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The main purpose of the present work as to study geometrical properties of closed, oriented 4-manifolds as domam n of generic maps into the plane. First, we get the local configurations for the 2-dimensional complex Wf of the Stein Factoriaation of a stable map f: M --> R2 and we study some examples where the maps are not stable. Then, we study the special generic maps where - all the singularities are definite folds, and we also classify the 4- manifolds which arise as the domain of such a map, generalizing the theorems of O. Burlet and G. de Rham for the 3-dimensional case. Finally, we get a relation between smooth families of special generic maps and image-homotopy tin the sense of L. Kauffman) of immersions of compact surfaces with boundary into the plane.