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

Complexified Morse functions 21
of the coordinates (8):
σt : T ∗ S n \ S n −→ T ∗ S n \ S n ,σt(u, v) = (cos tu + sin t v v
, cos t − sin tu).
|v| |v|
Given any function H : T ∗ S 3 \ S 3 −→ R we let φ H t denote the time t Hamiltonian
flow of XH (our convention is ω(·, XH) = dH). It is elementary to check that for any
k ∈ C ∞ (R, R),
(10)
Let
Then it is well-known that φ (1/2)µ2
t
φ k(H)
t (p) = φ H k ′ (H(p))t (p).
µ: T ∗ S 3 \ S 3 −→ R,µ(u, v) = |v|.
is the usual geodesic flow and so (10) implies
φ µ t is equal to the normalized geodesic flow, with the formula given by σt . Now let
h: R −→ R be any smooth function satisfying
(11)
h ′ (0) = 0,
h ′ (t) = 1/2, t ∈ [r/2, r],
h ′′ (t) > 0, t ∈ [0, r/2),
h(−t) = h(t) − t for small |t|
In §5.1 we will make a particular choice for h. Consider the map
defined by
F : Dr(T ∗ S 3 ) \ S 3 −→ Dr(T ∗ S 3 ) \ S 3
F(u, v) = φ h(µ)
π/2 (u, v) = σ h ′ (|v|)π(u, v).
Then F extends continuously over the zero-section because h ′ (0) = 0. To see that
the extension is smooth one applies [S03A, Lemma 1.8]. (This is why we need
h(−t) = h(t) − t for small |t|.) Call the extension
F : Dr(T ∗ S 3 ) −→ Dr(T ∗ S 3 ).
(Here, F plays the role of Φ before.) Now define
Notice that
T = F(Dr(ν ∗ K+)).
F(D [r/2,r](ν ∗ K+)) = D [r/2,r](ν ∗ K−).
(One can see this by the formula for σ π/2 .) It follows that
(12)
T ∩ Dr(ν ∗ K−) = D [r/2,r](ν ∗ K−).