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

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.

arXiv comment29 pages. Minor corrections

References in corpus (1)

• Holonomic approximation and Gromov's h-principle