La surface de Boy. (Boy's surface)
type: Item
instance of: scholarly article
title: La surface de Boy. (Boy's surface)
zbMATH Open document ID: 0663.57018
DOI: 10.1016/0001-8708(86)90080-0
author: François Apéry
published in: Advances in Mathematics
publication date: 1986-01-01T00:00:00Z
review text: Boy's surface (1901) is an immersion of the real projective plane, denoted \(P({\mathbb{R}}^ 3)\), into \({\mathbb{R}}^ 3\) having a three-fold axis of symmetry cutting the surface in a triple point and one other point (the ``pole''). This paper, which is very detailed, explicit and well- illustrated, answers a number of outstanding questions about this surface. The inspiration for the methods is a paper of \textit{B. Morin} [C. R. Acad. Sci. Paris Sér. A 287, 879-882 (1978; Zbl 0406.53001)]. Some of the results of the present paper are as follows: (1) If the image of a \(C^ 1\) immersion of \(P({\mathbb{R}}^ 3)\) into \({\mathbb{R}}^ 3\) is an algebraic surface then the degree is at least 6 (conjectured for his surface by Boy); (2) if a \(C^ 1\) immersion \(P({\mathbb{R}}^ 3)into\) \(P({\mathbb{R}}^ 3)\) is realized by \(S^ 2\to {\mathbb{R}}^ 3\) defined by three homogeneous polynomials of the same even degree d then \(d\geq 4\) (Hopf conjectured \(d\geq 6)\); (3) Boy's surface is not the projection to \({\mathbb{R}}^ 3\) of an embedding of the projective plane into \({\mathbb{R}}^ 4\); (4) Boy's surface is a real algebraic surface of degree 6 (explicitly written down), generated by a family of ellipses passing through the pole; (5) this surface lies in a continuous 1-parameter family of degree 6 surfces which also contains Steiner's Roman surface; (6) there is an explicit parametrization of Boy's surface by three homogeneous quartic polynomials defined on \(S^ 2\) (refuting Hopf's conjecture; the immersion condition is explicitly checked here). There is also some discussion of general immersions of the real projective plane into \({\mathbb{R}}^ 3\), regular homotopy classes, combinatorial models, etc.
Mathematics Subject Classification ID: 57R42
Mathematics Subject Classification ID: 14Pxx
zbMATH DE Number: 4084587
zbMATH Keywords: immersion of the real projective plane into \({\mathbb{R}}^ 3\)
zbMATH Keywords: algebraic surface
zbMATH Keywords: degree
zbMATH Keywords: Boy's surface
zbMATH Keywords: Steiner's Roman surface
zbMATH Keywords: parametrization of Boy's surface
zbMATH Keywords: regular homotopy classes
MaRDI profile type: Publication
full work available at URL: 0001-8708(86)90080-0
OpenAlex ID: W1993211540
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description: scientific article; zbMATH DE number 4084587