Topological triviality of families of map germs from $\mathbb R^3$ to $\mathbb R^3$

Abstract

We show that a one-parameter unfolding $ F:(\mathbb {R}^3 \times \mathbb {R}, 0) \rightarrow (\mathbb {R}^3 \times \mathbb {R}, 0) $ of a finitely determined map germ $f$, with $S(f)$ regular, is topologically trivial if it is excellent in the sense of Gaffney, and the family of the double point curves and cuspidal edges $D(f_t) \cup C(f_t)$ is topologically trivial.