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We shall apply the above lemma to the complexes
Ck = � M(ËΛ L1 , g1) ⊗ M(ËΛ L2 , g2) �
k+n , C′ k = F Θ k (H1 ⊕ H2, J1 ⊕ J2),
and to the chain maps K Λ 0 and K Λ α0 . The tensor products KΛ 0 ⊗K Λ α0 and KΛ α0 ⊗KΛ 0 are represented
by the coupling - in two different orders - of the corresponding elliptic boundary value problems.
4.7. Proposition. The chain maps K Λ 0 ⊗ KΛ α0 and KΛ α0 ⊗ KΛ 0
are homotopic.
Constructing a homotopy between the coupled problems is easier than dealing with the original
ones: we can keep α0 fixed and rotate the boundary condition on the initial part of the half-strip.
This argument is similar to an alternative way, due to Hofer, to prove the gluing statements in
standard Floer homology. Details of the proof of Proposition 4.7 are contained in section 6.5
below.
Here we just construct the cycle ǫ and the chain map δ required in Lemma 4.6. Since changing
the Lagrangians L1 and L2 (and the corresponding Hamiltonian) changes the chain maps appearing
in diagram (53) by a chain homotopy, we are free to choose the Lagrangians so to make the
construction easier.
We consider a Lagrangian of the form
where the potential V1 ∈ C ∞ (Ì×M) satisfies
L1(t, q, v) := 1
2 〈v, v〉 − V1(t, q),
V1(t, q) < V1(t, q0) = 0, ∀t ∈Ì, ∀q ∈ M \ {q0}, (58)
The corresponding Euler-Lagrange equation is
HessV1(t, q0) < 0, ∀t ∈Ì. (59)
∇tγ ′ (t) = −gradV1(t, γ(t)), (60)
where ∇t denotes the covariant derivative along the curve γ. By (58) and (59), the constant curve
q0 is a non-degenerate minimizer for the action functionalËΛ L1 on the free loop space (actually, it
is the unique global minimizer), so
m Λ (q0, L1) = 0.
Notice also that the equilibrium point q0 is hyperbolic and unstable. We claim that there exists
ω > 0 such that
every solution γ of (60) such that γ(0) = γ(1), other than γ(t) ≡ q0, satisfiesËL1(γ) ≥ ω. (61)
Assuming the contrary, there exists a sequence (γh) of solutions of (60) with γh(0) = γh(1) and
0