Gauss words and the topology of map germs from ${\mathbb{R}}^3$ to ${\mathbb{R}}^3$

Abstract

The link of a real analytic map germ $f:({\mathbb{R}}^3,0)\to ({\mathbb{R}}^3,0)$ is obtained by taking the intersection of the image with a small enough sphere $S_{ϵ}^2$ centered at the origin in ${\mathbb{R}}^3$. If $f$ is finitely determined, then the link is a stable map $\gamma$ from $S^2$ to $S^2$. We define Gauss words which contains all the topological information of the link in the case that the singular set $S(\gamma )$ is connected and we prove that in this case they provide us with a complete topological invariant.