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

52 Joe Johns
to Dr(T ∗Sj) ⊂ M2. By definition of ν2 −1/4 and ρ2 1/4 , and using (20) for the last step,
we have
ν 2 −1/4 ◦ (τ 2 −π ) j ◦ (ρ 2 1/4 )−1
= (ρ 0 1/4 ◦ m(i) ◦ α2) ◦ τ 2 −π ◦ (ρ 0 1/4
◦ α2) −1
= ρ 0 1/4 ◦ m(i) ◦ τ 0 −π ◦ (ρ0 1/4 )−1 = τ|Dr(T ∗ Sj)
This shows ν 2 −1/4 ◦τ 2 −π ◦ (ρ2 1/4 )−1 and τ agree on each neighborhood Dr(T ∗ Sj) ⊂ M2.
Since both maps equal the identity outside of ∪ j=k
j=1 Dr(T ∗ Sj), this shows
ν 2 −1/4 ◦ τ 2 −π ◦ (ρ2 1/4 )−1 = τ : M2 −→ M.
Remark 7.3 Just as a sanity check, let’s look at the other map τ 2 π (as opposed to τ 2 −π ).
In the last lemma we saw that
And the same calculation shows that
(ν 2 −1/4 ◦ τ 2 −π ◦ (ρ2 1/4 )−1 )| (E 2 loc ) j
= σπ/2 ◦ φ R1/4(µ) −π .
(ν 2 −1/4 ◦ τ 2 π ◦ (ρ2 1/4 )−1 )| (E 2 loc ) j
= σπ/2 ◦ φ R1/4(µ) π .
Then the total monodromy τ 2 2π = (τ 2 −π )−1 ◦ τ 2 π
as expected.
φ R 1/4(µ)
π
coresponds to
◦ σ −1
π/2 ◦ σπ/2 ◦ φ R1/4(µ) π = φ R1/4(µ) 2π
We now describe the vanishing spheres corresponding to γ0,γ2,γ4 .
Lemma 7.4 Under the canonical isomorphism
ν 2 −1/4 : π−1 (b) −→ M
j
the vanishing spheres Vγ0 , Vγ2 , Vγ4 correspond respectively to
L0, L j
2 , L4 ⊂ M.