Stable Bi-Maps on Surfaces and Their Graphs

ABSTRACT

In this paper we study stable bi-maps $F=(f_1,f_2):M\to \mathrm{ℝ}\times {\mathrm{ℝ}}^2$ from a global viewpoint, where M is a smooth closed orientable surface and $f_1:M\to \mathrm{ℝ},f_2:M\to {\mathrm{ℝ}}^2$ are stable maps. We associate a graph to F, so-called ℛℳ-graph and study its properties. The ℛℳ-graph captures more information about the topological structure of M than other graphs that appear in literature. Moreover, some graph realization theorems are obtained.

Keywords:
Stable maps; ℛℳ-graphs; closed surfaces