Immersions of the projective plane with one triple point
type: Item
instance of: scholarly article
title: Immersions of the projective plane with one triple point
zbMATH Open document ID: 1172.53003
author: Sue Goodman
author: Marek Kossowski
published in: Differential Geometry and its Applications
publication date: 2009-08-21T00:00:00Z
review text: In a famous paper published in 1903 [Math. Ann. 576, 151--184 (1903; JFM 34.0537.07)], \textit{W. Boy} described an immersion of the projective two-plane \(\mathbb P^2\) into the three-sphere \(\mathbb S^3\) by a sequence of planar curves which are level-curves of a hight function. \textit{U. Pinkall} [Topology 24, 421--434 (1985; Zbl 0583.57020)] showed that every \(\mathbb C^{\infty}\) immersion of the projective plane in \(\mathbb S^3\) is homotopic through immersions to Boy's immersion or to its mirror image. From the work of \textit{F. Apéry} [Adv. Math. 61, 185--266 (1986; Zbl 0663.57018)], there exists another immersion of the projective plane with self-intersection set equivalent to Boy's but whose image is not homeomorphic to Boy's. In the paper under review, the authors show that any \(\mathbb C^\infty\) generic immersion of the projective plane in \(\mathbb{S}^3\) whose self-intersection set is connected and has a single triple point is ambiently isotopic to one of the two models, the Boy model or the Apéry model, or to a mirror image of such a model. They also show that any generic immersion of the projective plane with one triple point can be obtained by a sequence of toral and spherical surgery operations on these models. Finally, the authors present some simple applications of their result, regarding discrete ambient automorphism groups; image-homology of immersions with one triple point; and almost tight ambient isotopy classes.
reviewed by: Athanase Papadopoulos
Mathematics Subject Classification ID: 53A05
Mathematics Subject Classification ID: 57R42
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 5597822
zbMATH Keywords: immersions
zbMATH Keywords: projective plane
zbMATH Keywords: triple point
zbMATH Keywords: isotropy
MaRDI profile type: Publication
full work available at URL: j.difgeo.2009.01.011
OpenAlex ID: W2060232068
Wikidata QID: Q56094068
cites work: La surface de Boy. (Boy's surface)
cites work: Q3761286
cites work: Triple Points and Surgery of Immersed Surfaces
cites work: Q4394850
cites work: Q4375394
cites work: Q3668507
cites work: Q3413661
cites work: Symmetric models of the real projective plane
cites work: The Euler Characteristic of a Generic Wavefront in a 3-Manifold
cites work: Stable surfaces in Euclidean three space.
cites work: Q4194070
cites work: Regular homotopy classes of immersed surfaces
cites work: Tight surfaces and regular homotopy
cites work: Q4107635
cites work: Q3840310
cites work: Knoten und Vollringe
DOI: 10.1016/J.DIFGEO.2009.01.011
description: scientific article; zbMATH DE number 5597822