

Thus, to each solution of (E) corresponds a set of trajectories, each of them starting from (x, v) with different v. However, if we require A_{H}(x, v) ∈ [a, b], i(x, v) ∈ [d_{1}, d_{2}] then we get only finitely many v ’s.

Now, due to the possible interference of holomorphic spheres in P , the space of trajectories P((x, v); (y, w)) is subject to a more complicated failure of compactness then described by lemma 1. Indeed, as it is now classical, a holomorphic sphere may “bubble off”. However, if we denote by α the sphere that bubbles off, from a trajectory with i(y, w)−i(x, v) = d, we must have 1 + 2 ⟨c_{1}(P), α⟩ ≤ d, since what is left after bubbling off is still a trajectory u from (x, v) to (y, w) with w − v = u and u = u^{#}α, thus

\begin{aligned}2 ⟨c_{1}(P), α⟩ & = 2 ⟨c_{1}(P), u − u⟩ = (i(y, w) − i(x, v)) − i(y, w) − i(x, v) = \\ & = d − dim P (x, v); (y, w)) . \end{aligned}

Since dim P((x, v), (y, w)) must be gretaer then one, we get the desired inequality.

Note : Our argument is not complete for two reasons. First together with bubbling off, we may have break up of the trajectory as in lemma 1. However this is not a problem, since the two phenomena may be treated separately.

Second we have to worry about orientations. This is more tricky than in the finite dimensional case, and we refer to [Fl-H] for this.

As a result, and as remarked by Floer, if c_{1}(P) > 0 on holomorphic spheres (this is in particular true for P monotone, i.e. c_{1}(P) = λω, λ > 0) we have that bubbling off does not occur on one dimensional trajectories.

Thus the δ operator is well defined and δ ∘ δ = 0 (this is based on the compactness of trajectory spaces of dimension 1, but 1+2c_{1}(P) is at least 3 on a holomorphic sphere).

Note that we may also define the maps P (σ) for σ ∈ H^{k}(Λ̃P) with k < 2 inf_{α} ⟨c_{1}(P), α⟩ = d(P) (we need an extra dimension to apply lemma 3, and show that P (σ) only depends on the cohomology class of σ, thus k < 1 + 2 inf_{α} ⟨c_{1}(P), α⟩ would not suffice).

Moreover we have P (σ_{1}, σ_{2}) = P (σ_{1}) . P (σ_{2}) provided deg(σ_{1}) + deg(σ_{2}) < d(P).

Remark. It is interesting to notice that if deg(σ_{1}) and deg(σ_{2}) are < d(P) but deg(σ_{1}) + deg(σ_{2}) > d(P), P (σ_{1}) . P (σ_{2}) is well defined. However we cannot prove that it depends only on σ_{1}.σ_{2} . We cannot even prove that P (σ_{1})P (σ_{2}) = ±P (σ_{2})P (σ_{1}). Thus if R_{∗} = H^{∗}(M) is defined as a ring though generators and relations, we get that R_{∗}^{(d(P))} operates on HF^{∗}, where R_{∗}^{(k)} is defined as the ring obtained from R_{∗} by forgetting all generators and relations of degree ≥ k.

Theorem. Let d(P) = 2 inf { ⟨c_{1}(P), α⟩ | α is a holomorphic } and R be the ring H^{∗}(Λ̃P). Then provided d(P) > 0, we may define HF^{∗}(Λ̃P, H; a, b). It has an R^{(d(P))} module structure.

Remark. If we want everything to be independent of the almost complex structure J, we essentially need to replace d(P) by d̃(P) = 2 inf { ⟨c_{1}(P), α⟩ | α ∈ π_{2}(P), ⟨ω, α⟩ > 0}. However there maybe a big difference between d and d̃ (d̃ may even vanish), thus it may sometimes be convenient to work with a given almost complex structure, and try to estimate d for this structure

Remark. Instanton Floer cohomology. In [Fl 5], Floer defined the instanton cohomology IH^{∗}(M) for a three dimensional homology sphere M. This is obtained by constructing the “TSW complex associated to the Chern-Simons functional”

CS(A) = ∫_{M} (\frac{1}{2}A ∧ dA + \frac{1}{3}A ∧ A ∧ A)

where A ∈ A(M) the space of SU(2) connections for the trivial bundle over M (note that every SU(2) bundle over a three dimensional manifold is trivial).

The space of maps from M to SU(2) is a group g, called the gauge group, acting on A(M) though

A ↦ gAg^{−1} + dg.g^{−1} .

Then CS(g ∗A) = CS(A)+2π deg(g), (g : M → SU(2) is a map between three dimensional manifolds, : deg(g) is its degree).

Thus CS is not well defined on A(M)/G, but on its universal cover A(M)/G_{0} with G_{0} = {g ∈ G | deg(g) = 0}.

At any rate the gradient flow of A is well defined, by

\frac{d}{dt}A(t) + ∗F_{A(t)} = 0 on M × R

where ∗ is the Hodge duality from two forms to one forms, and F_{A} is the curvature of A(t).

Because this is a Fredholm equation, the set of trajectories displays the usual compactness, except for “bubbling off” of a “point instanton”. But this can only happen on a family M(A, B) of dimension at least 8.

Now for any two flat connections on M : A, BM(A, B) has connected components of different dimensions, but they are all equal modulo 8 (in other words i(A) is only defined in Z/8Z.

Thus there is no problem to define IH^{∗}(M) since this only involves 1 and 2 dimensional trajectory spaces, as well as P (ω) for ω ∈ H^{k}(A/G_{0}) = H^{k}(BG_{0}) for k < 8 (since A(M) is contractible A/G_{0} has the homotopy type of the classifying space BG_{0} ).

Hence we totally ignored all difficulties in the construction of IH^{∗}(M), due in particular to the existence of reducible connections, and nontransversality of M(A, B).

However, it seems to us that there is no obstruction to defining an R_{∗}^{(8)} module structure on IH^{∗}(M), where

R_{∗} = H^{∗}(BG_{0}) .

Appendix . About the use of Stokes’ formula.

Lemma. Let X be a compact manifold with corners, and G : X → M a map such that

(1) G is Lipschitz

(2) G is C^{2} on X^{∘}, and G_{|X^{∘}} is a diffeomorphism on its image

(3) G is C^{2} on the regular part of X − X^{∘} = ∂X and G_{|∂X} is a diffeomorphism on its image.

Then we have, denoting G(X) = W , G(∂X) = ∂W

∫_{W} dα = ∫_{∂W} α

for any form α.

Proof: Let G_{n} be a sequence of C^{2} maps such that G_{n} → G in the Lipschitz topology, in other terms

|[G_{n}(x) − G(x)] − [G_{n}(y) − G(y)]| < ε_{n} |y − x| .

The sequence G_{n} may be obtained by convolution.

Then, for any form φ on M, and Z a submanifold of X,

∫_{Z} G_{n}^{∗}φ → ∫_{Z} G^{∗}φ .

Indeed, φ is a sum of forms of the type

a(x)dx^{1} ∧ ∧ dx^{p} , then
∫ G_{n}^{∗}φ = ∫ a(G_{n}(x))dG_{n}^{1} ∧ ∧ dG_{n}^{p} .

Now |a(G_{n}(x))dG_{n}^{1} ∧ ∧ dG_{n}^{p}| ≤ M(K + ε)^{p} where M = ∑ |a|, K is the Lipschitz constant for G. Since on Z, G_{n}^{∗}φ → G^{∗}φ pointwise, we have by Lebesgue’s dominated convergence that ∫_{Z} G_{n}^{∗}φ goes to ∫_{Z} G^{∗}φ.

The result is now clear, since

∫_{W} dα = ∫_{X} G^{∗}(dα) = ∫_{X} d(G^{∗}α) using (2)
= lim ∫_{X} d(G_{n}^{∗}α)
= lim_{n} ∫_{∂X} G_{n}^{∗}α Stokes
= ∫_{∂X} G^{∗}α = ∫_{∂W} α by (3)

References

[C-J-S] Cohen, R., Jones,J., Segal, G.,Personnal communication, see also this proceeding (Floer Memorial volume) [Fl 1] Floer,A., The unregularized Gradient Flow of the Symplectic Action; Comm. Pure Appl. Math., 41 , (1988), pp. 775–813. [Fl 2] Floer,A., Morse Theory for Lagrangian Intersections ; Journal of Differential Geometry, 28 , (1988), pp. 513–547. [Fl 3] Floer,A., Symplectic fixed points and Holomorphic spheres; Commun. in Math. Phys., 120 , (1988), pp. 575–611. [Fl 4] Floer,A., Witten’s complex and infinite-dimensional Morse theory; J. of Differential Geom. 30 (1989), pp. 207–221. [Fl 5] Floer,A., An instanton invariant for three manifolds; Communication in Math. Phys., 118 , (1988), pp. 215–240. [Fl 6] Floer,A., Cuplenght estimates on Lagrangian intersections; Comm. Pure Appl. Math., 42 , (1989), pp. 335–357. [Fl -H] Floer,A., Hofer, H., Coherent orientation for periodic orbits in symplectic geometry; Math. Zeitschrift, 1993, pp.13–38. [H -S] Hofer, H., Salamon, D.A.,Floer Homology and Novikov rings, preprint.

[L] Laudenbach, F., On the Thom-Smale complex, appendix to ” An extension of a theorem by Cheeger and Müller” by J.M. Bismut; Astérisque, Soc. Math. France, (1992).

[M] Milnor ”Lectures on the h-cobordism theorem”, Notes by L. Siebenmann and J. Sondow; Princeton University Press, 1965. [S-U] Sacks,J. Uhlenbeck,K., The existence of minimal immersions of 2spheres; Ann. of Math, 113 , (1981), pp. 1–24. [S-Z] Salamon, D. and Zehnder, E. Floer Homology, Maslov index and periodic orbits of Hamiltonian equations; Analysis et cetera, Papers in the honor of J. Moser 60th birthday, Academic Press 1987, pp. 573–600. [Th], Thom, R., Sur une partition en cellules associée à une fonction sur une variété. Comptes Rendus Acad. Sci. Paris Série I, (1949), pp. 973– 975 . [V 1] Viterbo,C., Intersections de sous-variétés lagrangiennes, fon ctionelles d’action et indice des systèmes hamiltoniens; Bull. Soc. Math. France, 115 , (1987), pp. 361–390. [V 2] Viterbo, C., Some remarks on Massey products and Lusternik-Shnirelman category; in preparation.

[Wi] Witten, E., Supersymmetry and Morse theory; J. Diff. Geometry, 17 , (1982), pp. 661–692.

U.R.A. 1169 Département de mathématique, Bâtiment 425 Université de Paris-Sud, F-91405 Orsay Cedex, FRANCE. et DMI, Ecole Normale Supérieure, 45 rue d’Ulm, 75230 Paris Cedex 05.