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

TL;DR

This paper extends Eliashberg's $h$-principle to classify smooth surface maps with prescribed fold and cusp singularities, providing conditions for their existence and homotopy classification.

Contribution

It generalizes the $h$-principle to include cusp singularities in surface maps and characterizes when a subset of a surface can be realized as a critical set.

Findings

Established a necessary and sufficient condition for homotoping maps to have prescribed singular loci.

Provided criteria for subsets of surfaces to be realized as critical sets of generic maps.

Extended the $h$-principle framework to include cusp singularities in surface mappings.

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 $N$.