An –move
is a homotopy of wrinkled fibrations which deforms images of indefinite fold
singularities like the Reidemeister move of type II. Variants of this move are
contained in several important deformations of wrinkled fibrations. In this paper, we
first investigate how monodromies are changed by this move. For a given fibration
and its vanishing cycles, we then give an algorithm to obtain vanishing cycles in a
single reference fiber of a fibration obtained by flip and slip, which is a sequence of
homotopies increasing fiber genera. As an application of this algorithm, we give
several examples of diagrams which were introduced by Williams to describe smooth
–manifolds
by a finite sequence of simple closed curves in a closed surface.
Keywords
wrinkled fibrations, homotopies of stable mappings, surface
diagrams of $4$–manifolds
