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The study of the this space reduces to the study of an operator of the form
which has index
∂A : X 1,p
∅,Ên ,(∆Ên) (Σ− ,�n ) → X p
∅ (Σ− ,�n ),
ind∂A = n
2 + µ(N ∗ ∆Ên, graphCΦ − ) − 1
2 (n + 2n − 2n) = µΛ (y),
by Theorem 5.25. For a generic choice of J, � MEv(y) is then a manifold of dimension
dim � MEv(y) = µ Λ (y).
For a generic choice of the Riemannian metric g on M, the map
�MEv(y) → M, u ↦→ π ◦ u(0, 0),
is transverse to the submanifold W s (x). For these choices of J and g, the space
�
MEv(y, x) = u ∈ � MEv(y) | π ◦ u(0, 0) ∈ W s �
(x)
is a manifold of dimension
dimMEv(y, x) = dim � MEv(y) − codimW s (x) = µ Λ (y) − m(x),
proving the second part of Proposition 3.14.
The space MI! . Let x ∈ PΛ (H) and y ∈ PΩ (H) (see section 3.6). The study of the space
(x, y) involves the study of an operator of the form
MI!
where
∂A : X 1,p
S ,W (Σ,�n ) → X p
S (Σ,�n ),
S = {0, i}, W = (∆Ên, (0)).
By Theorem 5.23 this operator is Fredholm of index
By (79),
ind∂A = µ(N ∗ ∆Ên, graphCΦ − + n
) − µ(iÊ2n
, graphCΦ ) − . (143)
2
µ(N ∗ ∆Ên, graphCΦ − ) = µCZ(Φ − ) = µ Λ (x). (144)
By the skew-symmetry of the Maslov index, by the fact that the action of C changes the sign of
the Maslov index and leaves iÊ2n invariant, and by (78),
µ(iÊ2n , graphCΦ + ) = −µ(graphCΦ + , iÊ2n ) = µ(graphΦ + C, iÊn × iÊn )
= µ(Φ + iÊn , iÊn ) = µ Ω (y) + n
2 .
Therefore by (143), (144), and (145),
This proves Proposition 3.16.
ind∂A = µ Λ (x) − µ Ω (y) − n.
85
(145)