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where Ã(z) = CA(z)C ⊕ A(z), C denoting complex conjugacy on�n .
By Theorem 5.9, the above operator ∂à - hence the original operator D + G - is Fredholm of
index
ind (∂à ) = µ(Φ− + ∗ n
iÊ2n
, iÊ2n
) − µ(Φ N ∆Ên, iÊ2n
) − , (129)
2
where Φ − , Φ + : [0, 1] → Sp(4n) solve the linear Hamiltonian systems
d
dt Φ− (t) = i Ã(−∞, t)Φ− (t),
d
dt Φ+ (t) = i Ã(+∞, t)Φ+ (t), Φ − (0) = Φ + (0) = I.
Since Ã(−∞, t) = CA1(−t)C ⊕ A2(t), we have Φ − (t) = CΨ − 1 (−t)C ⊕ Ψ2(t), and
µ(CΨ − 1 (−·)CiÊn , iÊn ) = µ(CΨ − 1 (−·)iÊn , iÊn ) = −µ(Ψ − 1 (−·)iÊn , CiÊn )
= −µ(Ψ − 1 (−·)iÊn , iÊn ) = µ(Ψ − 1 iÊn , iÊn ),
where we have used the fact that C is a symplectic isomorphism from (Ê2n , ω0) to (Ê2n , −ω0),
and the fact that the Maslov index changes sign when changing the sign of the symplectic form,
or when reversing the parameterization of the Lagrangian paths. By the additivity of the Maslov
index and by (128) we get
µ(Φ − iÊ2n , iÊ2n ) = µ(CΨ − 1 (−·)CiÊn , iÊn ) + µ(Ψ2iÊn , iÊn )
= µ(Ψ − 1 iÊn , iÊn ) + µ(Ψ − 2 iÊn , iÊn ) = µ Ω (x1) + µ Ω (x2) + n.
On the other hand, Ã(+∞, t) = CA+ ((1 − t)/2)C ⊕ A((t + 1)/2), which implies
Φ + (t) = CΨ + ((1 − t)/2)Ψ + (1/2) −1 C ⊕ Ψ + ((1 + t)/2)Ψ + (1/2) −1 .
Since N ∗ ∆Ên = graphC = {(z, z) | z ∈�n }, we easily find
Φ + (t)N ∗ ∆Ên = graphΓ(t)C, with Γ(t) := Ψ +
� 1 + t
2
�
Ψ +
The symplectic paths Γ and Ψ + are homotopic by the symplectic homotopy
(λ, t) ↦→ Ψ +
�
t + λ
(1 − t)
2
�
Ψ +
� �−1 λ
(1 − t) ,
2
� 1 − t
2
(130)
� −1
. (131)
which leaves the end-points Γ(0) = Ψ + (0) = I and Γ(1) = Ψ + (1) fixed. Therefore, by the
homotopy invariance of the Maslov index, by (78) and by (128) we have
µ(Φ + N ∗ ∆Ên, iÊ2n ) = µ(graphΓC, iÊ2n ) = µ(graphΨ + C, iÊ2n )
= µ(Ψ + iÊn , iÊn ) = µ Ω (y) + n
2 .
Therefore, by (129), (130), and (132), we conclude that
ind (∂à ) = µΩ (x1) + µ Ω �
(x2) + n − µ Ω (y) + n
�
−
2
n
2 = µΩ (x1) + µ Ω (x2) − µ Ω (y).
(132)
Hence we have proved that the fiberwise derivative of the section DJ,H : W Ω Υ → E Ω Υ is a Fredholm
operator of index µ Ω (x1) + µ Ω (x2) − µ Ω (y).
For a generic choice of the ω-compatible almost complex structure J, the section DJ,H is
transverse to the zero-section (the proof of transversality results of this kind is standard, see
[FHS96]). Let us fix such an almost complex structure J. Then, M Ω Υ - if non-empty - is a smooth
submanifold of W Ω Υ of dimension µΩ (x1)+µ Ω (x2) − µ Ω (y). This proves the Ω part of Proposition
3.4.
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