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The space M Υ GE . Let x1 ∈ P(H1), x2 ∈ P(H2), and z ∈ PΛ (H1#H2) (see the proof of
Theorem 3.11). The space M Υ GE (x1, x2; z) is the set of pairs (α, u) where α is a real positive
parameter and u is a solution of the Floer equation on the Riemann surface ΣΥ GE (α) with suitable
asymptotics and suitable non-local boundary conditions. Linearizing this Floer equation for a
fixed α ∈]0, +∞[ yields an operator of the form
where
∂A : X 1,p
S ,W (Σ,�2n ) → X p
S (Σ,�2n ),
S = {−α, α, −α + i, α + i}, W = (∆Ê2n, ∆ ΘÊn, ∆Ên × ∆Ên).
By Theorem 5.23 this operator is Fredholm of index
ind∂A = µ(N ∗ ∆Ê2n, graphCΦ − ) − µ(N ∗ (∆Ên × ∆Ên), graphCΦ + ) − n.
As in the discussion of M Λ Υ
(identities (136) and (137)),
µ(N ∗ ∆Ê2n, graphCΦ − ) = µ Λ (x1) + µ Λ (x2), µ(N ∗ (∆Ên × ∆Ên), graphCΦ + ) = µ Λ (z).
Therefore,
ind ∂A = µ Λ (x1) + µ Λ (x2) − µ Λ (z) − n.
Considering also the parameter α, we see that for a generic choice of J the space M Υ GE (x1, x2; z)
is a smooth manifold of dimension µ Λ (x1)+µ Λ (x2) −µ Λ (z) −n+1. This proves Proposition 3.12.
The spaces MC and MEv. Let f be a Morse function on M, let x be a critical point of f, and
let y ∈ P(Λ)(H). Given q ∈ M, let
�
�MC(x, y, q) := u ∈ C ∞ ([0, +∞[×Ì, T ∗ �
�
M) �u solves (32), π ◦ u(0, t) ≡ q ∀t ∈Ì,
lim u(s, t) = y(t) uniformly in t ∈Ì�
.
s→+∞
The study of such a space involves the study of an operator of the form
∂A : X 1,p
∅,(0),(∆Ên) (Σ+ ,�n ) → X p
∅ (Σ+ ,�n ).
By Theorem 5.24, the Fredholm index of the above operator is
Therefore, the space
has dimension
ind∂A = n
2 − µ(N ∗ ∆Ên, graphCΦ + ) − n
2 = −µΛ (y).
MC(x, y) = �
q∈W u (x)
�MC(x, y, q),
dimMC(x, y) = dimW u (x) − µ Λ (y) = m(x) − µ Λ (y),
for a generic choice of J and g, proving the first part of Proposition 3.14.
Consider the space of maps
�
�MEv(y) := u ∈ C ∞ (] − ∞, 0] ×Ì, T ∗ �
�
M) �u solves (32), u(0, t) ∈ÇM ∀t ∈Ì,
lim u(s, t) = y(t) uniformly in t ∈Ì�
.
s→−∞
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