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

Complexified Morse functions 63
For E4 and E2 we define conjugation maps ιE2 , ιE4 in a similar way: On M2, ιM2 is
defined the same way on D(T∗L0) but on D(T∗L j
2 ) it is defined to be
((x1, x2), (y1, y2)) ↦→ ((−x1, x2), (y1, −y2))
so that the fixed point set is D(ν ∗ K+), which is identified with part of L0 to form L ′ 4 .
Then by combining ιM2 × ιD2 with ιEloc 2 and ιEloc 4
and ιE4 on E4. To combine ιE0
that the gluing maps π −1
0
and ιE2
to ιE4 .
we get a conjugation maps ιE2 on E2
,ιE2 ,ιE4 to get a conjugation map ι on E one checks
(−1) and π−1(1)
−→ π−1
(1) −→ π−1
2
2
4
(−1) map ιE0 to ιE0
9 Construction of π : E −→ D 2 and N ⊂ E, dim N = 3
Consider the case when N is a closed 3-manifold, and f : N −→ R is a self-indexing
Morse function. The following discussion applies equally well to self-indexing Morse
functions f : N −→ R with four critical values 0, n, n+1, 2n+1. See §3.2 to see how
things are much the same from one dimension to the next. In §3.3 we explained how
to constructed a Weinstein manifold M with exact Lagrangian spheres L0, L j
1 , Lj
2 , L3,
one for each critical point of f . Assume for simplicity of notation there is only one
critical point of each index:
x3, x2, x1, x0.
Thus T = f −1 (3/2) ∼ = T 2 , a torus, and we have just one α curve and one β curve. For
each j = 0, 1, 2, 3 we will define a Lefschetz fibration πj : Ej −→ D(cj) over a disk
D(cj), where cj is the critical value of πj, and πj has just one critical point corresponding
to xj . Then we will fiber-connect sum the πj’s together to form π : E −→ S ∼ = D 2 .
Then we will show that π has regular fiber isomorphic to M and the vanishing spheres
for suitable paths correspond to L0, L2, L2, L3 ⊂ M. Finally, we will show there is an
exact Lagrangian embedding N ⊂ E such that π|N ∼ = f .
Remark 9.1 Of course, if there were several critical points of, say index 1, denoted
x j
1 , then π1 would have one critical value c1 and several critical points lying over c1
corresponding to the x j
1 . The treatment is much the same in this case, as one can see
from our treatment of 4−manifolds earlier.
Each πj will have a certain prescribed regular fiber Mj (which will be a “twist” of M
depending on j, as in §5.3) and it will have one prescribed vanishing sphere (which
will be a “twist” of Lj ⊂ M depending on j). The base point will be b = c1 + 1/4 and
the vanishing paths will be as in figure 11.