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where
S = {α0, α0 + i}, V0 = (0), W = (∆Ê2n, ∆ ΘÊn).
See the analysis for ME. By Theorem 5.24, the above operator is Fredholm of index
ind∂A = n − µ(N ∗ ∆ ΘÊn, graphCΦ + ) − n − n
2 = −µ(N ∗ ∆ ΘÊn, graphCΦ + ) − n
2 .
Then, by (139), we have
ind∂A = −µ Θ (x) − n.
Letting the elements of the unstable manifolds of γ1 and γ2 vary, we increase the index by
m Λ (γ1; L1) + m Λ (γ2; L2), and we find the formula
This proves Proposition 4.5.
dimM K α0 (γ1, γ2; x) = m Λ (γ1; L1) + m Λ (γ2; L2) − µ Θ (x) − n.
The space M K G . Let γ ∈ PΘ (L1 ⊕ L2) and x ∈ PΛ (H1#H2) (see section 4.5). The space
M K G (γ, 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 G (α), which is asymptotic to x for s → +∞, lies above
some element in the unstable manifold of γ for s = 0, and satisfies the figure-8 boundary condition
for s ∈ [0, α]. Linearizing the Floer equation for a fixed positive α and for a fixed curve in the
unstable manifold of γ yields an operator of the form
where
∂A : X 1,p
S ,V0,W (Σ+ ,�2n ) → X p
S (Σ+ ,�2n ),
S = {α, α + i}, V0 = (0), W = (∆ ΘÊn, ∆Ên × ∆Ên).
See the analysis for MG. By Theorem 5.24, the above operator is Fredholm of index
ind ∂A = n − µ(N ∗ (∆Ên × ∆Ên), graphCΦ + ) − n n
−
2 2 = −µ(N ∗ (∆Ên × ∆Ên), graphCΦ + ).
Hence, by (142), we have
ind∂A = −µ Λ (x; H1#H2).
Letting the elements of the unstable manifold of γ vary, we increase the index by m Θ (γ; L1 ⊕ L2).
Letting also α vary we further increase the index by 1, and we find the formula
This proves Proposition 4.8.
dim M K G (γ, x) = m Θ (γ; L1 ⊕ L2) − µ Λ (x; H1#H2) + 1.
6 Compactness and cobordism
6.1 Compactness in the case of jumping conormal boundary conditons
Compactness in the C∞ loc topology of all the spaces of solutions of the Floer equation considered
in this paper can be proved within the following general setting. Let Q be a compact Riemannian
manifold, and let Q0, Q1, . . . , Qk be submanifolds of Q × Q. We assume that there is an isometric
embedding Q ֒→ÊN and linear subspaces V0, V1, . . . , Vk ofÊN ×ÊN , such that Vj−1 is partially
orthogonal to Vj, for every j = 1, . . .,k, and
Qj = Vj ∩ (Q × Q).
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