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

Complexified Morse functions 35
The basic objects above become:
Φ 2 = Φ 0 ◦ α2,
ρ 2 s = ρ0s ◦ α2,
Σ 2 w = α2(Σ 0 w),
k 2 (z) = k 0 (α2(z)) = |z| 4 − |q2(z)| 2 ,
τ 2 θ
= α−1
2 ◦ τ 0 θ ◦ α2.
Then everything is as before. In particular, the formula for the transport map τ 2 θ
same as in (15):
(Φ 2
seiθ ◦ τ 2 θ ◦ (Φ2s )−1 )(u, v) = σθeR ′ (u, v).
s(|v|)
We could treat q4 in a similar way, but instead we will use the fact that
q −1
4
(−s) = q−1
0 (s), s > 0.
Namely, for q4 we have a canonical identification
q −1
4 (−s) −→ T∗ S 3
(rather than the the usual q −1 (s) −→ T ∗ S n ) given by
Thus, for w ∈ C, s > 0, z ∈ C 4 , we take
ρ 0 s : q −1
0 (s) = q−1
4 (−s) −→ T∗S 3 .
Φ 4 w = Φ0 −w ,
ρ 4 −s = ρ 0 s,
Σ 4 w = Σ0 −w ,
Σ 4 = Σ 0 ,
k 4 (z) = k 0 (z) = |z| 4 − |q4(z)| 2 .
is the
(Note that Φ 2 and Φ 4 are not defined analogously, but using this Φ 4 will save us a
little trouble later.)