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The space M Λ Υ . Let us study the space of solutions M Λ Υ (x1, x2; y), where x1 ∈ PΛ (H1), x2 ∈
PΛ (H2), and y ∈ PΛ (H1#H2) (see section 3.3). It is a space of solutions of the Floer equation
on the pair-of-pants Riemann surface ΣΛ Υ , described as a quotient of a strip with a slit in section
3.2.
Arguing as in the case of M Ω Υ , it is easily seen that the space M Λ Υ is the set of zeroes of a
smooth section of a Banach bundle, whose fiberwise derivative at some u ∈ M Λ Υ is conjugated to
an operator of the form
where
D + G : W 1,p (Σ Λ Υ ,�n 0,1
) → ΩLp(ΣΛΥ ,�n 1
), v ↦→ (Dv + iDv ◦ j) + Gv,
2
(Gv)(z) = 1 i
A(z)v(z)ds −
2 2 A(z)v(z)dt.
The smooth map A : Σ Λ Υ → L(Ê2n ,Ê2n ) has the following asymptotics
A(s + (t − 1)i) → A − 1 (t), A(s + ti) → A− 2 (t) for s → −∞, A(s + (2t − 1)i) → A+ (t), for s → +∞,
for any t ∈ [0, 1], where A − 1 (t), A− 2 (t), and A+ (t) are symmetric matrices such that the solutions
of the linear Hamiltonian systems
d
dt Ψ− 1 (t) = iA− 1 (t)Ψ− 1 (t),
d
dt Ψ− 2 (t) = iA− 2 (t)Ψ− 2 (t),
d
dt Ψ+ (t) = 2iA + (t)Ψ + (t), Ψ − 1 (0) = Ψ− 2 (0) = Ψ+ (0) = I,
are conjugated to the differential of the Hamiltonian flows along x1, x2, and y:
Ψ − 1 (t) ∼ Dxφ H1 (1, x1(0)) Ψ − 2 (t) ∼ Dxφ H2 (1, x2(0)) Ψ + (t) ∼ Dxφ H1#H2 (1, y(0)).
Then, by the relationship (79) between the Conley-Zehnder index and the relative Maslov index,
we have
µ Λ (x1) = µCZ(Ψ − 1 ) = µ(N ∗ ∆, graphCΨ − 1 ), µΛ (x2) = µCZ(Ψ − 2 ) = µ(N ∗ ∆, graphCΨ − 2 ), (133)
µ Λ (y) = µCZ(Ψ + ) = µ(N ∗ ∆, graphCΨ + ). (134)
Using again the transformation ˜v(z) := (v(z), v(z)), the operator D + G is easily seen to be
conjugated to the operator
where
∂ Ã
: X1,p
S ,W (Σ,�2n ) → X p
S (Σ,�2n ), u ↦→ ∂u + Ãu,
S = {0, i}, W = (W0, W1) = (∆Ê2n, ∆Ên × ∆Ên), Ã(z) = CA(z)C ⊕ A(z).
Notice that the intersection
W0 ∩ W1 = {(ξ, ξ, ξ, ξ) | ξ ∈Ên }
is an n-dimensional linear subspace ofÊ4n . Then by Theorem 5.23, the operator ∂Ã
of index
is Fredholm
ind∂ Ã = µ(N ∗ W0, graphCΦ − ) − µ(N ∗ W1, graphCΨ + ) − n, (135)
where the symplectic paths Φ − , Φ + : [0, 1] → Sp(4n) are related to Ψ − 1 , Ψ− 2 , Ψ+ by the identities
Φ − (t) = CΨ − 1 (−t)C ⊕ Ψ− 2 (t), Φ+ (t) = CΨ + ((1 − t)/2)Ψ + (1/2) −1 C ⊕ Ψ + ((1 + t)/2)Ψ + (1/2) −1 .
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