2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/75600In this paper we generalize the notion of regular homotopy of immersions of a closed connected n-manifold into R^{2n-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 R^4 whose projections into R^3 are locally generic.14 pages, 5 figuresGeometric TopologyAlgebraic Topology57R45 (Primary); 58K30; 57R42 (Secondary)text