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

Complexified Morse functions 23
L
Z
Figure 3: Schematic of W 0 = D(T ∗ L) ⊞ D(T ∗ S n ) embedded into W ′ = D(T ∗ L) ∪ H near
plumbing or handle-attachment region. Parts of L and the Lagrangian sphere Z are also labeled.
with exact symplectomorphisms
W1, W2 ∼ = D(T ∗ (K × D n−k )).
(Here, we assume that S(T ∗ (K × D n−k )) ⊂ S(T ∗ Li), i = 1, 2.) To define the plumbing
W 0 = D(T ∗ L1) ⊞ D(T ∗ L2)
we take the quotient of the disjoint union D(T ∗ L1) ⊔ D(T ∗ L2), where we identify W1
and W2 using a suitable exact symplectomorphism
η : D(T ∗ (K × D n−k )) −→ D(T ∗ (K × D n−k ))
which sends K × D n−k to D(ν ∗ K) and D(ν ∗ K) to K × D n−k . This means that in W 0
a tubular neighborhood of K in L1 is identified with the disk conormal bundle of K
in D(T ∗ L2), and vice-versa. (This condition is motivated by Pozniack’s local model
[P94, Proposition 3.4.1].)
To define η, let us pass for a moment to the noncompact model
T ∗ (K × R n−k ) ∼ = T ∗ K × T ∗ R n−k ∼ = T ∗ K × C n−k .