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

Complexified Morse functions 25
The construction of M in each case is identical as the dimension of N varies. For
this reason we will keep things slightly more concrete in the first two cases above by
focusing on the cases when dim N = 4 and dim N = 3 respectively. See §3.2 for
how things work in an arbitrary dimension in case (1).
3.1 Constructing M and the vanishing spheres in case (1), dim N = 4
Suppose N is a closed 4-manifold and
f : N −→ R
has critical points x0, x j
2 , x4, j = 1,... , k, where the subscript indicates the Morse
index. Let g be a Riemannian metric such that (f, g) is Morse-Smale.
First, (f, g) induces a handle decomposition of N , which determines k framed knots
Kj ⊂ S 3 , which are the attaching spheres of the 2-handles, together with a parameterization
of a tubular neighborhood of each Kj
(14)
φj : S 1 × D 2 −→ S 3 ,
determined by the framing for the 2-handle up to isotopy. Set
L0 = S 3 .
We start with the disk bundle D(T ∗ L0). To construct M, we attach a Morse-Bott handle
Hj = D(T ∗ (S 1 × D 2 ))
to D(T ∗ L0) for each j, where the gluing region is a neighborhood of S(ν ∗ Kj) in S(T ∗ L0),
and the framing φj determines the gluing map. This produces a Weinstein manifold
M = D(T ∗ L0) ∪ (∪jHj),
as we explained in §2.1. We have for each j an exact Lagrangian 3-sphere
L j
2 ⊂ M
which is the union of D(ν∗Kj) and S1 × D2 ⊂ Hj. (Each L j
2 corresponds to what we
called Z in §2.2.) See figure 3 in §2.3 for a schematic picture of the region near each
attaching region.
Now define L4 as the Lagrangian surgery of L0 and all the L j
2 ’s. In §2.2 we explain
how to define the Lagrangian surgery of L ⊂ D(T∗L) ∪ H and a single Lagrangian
’s simultaneously, where
sphere Z ⊂ D(T ∗ L) ∪ H. We use that definition for all L j
2