The Reeb graph of a map germ from \(\mathbb {R}^3\) to \(\mathbb {R}^2\) with non isolated zeros
This paper is part of a research programme devoted to the topological classification of finitely determined map germs $f:(\mathbb R^n,0)\rightarrow (\mathbb R^p,0)$ by studying the topological type of the link, carried out mainly by the third author and his collaborators. However, this is the first reference where the case of non-isolated zeros is treated (i.e. $f^{-1}(0)\neq\{0\}$). \par Following Fukuda's cone structure theorem, given a map germ $f:(\mathbb R^3,0)\rightarrow (\mathbb R^2,0)$, take a representative $f:U\subset\mathbb R^3\rightarrow \mathbb R^2$ and the intersection of the image with a small enough sphere $S^1_{\delta}$ centered at the origin of $\mathbb R^2$. When $f^{-1}(0)=\{0\}$, the link is a stable map $\gamma_f:S^2\rightarrow S^1$ and $f$ is topologically equivalent to the cone of $\gamma_f$, so two map germs are topologically equivalent if their links are topologically equivalent. In [Proc. Edinb. Math. Soc., II. Ser. 60, No. 2, 319--348 (2017; Zbl 1376.58016)] the authors gave a complete classification for this case. \par When $f^{-1}(0)\neq\{0\}$ the situation is more complicated since the link is a stable map $\gamma_f:N\rightarrow S^1$ where $N$ is diffeomorphic to $S^2$ minus $2L$ disks. The authors define extended Reeb graphs and show that they are a complete topological invariant for stable maps $\gamma:N\rightarrow S^1$. Finally they use their results to give a topological description of some map germs with Boardman symbol $\Sigma^{2,1}$.
scientific article; zbMATH DE number 6932352
scientific article