Eliashberg's h-principle and generic maps of surfaces with prescribed singular locus

Abstract

We extend Y. Eliashberg's h-principle to smooth maps of surfaces which are allowed to have cusp singularities, as well as folds. More precisely, we prove a necessary and sufficient condition for a given map of surfaces to be homotopic to one with given loci of folds and cusps. Then we use these results 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.