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

56 Joe Johns
A direct calculation shows that Φ 2 satisfies
(Φ 2 )(N j
2 ∩ (k2 ) −1 ([4(r/2) 2 , 4r 2 ]) = D [r/2,r](ν ∗ K+) × [c2 − 1/2, c2 + 1/2].
Indeed, if z = (x1, x2, x3, x4) + i(0, 0, 0, 0) ∈ R 4 ⊂ C 4 then
Let
If s < 0, then
Φ2(z) = Φ((x1, x2, 0, 0) + (0, 0, ix3, ix4)).
s = q2((x1, x2, 0, 0) + (0, 0, ix3, ix4)) ∈ [−1/4, 1/4].
Φ2(z) = (σ −π/2 ◦ ρs ◦ m(i))((x1, x2, 0, 0) + (0, 0, ix3, ix4))
= σ −π/2 ◦ ρs((0, 0, −x3, −x4) + (ix1, ix2, 0, 0))
= σ −π/2((0, u), (v, 0)) for some u, v ∈ R 2
= ((u ′ , 0), (0, v ′ ) for some u ′ , v ′ ∈ R 2
Similarly, if s ≥ 0 then Φ2(z) = ρs(α2(z)) and we get the same conclusion.
We recalled earlier in this section that φ (L j
2 )′ identifies D [r/2,r](ν ∗ K+) with
This shows that in the quotient space E2
is identified with
φj(S 1 × D 2 (r/2,r] ) ⊂ L0.
N j
2 ∩ (k2 ) −1 ((4(r/2) 2 , 4r 2 ]) ⊂ N j
2
φj(S 1 × D 2 (r/2,r] ) × [c2 − 1/2, c2 + 1/2] ⊂ N triv
2 .
Therefore the union N2 is smooth. In fact, N2 is diffeomorphic to f −1 ([1, 3]) ⊂ N ,
with π|N2 equivalent to f | f −1 ([1,3]). This essentially follows from the fact that we
are using the correct framing φj . (For more details the reader can consult Milnor,
[M65, pages 27-32] and especially Theorem 3.13. See remark 8.2 below where we
point out some small differences between what we have done here and Milnor’s set up.)
Next recall that E2 is fiber connect-summed to E4 using the natural identifications
and
For any s > 0 we have
ρ 2 s : π −1
2 (s) −→ M2, s > 0
ρ 4 −s : π −1
4 (−s) −→ M2, −s < 0.
ρ 4 −s (N4 ∩ π −1
4 (−s)) = L′ 4