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

Complexified Morse functions 53
Proof Let V0 ⊂ π −1
0 (c0 + 1/4) be the vanishing sphere corrresponding to [c0, c0 +
1/4]. In §5.2 we noted that
ρ 0 1/4 (V0) = L0.
The vanishing sphere Vγ0 is
Vγ0 = τ [c0+1/4,c2−1/4](V0).
At the end of §6 we mentioned that the transport map along [c0 + 1/4, c2 − 1/4] is
τ [c0+1/4,c2−1/4] = τ 2 [c2−1,c2−1/4] ◦ Ψ02 ◦ τ 0 [c0+1/4,c0+1] .
Now τ 0 [c0+1/4,c0+1] does not really affect V0 in the sense that τ 0 [c0+1/4,c0+1] (V0) satisfies
Set
Then, since Ψ02 = (ν2 −1 )−1 ◦ ρ0 1 , we have
ρ 0 1 (τ 0 [c0+1/4,c0+1] (V0)) = L0.
V2 = (Ψ02 ◦ τ 0 [c0+1/4,c0+1] )(V0).
ν 2 −1 (V2) = ρ 0 1 (τ 0 [c0+1/4,c0+1] (V0)) = L0 ⊂ M.
Now, again, τ 2 [c2−1,c2−1/4] does not really affect V2 in the sense that
ν 2 −1/4 (τ 2 [c2−1,c2−1/4] (V2)) = L0.
But τ 2 [c2−1,c2−1/4] (V2) = Vγ0 , so that proves ν2 (Vγ0 −1/4 ) = Vγ0 .
In §5.3 we saw that V j
γ2 ⊂ π−1
2 (c2 − 1/4) satisfies
By (6.2),
and so we have
V j γ2 = Σ2 −1/4 = α2( −1/4S 3 ) ⊂ [(π 2 loc ) j ] −1 (−1/4).
ν 2 −1/4 | [(π 2 loc ) j ] −1 (−1/4) = ρ0 1/4
because 1/4S 3 = Σ 0 1/4 ⊂ (π0 loc )−1 (1/4).
◦ m(i) ◦ α2,
ν 2 −1/4 (V j γ2 ) = ρ0 1/4 (1/4S 3 ) = L j
2 ⊂ M,
To analyze Vγ4 , first consider the vanishing sphere
V4 ⊂ π −1 (c4 − 1/4) = π −1
4 (c4 − 1/4)
corresponding to the path [c4 − 1/4, c4]. In §5.4 we saw that
ρ 4 −1/4 (V4) = (L4) ′ ⊂ M2.