La surface de Boy. (Boy's surface)
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.
immersion of the real projective plane into \({\mathbb{R}}^ 3\)
scientific article; zbMATH DE number 4084587
scientific article