Topological $${\mathcal{K}}$$ -classification of finitely determined map germs

Abstract

We consider smooth finitely C0-\({\mathcal{K}}\) -determined map germs \({f : (\mathbb{R}^n, 0) \to (\mathbb{R}^p, 0)}\) and we look at the classification under C0-\({\mathcal{K}}\) -equivalence. The main tool is the homotopy type of the link, which is obtained by intersecting the image of f with a small enough sphere centered at the origin. When f−1(0) = {0}, the link is a smooth map between spheres and f is C0-\({\mathcal{K}}\) -equivalent to the cone of its link. When f−1(0) ≠ {0}, we consider a link diagram, which contains some extra information, but again f is C0-\({\mathcal{K}}\) -equivalent to the generalized cone. As a consequence, we deduce some known results due to Nishimura (for n = p) or the first named author (for n < p). We also prove some new results of the same nature.