Generic Geometry of Stable Maps of 3-Manifolds into R4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^4$$\end{document}

We describe the generic geometry of the 3D-crosscap (image of a stable map of a 3-manifold into R4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^4$$\end{document}) by means of the simultaneous analysis of the generic singularities of height and squared distanced functions on the flag composed by the 3-manifold, the surface of double points and the crosscaps curve at any point of this curve.