Generic geometry of stable maps of 3-manifolds into \(\mathbb{R}^4\) (Q2042226)

In this paper, the authors study the generic geometry of the 3D-crosscap, image of a germ in the \(\mathcal{A}\)-orbit of \((x, y,z^2, yz)\), by means of the simultaneous analysis of the generic singularities of height and squared distance functions on the flag composed by the 3-manifold, the surface of double points and the crosscaps curve at any point of this curve. They use the same approach as in [\textit{J. W. Bruce} and \textit{J. M. West}, Math. Proc. Camb. Philos. Soc. 123, No. 1, 19--39 (1998; Zbl 1006.53005)] in order to study the geometry of the image of a stable map from a 3-manifold to \(\mathbb R^4\). They obtain all the possible geometrical phenomena and their interpretation given by contacts with hyperplanes and hyperspheres occurring at a generic point of a 3D-crosscap, as well as those concerning contacts with hyperplanes occurring in generic 1-parameter families of stable maps from a 3-manifold to \(\mathbb{R}^4\).