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The space M Θ Φ . Let γ ∈ PΘ (L1 ⊕ L2), and let x ∈ PΘ (H1 ⊕ H2). The study of the space
M Θ Φ (γ, x) reduces to the study of an operator of the form
∂A : X 1,p
∅,(0),(∆ ΘÊn) (Σ+ ,�2n ) → X p
∅ (Σ+ ,�2n ).
Indeed, for a generic choice of J, M Θ Φ (γ, x) is a manifold of dimension mΘ (γ) plus the Fredholm
index of the above operator. By Theorem 5.24, the index of this operator is
ind ∂A = n − µ(N ∗ ∆ ΘÊn, graphCΦ + ) − 1
2 dim∆ΘÊn = n
2 − µ(N ∗ ∆ ΘÊn, graphCΦ + ).
Since µ(N ∗ ∆ ΘÊn, graphCΦ + ) = µ Θ (x) + n/2, we conclude that
proving Proposition 4.2.
dimM Θ Φ (γ, x) = mΘ (γ) − µ Θ (x),
The space M K Υ . Let γ1 ∈ PΩ (L1), γ2 ∈ PΩ (L2), and x ∈ PΩ (H1#H2) (see section 4.2). The
space M K Υ (γ1, γ2; x) consists of pairs (α, u) where α is a positive number and u(s, t) is a solution
of the Floer equation on the Riemann surface ΣK Υ (α), which is asymptotic to x for s → +∞, lies
above some element in the unstable manifold of γ1 (resp. γ2) for s = 0 and −1 ≤ t ≤ 0− (resp.
0 + ≤ t ≤ 1), and lies above q0 at the other boundary points. Linearizing the Floer equation for a
fixed positive α and for fixed elements in the unstable manifolds of γ1 and γ2 yields an operator
of the form
where
∂A : X 1,p
S ,W,V ,V ′(Σ+ ,�2n ) → X p
S (Σ+ ,�2n ),
S = {α}, W = (0), V = ((0), ∆Ên), V ′ = (0).
See the analysis for M Ω Υ . By Theorem 5.21, the above operator is Fredholm of index
Hence, by (132), we have
ind∂A = n − µ(Φ + N ∗ n n
∆Ên, iÊ2n
) − =
2 2 − µ(Φ+ N ∗ ∆Ên, iÊ2n
).
ind∂A = −µ Ω (x; H1#H2).
Letting the elements of the unstable manifolds of γ1 and γ2 vary, we increase the index by
m Ω (γ1; L1) + m Ω (γ2; L2). Letting also α vary we further increase the index by 1, and we find
the formula
dimM K Υ (γ1, γ2; x) = m Ω (γ1; L1) + m Ω (γ2; L2) − µ Ω (x; H1#H2) + 1.
See [AS06b], section 3.1, for more details on how to deal with this kind of boundary data. This
proves Proposition 4.3.
The space M K α0 . Let γ1 ∈ P Λ (L1), γ2 ∈ P Λ (L2), and x ∈ P Θ (H1 ⊕ H2) (see section 4.4).
The space M K α0 (γ1, γ2; x) consists of solutions u = (u1, u2) of the Floer equation on the Riemann
surface Σ K α0 , which is asymptotic to x for s → +∞, u1 and u2 lie above some elements in the
unstable manifolds of γ1 and γ2 for s = 0, and u satisfies the figure-8 boundary condition for
s ≥ α0. Linearizing the Floer for a fixed pair of curves in the unstable manifolds of γ1 and γ2
yields an operator of the form
∂A : X 1,p
S ,V0,W (Σ+ ,�2n ) → X p
S (Σ+ ,�2n ),
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