prem-mappings, triple self-intersection points of oriented surfaces, and the Rokhlin signature theorem
Let \(f: M\to \mathbb{R}^3\) denote a smooth map in general position of the oriented surface \(M\) into the 3-dimensional Euclidean space. \(f\) is called a prem-mapping (an abbreviation for projected embedding) if there exists a smooth embedding \(\overline{f}: M\to \mathbb{R}^4\) such that \(f=\pi\cdot \overline{f}\), where \(\pi: \mathbb{R}^4\to \mathbb{R}^3\) is the standard projection. For a given orientation of the curves formed by the double points of the surface \(f(M)\), the triple points are classified into two classes \(A\) and \(B\). Using this classification the author proves a criterion for \(f\) to be a prem-mapping. The main result is that for any smooth \(f: S^2\to \mathbb{R}^3\) in general position and any choice of the orientations of the curves of double points, the number of triple points of type \(B\) is even. The proof uses the Rokhlin signature theorem.
singularities of differentiable mappings
scientific article; zbMATH DE number 1037770
scientific article