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

62 Joe Johns
⋍ ∆0 ∪ [∪jH j
2 ].
Here, ∆0 ∪ [∪jH j
2 ] is a partial handle decomposition (of N ) given by D4 with k 2handles
attached using the framings φj. Indeed, recall that H j
2 ∼ = D2 × D2 in such a
way that
∂D 2 × D 2 ∼ = Dr0 (ν∗K j
− ),
and φ j
L |
2
Dr0 (ν∗K j
− ) = φj. Next, since D2 × ∂D2 corresponds to
Γ(α j
2 ) ∪ Sr0 (ν∗ K j
− ),
it follows that the boundary of ∆0 ∪ H j
2 is equal to L4, which is the sphere where ∆4
is attached. To finish, we write
E ⋍ [DR(T ∗ L0) ∪ ∆0] ∪ [∪j(Dr(T ∗ L j
2
⋍ ∆0 ∪ [∪jH j
2 ] ∪ ∆4 ∼ = N.
j
) ∪ ∆ )] ∪ ∆4
For the last diffeomorphism, note that the left hand side is a handle decomposition
using the attaching maps φj, hence it is a handle decomposition of N .
Remark 8.5 Theorem 8.1 can be refined in a couple of ways. First if ξ is any gradientlike
vector field for f on N we can construct a corresponding vectorfield on N , which
is the symplectic lift of ∂t (up to a scaling function). To see the correspondence one
compares the two vector fields on each handle, just like we did for π|eN and f . Second,
there is an anti-symplectic involution on E, say ιE , for which N is the fixed point set
(as one would be the case if E = T∗N and N = N ).
Here is a sketch of the construction of ιE . (Instead of anti-symplectic involution,
denote
we will say conjugation map.) On each local fibration E loc
0
2
, Eloc
2
, Eloc
4 let ι E loc
i
the standard conjugation. Also, let ι D 2 be the usual conjugation map on D 2 ⊂ C. First
we define a conjugation map ιE0 on E0 as follows. Think of M as the plumbing (see
§2.3) of D(T ∗ L0) and D(T ∗ L j
2 ), where L0 = L j
2 = S3 . Then we define a conjugation
map ιM on M as follows. The guiding idea is that ιM is to have fixed point set equal to
L0. On D(T ∗ L0) it is defined to be (x, y) ↦→ (x, −y); and on each D(T ∗ L j
2
to be
((x1, x2), (y1, y2)) ↦→ ((x1, −x2), (−y1, y2))
) it is defined
so that the fixed point set in D(T ∗ L j
2 ) is D(ν∗ K−), which is identified with part of L0.
Then, when we glue M × D 2 and E loc
0 together to get E0, the two conjugation maps
ιM × ι D 2 and ι E loc
0
will patch together to give a conjugation map on ιE0 on E0.