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

Complexified Morse functions 17
In our case, one has the same splitting
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N∂W(S) ∼ = τ 1 S ⊕ T∗ S ⊕ TS ω /TS,
but, since S is Legendrian, we have TS ω /TS = 0. On the other hand S ∼ = S k × S n−k−1
is not a sphere, so
τ 1 S ⊕ T∗ S ∼ = τ 1 S ⊕ T∗ S k × T ∗ S n−k−1
is usually not trivial. (Here T ∗ S k × T ∗ S n−k−1 −→ S k × S n−k−1 is just the Cartesian
product of the total spaces.) There is, however, a canonical isomorphism
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τ 1 S ⊕ T ∗ S ∼ = T ∗ (S k × D n−k )| S k ×∂D n−k.
So, for us, we do not need to choose any framing data; we only need to choose the
identification S ∼ = S k × S n−k . See §2.1 below for how this identification is chosen in
some special situations.
Note that we have only extended the Weinstein construction to a very particular Morse-
Bott situation, namely the case where the critical manifold C is a sphere, and the
normal bundle of C has a certain form. In general, a Morse-Bott function f : X −→ R
can have an arbitrary connected manifold C as critical manifold, and the normal bundle
of C in X , say E −→ C, can be arbitrary. In that situation the Morse-Bott handle
would be modeled on the bundle D(E+) × D(E−) −→ C, with fiber D k × D n−k , where
E ∼ = E+ ⊕E− is the splitting of E into positive and negative eigenspaces of the Hessian
of f at C. It might be interesting to extend the Weinstein construction to the general
Morse-Bott case.
2.1 How this construction is applied
When we construct the regular fiber M in various cases (see §3) we will repeatedly
apply the above handle-attachment construction in the following set up. We take
W = D(T ∗ L),
i.e. the disk bundle cotangent bundle of some manifold L = L n , with respect to some
metric on T ∗ L. Then we take an embedded sphere
S n−k−1 ⊂ L
with a chosen parameterization of tubular neighborhood of S n−k−1 ⊂ L,
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φ: S n−k−1 × D k+1 −→ L