Aplicações estáveis entre superfícies fechadas

Abstract

This dissertation’s main theme is the graphs associated with stable maps between closed surfaces, orientable and non-orientable, as an invariant regarding topological equivalence, intending to determine whenever two stable maps with the same image are not topologically equivalent and codify informations about the map’s domain. The process of associating a graph to a stable map is done over the regular regions and singular curves of the map, respectively assigning a vertex and an edge to each of these; the inverse process is the realization of graphs by stable maps, which consists in constructing a map from a given graph. In addition, the tools used in the classification of the invariants are the transitions, which are homotopies between maps in distinct stability neighbourhoods; and the surgeries of maps and graphs resulting from manifold surgeries — a technique of obtaining new manifolds from a given manifold. The effects of a surgery over a manifold induce effects on the regular and singular sets which, in turn, induces changes to the associated graph; in this way, it is possible to obtain a stable map from other stable maps, which are simpler to study.