We describe the generic geometry of the 3D-crosscap (image of a stable map of a 3-manifold into $$\mathbb {R}^4$$) 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.

中文翻译：

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$${\mathbb {R}}^4$$ 的三流形稳定映射的通用几何

我们通过同时分析高度和平方距离函数的通用奇点来描述 3D-crosscap 的通用几何形状（3-流形的稳定映射到 $$\mathbb {R}^4$$）在由3-流形组成的旗帜上，双点的表面和该曲线任意点的crosscaps曲线。