Generic maps of the projective plane with a single triple point (Q2883238)

We all know that the projective plane \(P\), as a non-orientable closed surface, does not live in \(\mathbb R^3\). In order to imagine \(P\) in \(\mathbb R^3\) we draw different models, i.e surfaces with self-intersections, which are images of the maps of \(P\) into \(\mathbb R^3.\) A famous model is the so called Boy's surface [\textit{W. Boy}, Über die Curvatura integra und die Topologie geschlossener Flächen. Göttingen: Universitäts-Buchdruckerei (W. Fr. Kaestner) (1901; JFM 32.0488.02); Math. Ann. 57, 151--184 (1903; JFM 34.0537.07)], which is an immersion of \(P\) having single triple point and a connected self-intersection set. Recently, \textit{F. Apéry} [Models of the real projective plane. Computer graphics of Steiner and Boy surfaces. Braunschweig/Wiesbaden: Friedr. Vieweg \& Sohn (1987; Zbl 0623.57001)], presented another (non-homeomorphic to Boy's) model of \(P,\) also an immersion of \(P\) having a connected self-intersection set with a single triple point. These are the only two immersions of \(P\) that have the above type of the self-intersection set and furthermore, an immersion of the projective plane must have at least one triple point.NEWLINENEWLINEFor a closed surface \(M\), a map \(f:M\to \mathbb R^3\) is called generic (semi regular) if it is an immersion except for a finite number of pinch points. Models of surfaces obtained by such maps under the additional assumption that the self-intersection set is connected and has a single triple point have been analyzed by \textit{P. R. Cromwell} and \textit{W. L. Marar} [Geom. Dedicata 52, No. 2, 143--153 (1994; Zbl 0842.57027)]. The present paper is restricted to \(M=P\) and is based on the just mentioned work by Cromwell and Marar. By incorporating twists similar to that of Apéry's immersion of the projective plane (mentioned above) they identify all such generic maps of the projective plane. In the last section of the paper homotopies relating these models are described.