Introdução

In this work, we study the problem of separating the range of a given map, by its image; we also study the topology of certain stable maps. By working with immersions with normal crossings, Min --+ we firstly obtain interesting results which guarantee the separation of N by f(M), under certain conditions. Then, by using the Stein Factorization of a given stable map and some results on the classification of 1-connected 4-manifolds, we obtain interesting information on the topologies of the singular set and of the domain of such a map, particularly in the special generic case. Finally, by working with stable maps whose only singularities are fold points and whose domain has low dimension, we simplify its singular set, by cancelling components by local deformations, in order to get topological information of the source.