Stable maps from surfaces to the plane with prescribed branching data

Abstract

We consider the problem of constructing stable maps from surfaces to the plane with branch set a given set of curves immersed (except possibly with cusps) in the plane. Various constructions are used (1) piecing together regions immersed in the plane (2) modifying an existing stable map by a sequence of codimension one transitions (swallowtails etc) or by surgeries. In (1) the way the regions are pieced together is described by a bipartite graph (an edge C* corresponds to a branch curve C with the vertices of C* corresponding to the two regions containing C). We show that any bipartite graph may be realized by a stable map and we consider the question of realizing graphs by fold maps (i.e. maps without cusps). For example, using Arnol'd's classification of immersed curves, we list all branch sets with at most two branch curves and four double points realizable by planar fold maps of the torus.

Teorema de Quine em Teoria de Singularidades

“…A Figura 3 ilustra uma aplicação estável f da esfera no plano, obtida de uma aplicação g, através de uma transição rabo de andorinha (S), e de uma transição bicos no sentido Um invariante da classe de homotopia de aplicações entre superfícies fechadas e orientadasé o grau da aplicação. Uma forma de construir aplicações estáveis em diferentes classes de homotopiaé através dos dois tipos de somas de aplicações, definidas em [2], chamadas de cirurgia horizontal e cirurgia vertical, ilustradas na Figura 4 itens (a) e (b), respectivamente. Em ambas as cirurgias, remove-se um par de discos disjuntos na superfície de domínio, os quais são substituídos por um tubo limitado (conectando os bordos da superfície) ao longo do qual as aplicações envolvidas são estendidas estavelmente.…”

A Survey on Polygonal Portraits of Manifolds

“…For these maps from surfaces, important tools have been developed by S. Blank [Bl] (as exposed in [L2]) and Francis-Troyer [FT], etc. Recently, HMF2,HMF3] introduced an 434 M. Kobayashi approach using a graph to represent a stable map of a surface, developed surgery techniques to calculate invariants, and obtained some classification results of fold maps. In contrast, our approach here are intended to treat manifolds which are not necessarily surfaces, and much more stress is placed on critical loci of uncertain manifolds.…”

Invariants of Stable Maps from the 3-Sphere to the Euclidean 3-Space

“…This local study gives a step towards the global study of stable maps from compact, oriented 3manifolds to the space R 3 . Mendes de Jesus, Shina and Romero-Fuster [8] introduced weighted graphs associated to stable maps, as global topological invariants, in parallel with the study of maps between surfaces done by Hacon, Mendes de Jesus and Romero-Fuster in [5,6]. The aforementioned authors studied the global point of view of stable maps from closed surfaces to the plane, and provided techniques for constructing maps, as well as introducing graphs with weights on its vertices associated to stable maps between surfaces, as global invariants.…”