Volume 14, issue 4 (2014)

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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Modification rule of monodromies in an $R_2$–move

Kenta Hayano

Algebraic & Geometric Topology 14 (2014) 2181–2222

An R2–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.

wrinkled fibrations, homotopies of stable mappings, surface diagrams of $4$–manifolds
Mathematical Subject Classification 2010
Primary: 57R45
Secondary: 30F99
Received: 16 May 2013
Accepted: 17 December 2013
Published: 28 August 2014
Kenta Hayano
Department of Mathematics
Graduate School of Science
Hokkaido University
Kita 10
Nishi 8, Kita-ku
Sapporo, Hokkaido 060-0810