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

Complexified Morse functions 41
defined as before. For fixed 0 < s ≤ 1, let
γ2(t) = s(1 − t)
so that γ2(0) = s, γ2(1) = 0. There are k disjoint Lefschetz thimbles corresponding
to this one path γ2, one in each local model (E 2 loc ) j . Namely,
The vanishing spheres
satisfy
∆ j γ2 = ∪tΣ 2 γ2(t) = ∪t α −1
2 (Σ0 γ2(t) ) ⊂ (E2 loc) j ⊂ E2.
5.4 Construction of π4 : E2 −→ D 2
V j γs = Σ2 s ⊂ [(π 2 loc) j ] −1 (s) ⊂ π −1
2 (s)
ρ 2 s (V j j
γs ) = (L2 )′ .
π4 will also have regular fiber M2, but at s = −1 rather than s = 1. It remains to
specify the vanishing sphere. We have already seen M2 has exact Lagrangian spheres
(L j
. There are also exact Lagrangian spheres
2 )′ corresponding to L j
2
L ′ 0, L ′ 4 ⊂ M2
which correspond to L0, L4. (in the sense of lemma 7.2 below). Namely, define L ′ 4 as
a union in M2:
L ′ 4 = [L0 \ (∪ j=k
j=1φj(S 1 × D 2 j=k
r/2 ))] ∪ [∪j=1φ (L j
2 )′(Dr(ν ∗ K+))].
Here, φ j
(L2 )′ identifies
D [r/2,r](ν ∗ K+) ⊂ Dr(T ∗ S 3 )
with
(17)
φ L j
2
◦ σπ/2(D [r/2,r](ν ∗ K+)) = φ j(D
L [r/2,r](ν
2
∗ K−)) = φj(S 1 × D 2 [r/2,r] ) ⊂ L0.
The definition of L ′ 0 is analogous to that of L4 in §3.1, 2.2, as follows. First set
Note that
Then L ′ 0
is defined to be
L ′ 0 = φ h(µ)
−π (Dr(ν ∗ K−)) ⊂ Dr(T ∗ S 3 ).
L ′ 0 ∩ Dr(ν ∗ K+) = D (r/2,r](ν ∗ K+).
L ′ 0 = [L0 \ (∪ j=k
j=1 φj(S 1 × D 2 r/2 ))] ∪ [∪ j=k
j=1 φ (L j
2 )′(L ′ 0)] ⊂ M2.