Dissecting the 2-sphere by immersions

The self intersection of an immersion \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i: S^2 \to \mathbb {R}^3$$\end{document} dissects S2 into pieces which are planar surfaces (unless i is an embedding). In this work we determine what collections of planar surfaces may be obtained in this way. In particular, for every n we construct an immersion \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i: S^2 \to \mathbb {R}^3$$\end{document} with 2n triple points, for which all pieces are discs.