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

Complexified Morse functions 47
ψ|D 1/2 is just the inclusion D 1/2 −→ D 2 ;
ψ| D2 maps D
3/4
2 3/4 diffeomorphically onto D2 , by a radial map;
ψ| D2 [3/4,1]
radially collapses D 2 [3/4,1] onto ∂D2 . Here, D 2 [a,b] = {x ∈ D2 : |x| ∈ [a, b]}.
Then, the pull back ψ ∗ πi is flat near the boundary of D(ci), and we replace πi by ψ ∗ πi
but keep the same notation. Note that, since ψ|D 1/2 is just the inclusion, Ei|D 1/2(ci) has
not been modified at all. So, in particular, the transport map τ i θ along γ(t) = ci + se iθ ,
for 0 < s ≤ 1/2 is the same as before.
Concretely, when we do the boundary connect sums of the base manifolds D(ci),
the segments [ci − 1, ci + 1] ⊂ D(ci) are glued together to form an interval I, and we
think of S as embedded in C so that I ⊂ R. In fact, parts of [ci −1, ci +1] are chopped
off before we glue, so I is shorter than [c0 −1, c4 +1] = [−1, 5]. Nevertheless we will
refer to intervals such as [c0 − 1/4, c2 + 1/4] with the understanding that this means
the corresponding sub-interval of I, i.e. [c0 − 1/4, c2 + 1]#[c2 − 1, c2 + 1/4]. Also
note that, for example, the transport map along [c0 − 1/4, c2 + 1/4],
is equal to the composite
τ [c0−1/4,c2+1/4] : π −1 (c0 − 1/4) −→ π −1 (c2 + 1/4)
τ 2 [c2−1,c2+1/4] ◦ Ψ02 ◦ τ 0 [c0−1/4,c0+1] ,
where τ 2 [c2−1,c2+1/4] and τ 0 [c0−1/4,c0+1] are the transport maps for π0 and π2 respectively.
7 Computing the vanishing spheres of π : E −→ D 2
Fix the base point b ∈ D 2 (c2) to be
(19)
b = c2 − 1/4.
Then π −1 (b) = π −1
2 (c2 − 1/4) and so, by lemma 6.1, there is a canonical isomorphism
ν 2 −1/4 : π−1 (b) −→ M.
In this section we show that for suitable vanishing paths γ0,γ2,γ4, the vanishing
spheres
j
Vγ1 , Vγ2 , Vγ4 ⊂ π−1 (b), j = 1,... , k
correspond precisely to L0, L j
2 , L4, under the map ν2 −1/4 . Here, γ2 will give rise to k
disjoint vanishing spheres V j
γ2 ⊂ π−1 (b) one for each critical point x j
2 .