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

Complexified Morse functions 39
For any 0 < s ≤ 1 we have the canonical identification
(ρ 0 loc)s : π −1
loc (s) −→ Dr(T ∗ S n )
which gives rise to a canonical identification
ρ 0 s : π −1
0 (s) −→ M
which is defined to be the identity map on (M \ φL0 (D r/2(T ∗ S n )) × {s}, and on π −1
loc (s)
it is defined to be
For any embedded path
φL0 ◦ (ρ0 loc)s = φL0 ◦ Φ0 | π −1
loc (s).
γ : [0, 1] −→ D 2
such that γ(0) = s > 0, γ(1) = 0, the Lefschetz thimble of γ is
The vanishing sphere in π −1
0 (s) is
Note that
5.3 Construction of π2 : E2 −→ D 2
∆γ = ∪tΣ 0 γ(t) ⊂ E 0 loc ⊂ E0.
Vγ = Σ 0 s ⊂ (π 0 loc )−1 (s) ⊂ (π0) −1 (s).
ρ 0 s (Vγ) = L0 ⊂ M.
First, we modify M to get a new manifold M2 which will be the regular fiber of π2.
For each j = 1,... , k take an exact Weinstein embedding for L j
2 ⊂ M
satisfying
φ j : D L2 gS3 r (T ∗ S 3 ) −→ M
φ j|Dr(ν
L2 ∗K−) = φj| S1 ×D2 r
as in section 3.1 for some r. Note this is the same r as in 4.3, and 3.1 (shrink r if
necessary). We want to describe a twist operation which will happen near each L j
2 .
But it is clearer to describe it near just one L j
2
quotient of the disjoint union
where we identify
[M \ φ j(D
L r/2(T
2
∗ S 3 )] ⊔ Dr(T ∗ S 3 ),
D (r/2,r](T ∗ S 3 ) and φ j
L (D (r/2,r](T
2
∗ S 3 )) ⊂ M
j
2
to start: Define TL
π/2 (M) to be the be the