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

Complexified Morse functions 5
indices 0, n, n + 1, 2n + 1, focusing on the case dim N = 3. In general, for any fixed
number of Morse indices for f , all three steps in the proof are essentially the same as
the dimension of N varies. (See §3.2, for example, to see why this is so.) That is why
we look at the slightly more concrete cases where dim N = 4 and dim N = 3. (We
focus on the case dim N = 4 rather than dim N = 2 because the latter case is actually
a little confusing from the general point of view, because of its very low dimension.)
Remark 1.4 As it happens, closed 4-manifolds admitting handle decompositions
without 1- and 3-handles form a fairly rich class of examples. This class includes CP 2 ,
CP 2 , and elliptic surfaces (assuming simply-connected with sections, see [GS99]), and
it is closed under connected sum (so blow up in particular). It is not known if it includes
all simply-connected closed 4-manifolds, or even all hypersurfaces in CP 3 ; this has
been a long standing question. (Recently, however, a famously conjectured counterexample,
the Dolgachef surface [HKK89], has been proved by Akbulut [A08] to be in
this class as well.) See [HKK89], [K89], [GS99] for more extensive discussions.
1.1.1 Sketch of the construction of (E,π)
Given (M, L1,...,Lm) there is a standard (and fairly easy) construction for producing
a Lefschetz fibration with fiber M and vanishing spheres L1,... , Lm. (See [S08A,
§16e].) We, however, construct (E,π) in a different, more explicit, way in order to
facilitate the construction of N ⊂ E. Actually, most of the difficulty for us is concentrated
in this non-standard construction of (E,π) (see §4-7) after which the contruction
of N ⊂ E is relatively easy (see §8).
Our strategy for constructing (E,π) is as follows. For simplicity of notation assume
that f has just one critical point of each index, say x0, x2, x4 . Our aim is for
π : E −→ D 2 to have exactly three critical values c0 < c2 < c4; these will lie on the
real line and correspond to the critical values of f . We first construct three Lefschetz
fibrations, one for each critical value ci :
πi : Ei −→ Ds(ci), i = 0, 2, 4,
where Ds(ci) is a small disk around ci of radius s > 0. The aim is to construct
(E,π) so that (Ei,πi) is equal to the restriction of π to a small disk around ci , namely
(E|Ds(ci),π|Ds(ci)) = (Ei,πi). Each πi will have a unique critical point, say xi , with just
one vanishing sphere corresponding to xi . In this case there is a standard construction
(namely [S03A, lemma 1.10]) which we use to produce each πi (see §5). The regular