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

Complexified Morse functions 27
3.3 Constructing M and the vanishing spheres in case (2), dim N = 3
In this section we explain how, for a closed 3-manifold N , one can construct the
regular fiber M and vanishing spheres L0, L1, L2, L3 in M. This discussion applies
equally well to self-indexing Morse functions f : N −→ R with four critical values
0, n, n + 1, 2n + 1. See section 3.2 to see how things are much the same from one
dimension to the next.
Let (N, f, g) be a triple consisting of a closed 3-manifold N , and a self-indexing
Morse function f : N −→ R, together with a Morse-Smale metric g on N . Then
(N, f, g) determines a Heegard diagram for N as follows. Let
T = f −1 (3/2).
This is a closed 2-manifold of some genus h. We may assume f has h critical points
, j = 1,... , h. Then there are h circles
of index 1 and 2, x j j
1 , x2 namely
where S(x j
1
αj,βj ⊂ T, j = 1,... g,
αj = S(x j
) is the stable manifold of x j
1
1 ) ∩ T,βj = U(x j
2 ) ∩ T,
and U(x j
1
) is the unstable manifold of x j
1 .
The data of T together with the circles αj,βj is called a Heegard diagram for N ; it
determines the diffeomorphism type of N . In addition, (f, g) determine framings
φ α j : S1 × D 1 −→ T
φ β
j : S1 × D 1 −→ T
for αj , βj respectively. (Of course, in this dimension, there are only two possible
framings for each αj or βj and they give rise to diffeomorphic manifolds. But in higher
dimensions the analogue of these framings are important.)
Then M is defined as follows. Set
L j j
1 , L2 = S2 , j = 1,... , h.
Now consider the disk bundle D(T ∗ T) and consider the disk conormal bundles D(ν ∗ (αj))
and D(ν ∗ (βj)), each being diffeomorphic to S 1 × D 1 . Since (f, g) is Morse-Smale, it
follows that αj and βk are transverse for any j, k. Therefore we may assume that the
boundaries of the disk conormal bundles S(ν ∗ (αj)) and S(ν ∗ (βk)) are disjoint for every