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

Complexified Morse functions 51
And τ(L ′ 4 ) = L4 because
To prove (3), consider
φ h(µ)
π (T ′ ) = φ h(µ)
π (Dr(ν ∗ K+)) = T.
τ 2 −π : π −1
2 (c2 + 1/4) −→ π −1
2 (c2 − 1/4)
and focus on its action near one of the k local models (E 2 loc ) j (where it is supported).
Thus we consider the restrictions
(τ 2 −π ) j = (τ 2 −π )| [(π 2 loc ) j ] −1 (c2+1/4) : [(π2 loc ) j ] −1 (c2 + 1/4) −→ [(π 2 loc ) j ] −1 (c2 − 1/4).
Now (16) and (10) imply that
(Φ 2 −1/4 ◦ (τ 2 −π ) j ◦ (Φ 2 1/4 )−1 )(u, v) = σ−πR ′ (|v|)(u, v) = φ
1/4 R1/4(µ) −π (u, v).
(Here (τ 2 −π) j denotes the restriction away from the vanishing sphere.) Expanding out
the definitions of Φ2 ±1/4 , and using τ 2 −π = α2 ◦ τ 0 −π ◦ α−1
2 , we get
Note that φ R 1/4(µ)
−π
(σ −π/2 ◦ ρ 0 1/4 ◦ m(i) ◦ α2) ◦ τ 2 −π ◦ (ρ0 1/4
◦ α2) −1
= σ−π/2 ◦ ρ 0 1/4 ◦ m(i) ◦ τ 0 −π ◦ (ρ 0 1/4 )−1 = φ R1/4(µ) −π .
does not extend continuously over the zero section, but composing
on the left with σ π/2 yields a map which does extend over the section, namely φ h(µ)
π =
τ|Dr(T ∗ Sj). Indeed:
(20)
ρ 0 1/4 ◦ m(i) ◦ τ 0 π ◦ (ρ0 1/4 )−1
= σπ/2 ◦ φ R1/4(µ) −π
= φ µ/2
π ◦ φ −R1/4(µ) π
= φ h(µ)
where we recall h(t) = t/2 − R 1/4(t).
Now conjugate
to get
which is the restriction of
π = τ|φ j (Dr(T
L
2
∗Sj)), (τ 2 −π) j : [(π 2 loc) j ] −1 (1/4) −→ [(π 2 loc) j ] −1 (−1/4)
ν 2 −1/4 ◦ (τ 2 −π) j ◦ (ρ 2 1/4 )−1 : Dr(T ∗ S 3 ) −→ Dr(T ∗ S 3 ),
ν 2 −1/4 ◦ τ 2 −π ◦ (ρ2 1/4 )−1 : M2 −→ M