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

Abstract

Let $$f:(\mathbb R^3,0)\rightarrow (\mathbb R^5,0)$$ f : ( R 3 , 0 ) → ( 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 $$S^4_\epsilon $$ S ϵ 4 centered at the origin in $$\mathbb R^5$$ 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 $$\mathcal {A}^2$$ A 2 -class $$(x,y,z^2,xz,0)$$ ( x , y , z 2 , x z , 0 ) .