Regular homotopy classes of locally generic mappings

In this paper we generalize the notion of regular homotopy of immersions of a closed connected n-manifold into R2n−1 to locally generic mappings. The main result is that if n=2 then two mappings with singularities are regularly homotopic if and only if they have the same number of cross-cap (or Whitney-umbrella) singularities. As an application, we get a description of the path-components of the space of those immersions of a surface into R4 whose projections into R3 are locally generic.