In this work, we study families of singular surfaces in $\mathbb{ C}^3$ parametrized by $\mathcal{ A}$-finitely determined map germs. We consider the topological triviality and Whitney equisingularity of an unfolding $F$ of a finitely determined map germ $f:(\mathbb{ C}^2,0)\rightarrow(\mathbb{ C}^3,0)$. We investigate the following conjecture: topological triviality implies Whitney equisingularity of the unfolding $F$? We provide a complete answer to this conjecture, given counterexamples showing how the conjecture can be false.
Categories: math.CV