Modification rule of monodromies in R_2-move

TL;DR

This paper studies how monodromies change under R_2-moves in wrinkled fibrations, providing an algorithm to compute vanishing cycles after flip and slip deformations, aiding in the diagrammatic understanding of smooth 4-manifolds.

Contribution

It introduces an algorithm to determine vanishing cycles after R_2-moves and flip and slip deformations in wrinkled fibrations, linking monodromy changes to mapping class groups.

Findings

Derived an explicit method to track monodromy changes under R_2-moves.

Provided examples of diagrams representing smooth 4-manifolds using simple closed curves.

Connected monodromy modifications to surface diagram representations.

Abstract

An R_2-move is a homotopy of wrinkled fibrations which deforms images of indefinite fold singularities like Reidemeister move of type II. Variants of this move are contained in several important deformations of wrinkled fibrations, flip and slip for example. 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 one reference fiber of a fibration, which is obtained by applying flip and slip to the original fibration, in terms of mapping class groups. As an application of this algorithm, we give several examples of diagrams which were introduced by Williams to describe smooth 4-manifolds by simple closed curves of closed surfaces.