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

Complexified Morse functions 61
Figure 10: In the case dim N = 2, from left to right: Dr(T ∗ S 1 ) ∪ D 2 ⋍ H2 ∼ = a 2-handle.
where L j
4 = φL j(T).
Now the definition of T (see §2.2.2) shows that T \ Dr(ν
2
∗K−) is
the graph of a 1-form
Meanwhile
α4 : S 3 \ K− −→ T ∗ (S 3 \ K−).
T ∩ Dr(ν ∗ K−) = D[r0,r](ν ∗ K−)
for some 0 < r0 < r. Define H j
2 ⊂ Dr(T ∗ Sj) ∪ ∆ j
2 by
H j j
2 = ∆2∪{(p, v) ∈ T∗ (S 3 \K−) : p ∈ S 3 \K−, v = sα4(p), for some s ∈ [0, 1]}∪Dr0 (ν∗K j
− ).
Set
H j
2 = φ L j
2
(H j
2 ).
Then we claim that H j
2 is homeomorphic to a 2-handle D2 × D 2 , where ∂D 2 × D 2
corresponds to Dr0 (ν∗ K j
− ) and D2 × ∂D 2 corresponds to
φ j
L (Γ(α4)) ∪ Sr0
2
(ν∗K j
− ).
(See figure 10 for the picture corresponding to the case dim N = 2, dim M = 2,
j
) ∪ ∆2 is homotopy equivalent to
(by a retraction). (We omit the proofs of these claims.)
dim ∆ = 2.) Furthermore we claim that Dr(T ∗L j
2
H j
2
Note the restriction
φ j
L |
2
Dr0 (ν∗K j
− ) : Dr0 (ν∗K j
− ) −→ L0,
still makes sense on the retract H j
2 . And, as we noted before, this restriction is equal
to the framing
φj : S 1 × D 2 r0
−→ L0.
Now, using [DR(T ∗L0) ∪ ∆0] ⋍ ∆0 and (Dr(T ∗L j j
2 ) ∪ ∆2 [DR(T ∗ L0) ∪ ∆0] ∪ [∪j(Dr(T ∗ L j j
2 ) ∪ ∆2 )]
j
⋍ H2 , we see that