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

On bifurcations of cusps

“…There are several results concerning relations between the topology of surfaces and the topology of the critical locus of a mapping (see [17], [26], [27]). Singularities of map germs of the plane into the plane were studied in [9], [10], [12], [19], [20], [23].…”

“…Under some natural assumptions there is a finite family of cusp points of f t bifurcating from the origin. There are important results [9,Theorem 3.1], [20,Proposition 7.1] concerning the parity of the number of those points.…”