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

Complexified Morse functions 43
Figure 8: A 2 dimensional version of the regular fiber M2 of π2 at c2 + 1. L ′ 0
the (twisted) red curve; (L j
2 )′ j = 1, 2 correspond to the two green curves; L ′ 4
the blue curve.
satisfies
ρ 4 −s(Vγ) = (L4) ′ .
5.5 The transport map for πi : Ei −→ D 2
We denote the transport map for πi along γ(t) = se iθt , t ∈ [0, 1] by
τ i θ : π−1 i (s) −→ πi(se iθ ).
corresponds to
corresponds to
This is the same notation as for the local models Ei loc , but since the transport map for
Ei is just the transport map for Ei loc extended by the identity map, this should not cause
confusion.
6 Construction of the Lefschetz fibration π : E −→ D 2
We now have three Lefschetz fibrations π0,π2,π4 over the disk D 2 . It would be
convenient to have notation which distinguishes between the different critical points
which at the moment are all labeled as 0 ∈ D 2 . Thus, let c0 = 0, c2 = 2, c4 = 4 and