Maps of manifolds of the same dimension with prescribed Thom-Boardman singularities

TL;DR

This paper extends Eliashberg's h-principle to characterize when a continuous map between same-dimensional manifolds can be homotoped to a generic map with specified Thom-Boardman singularities, using topological invariants.

Contribution

It provides a necessary and sufficient topological condition for realizing prescribed singularities in generic maps between manifolds of the same dimension.

Findings

Characterization of maps with prescribed Thom-Boardman singularities.

Conditions expressed via Stiefel-Whitney classes and cohomology.

Extension of h-principle to arbitrary generic smooth maps.

Abstract

In this paper we extend Y.Eliashberg's $h$-principle to arbitrary generic smooth maps of smooth manifolds. Namely, we prove a necessary and sufficient condition for a continuous map of smooth manifolds of the same dimension to be homotopic to a generic map with a prescribed Thom-Boardman singularity $\Sigma^I$ at each point. In dimension 3 we rephrase these conditions in terms of the Stiefel-Whitney classes and the cohomology classes of the given loci of folds, cusps and swallowtail points.