Topological classification of corank 1 map germs from $$\mathbb {R}^{3}$$ R 3 to $$\mathbb {R}^{3}$$ R 3

Abstract

The link of a real analytic map germ \(f: (\mathbb {R}^{3}, 0) \rightarrow (\mathbb {R}^{3}, 0)\) is obtained by taking the intersection of the image with a small enough sphere \(S^2_\epsilon \) centered at the origin in \(\mathbb {R}^3\). If \(f\) is finitely determined, this link becomes a stable map from \(S^2\) to \(S^2\). In a previous work, we defined the Gauss paragraph which contains all the topological information of the link when the singular set \(S(\gamma )\) is connected. Now, starting from this point, we give a classification of some finitely determined weighted homogeneous map germs with two-jet equivalent to \((x,y,xz)\). In particular, we classify all 2-ruled map germs from \(\mathbb {R}^3\) to \(\mathbb {R}^3\).