Immersions of the projective plane with one triple point (Q838020)

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.