Modification rule of monodromies in an $R_2$–move

Abstract

An $R_2$–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 $4$–manifolds by a finite sequence of simple closed curves in a closed surface.