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

54 Joe Johns
Now
Using
V γ 1 4 = τ −1
[c2+1/4,c4−1/4] (V4) ⊂ π −1
2 (c4 + 1/4).
τ −1
[c2+1/4,c4−1/4] = (τ 2 [c2+1/4,c2+1] )−1 ◦ Ψ −1
24 ◦ (τ 4 [c4−1,c4−1/4] )−1
and arguing as we did for Vγ0 one sees that V γ 1 4 satisfies
Then
Vγ4
Therefore, using lemma 7.2, we get
ρ 2 1/4 (V γ 1 4 ) = L′ 4 ⊂ M2.
−1
= τ
γ0 (Vγ1) = τ
4 4 2 −π (Vγ1 4 )
ν 2 −1/4 (Vγ4 ) = (ν2 −1/4 ◦ τ 2 −π ◦ (ρ 2 1/4 )−1 )(L ′ 4) = τ(L ′ 4) = L4 ⊂ M.
8 Construction of N ⊂ E
In this section we construct an exact Lagrangian embedding N ⊂ E and prove Theorem
A (see Theorem 8.1). We also discuss some ways of refining Theorem A in remark
8.5, and we give a detailed sketch of the proof that E is homotopy equivalent to N in
Proposition 8.4.
We first construct a Lagrangian submanifold N ⊂ E and then check that N is diffeomorphic
to N . N ⊂ E will be defined as the union of several Lagrangian manifolds,
say Ni, with boundary (sometimes with corners). This decomposition of N is essentially
the same as the handle-type decomposition which appears in [M65, pages 27-32].
The fact that the union of the Ni is diffeomorphic to N is essentially Theorem 3.13 there.
Let
N0 = ∆ [c0,c2−1/10],
N4 = ∆ [c2+1/10,c4].
These are the Lefschetz thimbles over the indicated intervals. They correspond to the
0− and 4− handles of N . Let
f2 = q2| E 2 loc ∩R 4 : E2 loc ∩ R 4 −→ R,