Generic smooth maps of surfaces (Q760058)

The author proves geometrically a generalization of Levine's theorem on the elimination of cusps of a smooth mapping \(M\to N\), where M is a compact connected surface and N is an oriented connected surface. [This result appears in \textit{Ya. M. Ehliashberg}, Math. USSR, Izv. 4 (1970), 1119-1134 (1971) Corollary 4.10; translation from Izv. Akad Nauk SSSR, Ser. Mat. 34, 1110-1126 (1970; Zbl 0202.548)]. Further he gives conditions under which a generic map is the composition of an immersion of M into \(N\times {\mathbb{R}}\) followed by projection to N. The author gives an interesting example of immersion of the projective plane \(P^ 2\) into \({\mathbb{R}}^ 3\), whose projection to \({\mathbb{R}}^ 2\) has a connected fold locus with a single cusp.