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

Complexified Morse functions 49
γ4, is smooth, provided we parametrize both paths so they have unit speed. We express
Veγ4 in away similar to Vγ4 above:
Since γ 0 4 is smoothly isotopic to γ0 4
Veγ4
−1
= τ
eγ 0 (Vγ1). 4 4
it follows that τ −1
eγ 0 4
is smoothly isotopic to τ −1
γ 0 4
through exact symplectomorphisms and hence Veγ4 is exact isotopic to Vγ4 . Moreover,
by making γ 0 4 C0-close to γ0 4 , we can arrange that Veγ4 is C0-close to Vγ4 , for
any desired closeness. (C0 close is best possible since γ 0 2 and γ0 2 neccesarliy have
orthogonal derivatives at the right end-point.) With this understood, we stick with our
definition of Vγ4 for convenience.
We point out that τ −1
γ 0 2
coincides with the transport map
τ 2 −π : π−1
2 (c2 − 1/4) −→ π −1
2 (c2 + 1/4)
along γ(t) = c2 + 1
4 e−πit , t ∈ [0, 1]. The next lemma describes this map.
Lemma 7.2 (1) There is a canonical exact symplectic isomorphism
such that
τ : M2 −→ M
(2) τ(L ′ 0 ) = L0, τ((L j
2 )′ ) = L j
2 and τ(L′ 4 ) = L4.
(3) Under the canonical identifications π −1
2 (c2 + 1/4) ∼ = M2, π −1
2 (c2 − 1/4) ∼ = M,
τ becomes the transport map
that is,
τ 2 −π : π−1
2 (c2 + 1/4) −→ π −1
2 (c2 − 1/4),
ν 2 −1/4 ◦ τ 2 π/2 ◦ (ρ2 1/4 )−1 = τ.
Proof To prove (1), recall that M2 is defined to be the quotient of
[M \ (∪jφ j(D
L r/2(T
2
∗ S 3 ))] ⊔ [∪jDr(T ∗ Sj)]
where Sj = S 3 for all j and we have gluing maps
φ L j
2
◦ σπ/2 : D [r/2,r](T ∗ Sj) −→ φ j(D
L [r/2,r](T
2
∗ S 3 )).
We can think of M as a quotient of the same space where we use the gluing map φ L j
2
instead of φ j
L ◦ σπ/2. 2