Eliashberg's \(h\)-principle and generic maps of surfaces with prescribed singular locus (Q1987226)

The author extends \textit{Ya. M. Ehliashberg}'s \(h\)-principle [Math. USSR, Izv. 4, 1119--1134 (1971; Zbl 0226.57012)] to smooth maps of surfaces which are allowed to have cusp singularities, as well as folds. The main result gives a necessary and sufficient condition for a given map of surfaces to be homotopic to one with given loci of folds and cusps (Theorem 1). These results are used to obtain a necessary and sufficient condition for a subset of a surface \(M\) to be realizable as the critical set of some generic smooth map from \(M\) to a given surface \(N\) (Theorem 2).