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and the asymptotic condition
lim u(s, t) = x(t), (51)
s→+∞
uniformly in t ∈ [0, 1]. Similarly, for γ ∈ PΛ (L) and x ∈ PΛ (H), we denote by M Λ Φ (γ, x)
the space of maps u ∈ C∞ ([0, +∞[×Ì, T ∗M) solving the Floer equation (50) with the boundary
condition
π ◦ u(0, ·) ∈ W u (γ, −gradËΛ L),
and the asymptotic conditon (51) uniformly in t ∈Ì. For a generic choice of H, these spaces of
maps are manifolds of dimension
dim M Ω Φ (γ, x) = mΩ (γ) − µ Ω (x), dimM Λ Φ (γ, x) = mΛ (γ) − µ Λ (x).
The inequality (49) provides us with the energy estimates which allow to prove suitable compactness
properties for the spaces M Ω Φ (γ, x) and M Λ Φ (γ, x). When µΩ (x) = mΩ (γ), resp. µ Λ (x) =
mΛ (γ), the space M Ω Φ (γ, x), resp. M Λ Φ (γ, x), consists of finitely many oriented points, which add
up to the integers nΩ Φ (γ, x), resp. nΛΦ (γ, x). These integers are the coefficients of the homomorphisms
Φ Ω L : Mk(ËΩ
L , g Ω ) → F Ω �
k (H, J), γ ↦→ n Ω Φ (γ, x)x,
x∈P Ω (H)
µ Ω (x)=k
Φ Λ L : Mk(ËΛ L, g Λ ) → F Λ k (H, J), γ ↦→ �
x∈P Λ (H)
µ Λ (x)=k
n Λ Φ(γ, x)x,
which are shown to be chain maps. The inequality (49) together with its differential version implies
that nΦ(γ, x) = 0 if�H(x) ≥ËL(γ) and γ �= π ◦ x, while nΦ(γ, x) = ±1 if γ = π ◦ x. These facts
imply that Φ Ω and Φ Λ are isomorphisms. We summarize the above facts into the following:
4.1. Theorem. For a generic metric g Ω , resp. g Λ , on the based loop space Ω 1 (M, q0), resp. on the
free loop space Λ 1 (M), and for a generic Lagrangian L ∈ C ∞ ([0, 1]×TM), resp. L ∈ C ∞ (Ì×TM),
satisfying (L0) Ω , resp. (L0) Λ , (L1), (L2), the above construction produces an isomorphism
Φ Ω L : Mk(ËΩ L, g Ω ) → F Ω k (H, J), resp. Φ Λ L : Mk(ËΛ L, g Λ ) → F Λ k (H, J),
from the Morse complex of the Lagrangian action functional to the Floer complex of (H, J), where
H is the Fenchel transform of L.
The same idea produces an isomorphism
Φ Θ L1⊕L2 : Mk(ËΘ L1⊕L2 , g Θ ) → F Θ k (H1 ⊕ H2, J),
between the Morse and the Floer complex associated to the figure-8 problem (see sections 2.5 and
3.4). Indeed, given γ ∈ PΘ (L1 ⊕ L2) and x ∈ PΘ (H1 ⊕ H2), we consider the space M Θ Φ (γ, x) of
maps u ∈ C∞ ([0, +∞[×[0, 1], T ∗M2 ) solving the Floer equation
with non-local boundary conditions
and asymptotic condition
∂J,H1⊕H2(u) = 0,
(u(s, 0), −u(s, 1)) ∈ N ∗ ∆ Θ M , ∀s ≥ 0,
π ◦ u(0, ·) ∈ W u (γ, −gradËΘ
L1⊕L2 ),
lim u(s, t) = x(t),
s→+∞
uniformly in t ∈ [0, 1]. The following fact is proved in section 5.10:
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