The Reeb Graph of a Map Germ from $$\mathbb {R}^3$$ R 3 to $$\mathbb {R}^2$$ R 2 with Non Isolated Zeros

Abstract

We consider the topological classification of finitely determined map germs $$[f]:(\mathbb {R}^3,0)\rightarrow (\mathbb {R}^2,0)$$ with $$f^{-1}(0)\ne \{0\}$$ . The case $$f^{-1}(0) = \{0\}$$ was treated in another recent paper by the authors. The main tool used to describe the topological type is the link of [f], which is obtained by taking the intersection of its image with a small sphere $$S^1_\delta $$ centered at the origin. The link is a stable map $$\gamma _f:N\rightarrow S^1$$ , where N is diffeomorphic to a sphere $$S^2$$ minus 2L disks. We define a complete topological invariant called the generalized Reeb graph. Finally, we apply our results to give a topological description of some map germs with Boardman symbol $$\Sigma ^{2,1}$$ .