2. Universidad Cardenal Herrera-CEU

In this paper we will study the topology of the singular set of stable Gauss maps from closed orientable surfaces immersed in the 3-space from a global point of view and we will obtain a relationship between the Euler Characteristic of the regular components and the signs of the cusps.

In this paper, we will consider the problem of constructing stable maps between two closed orientable surfaces M and N with a given branch set of curves immersed on N. We will study, from a global point of view, the behavior of its families in different isotopies classes on the space of smooth maps. The main goal is to obtain different relationships between invariants. We will provide a new proof of Quine’s Theorem.

The aim of this paper is to introduce a computational tool that checks theoretical conditions in order to determine whether a weighted graph, as a topological invariant of stable maps, can be associated to stable maps without cusps (i.e. fold maps) from closed surfaces to the projective plane.