The Picard-Lefschetz theory of complexified Morse functions 1 ...

Complexified Morse functions 55
and let
N j
2 = ((E2 loc ) j ∩ R 4 ) ∩ f −1
2 ([c2 − 1/2, c2 + 1/2]).
This corresponds to the jth 2−handle of N .
Recall from §5.3 that E2 is a certain quotient of the disjoint union
[M2 \ (∪ j=k
j=1 φ (L j
2 )′(D r/2(T ∗ S 3 )))] × D 2 (c2) ⊔ ⊔ j=k
j=1 (E2 loc) j ) .
Recall L ′ 4 ⊂ M2 is defined as
L ′ 4 = [L0 \ (∪ j=k
j=1 φj(S 1 × D 2 r/2 ))] ∪ φ (L j
2 )′(Dr(ν ∗ K+)),
where φ j
(L2 )′ identifies D (r/2,r](ν ∗K+) with φj(S1 × D2 (r/2,r] ). Define Ntriv
2 ⊂ E2 by
N triv
2 = [L0 \ (∪ j=k
j=1 φj(S 1 × D 2 r/2 ))] × [c2 − 1/2, c2 + 1/2]
Then, N ⊂ E is defined to be the union
⊂ [M2 \ (∪ j=k
j=1 φ (L j
2 )′(D r/2(T ∗ S 3 )))] × D 2 1/2 (c2).
N = N0 ∪ (∪j N j
2 ) ∪ Ntriv 2 ∪ N4.
See figure 1 in §1 for the 2-dimensional version of this. (Note that the pieces over-lap.)
Theorem 8.1 (1) N ⊂ E is a smooth closed exact Lagrangian submanifold.
(2) There is a diffeomorphism α: N −→ N .
(3) π(N) = [c0, c4].
(4) All critical points of π lie on N , and in fact Crit(π) = α(Crit(f )).
(5) There is a diffeomorphism β : R −→ R such that
β ◦ π|eN ◦ α = f : N −→ R.
Proof First we prove that N is a smooth manifold diffeomorphic to N . Define
N2 = (∪ j=k j
j=1 N2 ) ∪ Ntriv 2 .
The gluing map in the definition of E2 is the composite of
and
Φ 2 : (E 2 loc ) j ∩ (k 2 ) −1 ([4(r/2) 2 , 4r 2 ]) −→ D [r/2,r](T ∗ S 3 ) × D 2 (c2)
φ (L j
2 )′ : D [r/2,r](T ∗ S 3 ) −→ M2.