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

Complexified Morse functions 67
Moreover, under the canonical isomorphism π −1
1 (c1 − 1) ∼ = T L1
−π/2 (M), the boundary
component of N1 which lies in π −1
1 (c1 − s) corresponds precisely to L ′ 0 . Indeed,
that boundary component of N1 is by definition equal to the union of
and
[T \ φ α (S 1 × D 1 r/2 )] × {c1 − 1/2}
N loc
1 ∩ f −1 (c1 − 1/2) ∼ = S 0 × D 2 .
Under the isomorphism π −1
1 (c1 − 1) ∼ = T L1
−π/2 (M), this union corresponds to the union
of [T \ φα (S1 × D1 r/2 )] × {c1 − 1/2} and Dr(ν ∗K+) ⊂ Dr(T ∗L1). This is precisely
L ′ 0 , by definition. This implies that N1 and N0 glue up smoothly (with over-lapping
boundary) to yield a manifold diffeomorphic to f −1 ([0, 1 + s]) for some small s > 0.
We define N loc
2
way. Then
is defined to be
⊂ E1 loc ⊂ E1 and f2 : N loc
2
−→ [c2 − 1/2, c2 + 1/2] in a similar
N triv
2 ⊂ [M2 \ φ L ′ 2 (D r/2(T ∗ S 2 ))] × [c2 − 1/2, c2 + 1/2]
N triv
2 = L′ 3 \ φ L ′ 2 (D r/2(ν ∗ K+)) × [c2 − 1/2, c2 + 1/2].
As before N triv
2 and N loc
2 glue together to form a manifold diffeomorphic to f −1 ([c2 −
s, c2 + s]). The boundary component of
N2 = N loc
2
∪ Ntriv
2
which lies in π −1
2 (c2 + 1/2) corresponds to L ′ 3 . (This is immediate in this case.) This
means N2 and N3 glue up smoothly to form a manifold diffeomorphic to f −1 ([2−s, 3]),
for some small s > 0. And, as before, the boundary component of
N2 = N loc
2
∪ Ntriv
2
which lies in π −1
2 (c2 − 1/2) corresponds to T under the canonical isomorphism
π −1
2 (c2 − 1/2) ∼ = M. Therefore N0 ∪ N1 and N2 ∪ N3 glue up smoothly along T
to form a manifold
N = N0 ∪ N1 ∪ N2 ∪ N3
diffeomorphic to N . To see that π is equivalent to f we just compare π|Nj to f on the
corresponding handle in the Milnor type decomposition of N .