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

Complexified Morse functions 37
5 Construction of three Lefschetz fibrations
In this section we review how to construct a Lefschetz fibration with any prescribed
symplectic manifold M ′ as the fiber, and with a single vanishing sphere consisting
of any prescribed exact Lagrangian sphere L ′ ⊂ M ′ (see lemma 1.10 in [S03A] for
details). We apply this construction for each of our local models πi loc , i = 0, 2, 4
to produce three Lefschetz fibrations πi, i = 0, 2, 4. If we think of the target of
π : E −→ D2 as containing three real critical values c0 < c2 < c4 corresponding to
those of f , then (E,π) will be constructed so that (Ei,πi) is equal to the part of (E,π)
lying over a small disk around ci :
(E|Ds(ci),π|Ds(ci)) ∼ = (Ei,πi).
The regular fiber of πi, say Mi, is a exact symplectomorphic to M (the regular fiber
of π) but Mi is in some cases obtained from M by a key twisting operation, which
models the transport map π −1 (c2 − s) −→ π −1 (c2 + s) along a half circle in the lower
half plane.
5.1 A particular choice for h in the definition of L4
Before proceding we specify our choice of the function h in the definition of L4 in
§3.1. Namely we set
h(t) = t/2 − R 1/4(t)
where R 1/4 is from (16). (The choice s = 1/4 comes from the choice of basepoint
(19), which comes later.)
5.2 Construction of π0 : E0 −→ D 2
In this section we construct a Lefschetz fibration
π0 : E0 −→ D 2
such that for any 0 < s ≤ 1 there is a canonical exact symplectic identification
ρ 0 s
: π−1
0 (s) −→ M, 0 < s ≤ 1
with the vanishing sphere corresponding to L0. By construction of M, we have a
canonical exact Weinstein embedding
D(T ∗ S 3 ) −→ M.