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

Complexified Morse functions 45
Then we will define
π = π0#π2#π4, E = E0#E2#E4.
The main step is to specify exact symplectic identifications
Ψ02 : π −1
0 (c0 + 1) −→ π −1
2 (c2 − 1), and
Ψ24 : π −1
2 (c2 + 1) −→ π −1
4 (c4 − 1).
First, Ψ24 is easy because, for 0 < s ≤ 1, there are canonical maps
So we set
ρ 2 s : π −1
2 (c2 + s) −→ M2,
ρ 4 −s : π −1
4 (c4 − s) −→ M2.
Ψ24 = (ρ 4 −1 )−1 ◦ ρ 2 1 .
The next lemma will tell us how to define Ψ02.
Lemma 6.1 For 0 < s ≤ 1, there is a canonical exact symplectic isomorphism
Proof Note that
ν 2 −s : π −1
2 (c2 − s) −→ M.
M2 \ (∪ j=k
j=1 φ (L j
2 )′(D r/2(T ∗ S 3 ))
is canonically isomorphic to
M \ (∪ j=k
j=1φL j(Dr/2(T
2
∗ S 3 )).
From now on we will identify these. Then, by definition, π −1
2 (c2 − s) is equal to the
quotient manifold obtained from the disjoint union:
M \ (∪ j=k
where for each j we identify
with
using the gluing maps
j=1φL j(Dr/2(T
2
∗ S 3 )) ⊔ [⊔ j=k
j=1 [(π2 loc) j ] −1 (c2 − s)]
Uj = (Φ 2 −s) −1 (D (r/2,r](T ∗ S 3 )) ⊂ [(π 2 loc) j ] −1 (c2 − s)
Wj = φ j(D
L2 gS3 [r/2,r] (T∗S 3 )) ⊂ M,
(φ j
L ◦ σπ/2 ◦ Φ
2
2 −s)|Uj : Uj −→ Wj.