Topology of Fold Map Germs from mathbb R^3 to mathbb R^5

Abstract

Let f:(mathbb R^3,0)rightarrow (mathbb R^5,0) be an analytic map germ with isolated instability. Its link is a stable map which is obtained by taking the intersection of the image of f with a small enough sphere S^4_epsilon centered at the origin in mathbb R^5. If f is of fold type, we define a labeled tree associated to its link and prove that is a complete topological invariant for it. As an application we obtain the complete topological classification of map germs contained in the mathcal {A}^2-class (x,y,z^2,xz,0).