Whitney's Umbrellas in Stable Perturbations of a Map Germ $({\bf R}^n, 0)\to ({\bf R}^{2n- 1}, 0)$

Abstract

Let $f:({\bf R}^n, 0)\to ({\bf R}^p, 0)$ be a $C^{\infty}$ map-germ. We are interested in whether the number modulo 2 of stable singular points of codimension $n$ that appear near the origin in a generic perturbation of $f$ is a topological invariant. In this paper we concentrate on investigating the problem when $p$ is $2n- 1$, where stable singular points of codimension $n$ are only Whitney's umbrellas, and give a positive answer to the problem.