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

60 Joe Johns
As we explained in remark 1.3 in the introduction, we expect to prove in a future
paper that E is conformally exact symplectomorphic to D(T ∗ N). For now, we give a
detailed sketch of the proof of the following proposition, which states that E is at least
homotopy equivalent to N .
Proposition 8.4 E is homotopy equivalent to N .
Sketch of proof First it is well-known that E is homotopy equivalent to the result of
attaching one 4-disk to M at each vanishing sphere. (In fact, E is diffeomorphic to the
result of attaching one 4-handle to M × D2 at each vanishing sphere. See [GS99] for
the corresponding statement when dim E = 4.) Let us denote the disks ∆0,∆ j
2 ,∆4,
where ∂∆0 is attached to L0 and so on.
Recall from §3.1 that there are exact Weinstein embeddings
such that
(21)
φ j : Dr(T L2 ∗ Sj) −→ M, Sj = S 3 , j = 1,... , k
φ j|Dr(ν
L2 ∗K−) = φj : S 1 × D 2 r −→ L0,
where we use the canonical identification Dr(ν ∗K−) ∼ = S1 × D2 r . Let
M0 = DR(T ∗ L0) ∪ (∪jφ j
L (Dr(T
2
∗ Sj)) ⊂ M.
Then M0 is a retract of M. (In fact the homeomorphism M −→ M0 we mentioned in
§2.3 is a retract.)
Now, writing Dr(T ∗ L j
2 ) for φL j
2
(Dr(T ∗ Sj)), we have
E ⋍ M0 ∪ (∆0 ∪ ∆ j
2 ∪ ∆4)
= [DR(T ∗ L0) ∪ (∪jDr(T ∗ L j
2 ))] ∪ (∆0 ∪ ∆ j
2 ∪ ∆4)
⋍ [DR(T ∗ L0) ∪ ∆0] ∪ [∪j(Dr(T ∗ L j
2
j
) ∪ ∆ )] ∪ ∆4.
Note that DR(T ∗ L0) ∪ ∆0 is homotopy equivalent to ∆0, and we have
Kj ⊂ ∂∆0.
Now we define a certain subset of Dr(T ∗L j
2 )∪∆j 2 which is diffeomorphic to a 2-handle
D2 × D2 . Recall from section 3.1 that L4 is the union of
L0 \ (∪jφj(S 1 × D 2 r/2 )) and ∪j L j
4 ,
2