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The following compactness statement follows from the general discussion of section 6.1.
3.17. Proposition. The space MI! (x, y) is pre-compact in C∞ loc (ΣI! , T ∗ M).
When µ Ω (y) = µ Λ (x) − n, the space MI! (x, y) consists of finitely many oriented points. The
algebraic sum of these orientations is denoted by nI! (x, y), and defines the homomorphism
I! : F Λ k (H, J) → F Ω k−n(H, J), x ↦→ �
nI! (x, y)y.
A standard gluing argument shows that I! is a chain map.
y∈P Ω (H)
µ Ω (y)=k−n
4 Isomorphisms between Morse and Floer complexes
4.1 The chain complex isomorphisms
Let L ∈ C∞ ([0, 1] × T ∗M) or L ∈ C∞ (Ì×T ∗M) be a Lagrangian satisfying (L1) and (L2). Let
H be the Fenchel transform of L, that is
�
�
H(t, q, p) := max p(v) − L(t, q, v) .
v∈TqM
It is easy to see that H satisfies (H1) and (H2). If v(t, q, p) ∈ TqM is the (unique) vector where
the above maximum is achieved, the map
[0, 1] × T ∗ M → [0, 1] × TM, (t, q, p) ↦→ (t, q, v(t, q, p)),
is a diffeomorphism, called the Legendre transform associated to the Lagrangian L. The Legendre
transform induces a one-to-one correspondence x ↦→ π ◦ x between the orbits of the Hamiltonian
vector field XH and the solutions of the second order Lagrangian equation given by L. When H
is the Fenchel transform of L, we have the fundamental inequality between the Hamiltonian and
the Lagrangian action functionals:
�H(x) ≤ËL(π ◦ x), ∀x : [0, 1] → T ∗ M, (49)
with the equality holding if and only if x is related to (π ◦ x, (π ◦ x) ′ ) by the Legendre transform.
In particular, the equality holds if x is an orbit of the Hamiltonian vector field XH. In this section
we recall the definition of the isomorphisms
Φ Ω L : Mk(ËΩ L, g Ω ) → F Ω k (H, J), Φ Λ L : Mk(ËΛ L, g Λ ) → F Λ k (H, J),
between the Morse complex of the Lagrangian action functional and the Floer complex of the
corresponding Hamiltonian system. See [AS06b] for detailed proofs.
We assume that L satisfies (L0) Ω , resp. (L0) Λ , equivalently that H satisfies (H0) Ω , resp. (H0) Λ .
Consider a Riemannian metric g Ω , resp. g Λ , on Ω 1 (M, q0), resp. Λ 1 (M), such that the Lagrangian
action functionalËL satisfies the Palais-Smale and the Morse-Smale conditions. Given γ ∈ P Ω (L)
and x ∈ P Ω (H), we denote by M Ω Φ (γ, x) the space of maps u ∈ C∞ ([0, +∞[×[0, 1], T ∗ M) which
solve the Floer equation
and satisfy the boundary conditions
∂J,H(u) = 0, (50)
u(s, 0) ∈ T ∗ q0M, u(s, 1) ∈ T ∗ M, ∀s ≥ 0,
q0
π ◦ u(0, ·) ∈ W u (γ, −gradËΩ
L ),
38