prem-mappings, triple self-intersection points of oriented surfaces, and the Rokhlin signature theorem
type: Item
instance of: scholarly article
title: prem-mappings, triple self-intersection points of oriented surfaces, and the Rokhlin signature theorem
zbMATH Open document ID: 0890.57046
DOI: 10.1007/BF02307206
published in: Mathematical Notes
publication date: 1997-07-22T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 57R45
Mathematics Subject Classification ID: 57R40
Mathematics Subject Classification ID: 57Q35
zbMATH DE Number: 1037770
zbMATH Keywords: singularities of differentiable mappings
author: Petr Akhmet'ev
reviewed by: K. H. Mayer
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1037770