Finiteness theorem for topological contact equivalence of map germs
Let \(P^{k}(n,m)\) be the set of all real polynomial map germs \(f:(\mathbb{R} ^{n},0)\rightarrow (\mathbb{R}^{m},0)\) such that \(\deg f\leq k.\) A weaker than known topological equivalence \(f\sim _{\text{top}}g\) in \(P^{k}(n,m)\) is the following topological contact equivalence \(f\sim _{\text{con}}g\): there exist germs of homeomorphisms \(h:(\mathbb{R}^{n},0)\rightarrow (\mathbb{R}^{n},0)\) and \(H:(\mathbb{R}^{n}\times \mathbb{R}^{m},0)\rightarrow (\mathbb{R}^{n}\times \mathbb{R}^{m},0)\) such that: {\parindent5mm \begin{itemize}\item[1.] \(H(\mathbb{R}^{n}\times \{0\})=\mathbb{R}^{n}\times \{0\}\), \item[2.] \((\text{Id},g)\circ h=H\circ (\text{Id},f):(\mathbb{R} ^{n},0)\rightarrow (\mathbb{R}^{n}\times \mathbb{R}^{m},0)\), \item[3.] \(h\circ \pi =\pi \circ H:(\mathbb{R}^{n}\times \mathbb{R} ^{m},0)\rightarrow (\mathbb{R}^{n},0),\) \end{itemize}} where \(\text{Id}:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}\) is the identity mapping and \(\pi :\mathbb{R}^{n}\times \mathbb{R} ^{m}\rightarrow \mathbb{R}^{n}\) is the canonical projection. The authors prove that the set of equivalence classes of topological contact equivalence in \(P^{k}(n,2),\) \((m=2)\) is finite (this is not true for topological equivalence).
scientific article; zbMATH DE number 5606278