Constructing generic smooth maps of a manifold into a surface with prescribed singular loci

Saeki, Osamu

Annales de l'Institut Fourier, Tome 45 (1995) no. 4, pp. 1135-1162.

On sait que l’ensemble des points singuliers $S\left(f\right)$ d’une application différentiable générique $f:M\to N$ d’une variété à $n$ dimensions dans une surface est une sous-variété à une dimension de $M$ et que $S\left(f\right)$ a une stratification naturelle introduite par l’indice absolu. Dans cet article nous donnons une caractérisation complète des sous-variétés à une dimension (stratifiées) qui se présentent comme un ensemble de points singuliers d’une application générique, en termes de classes d’homologie représentées par les sous-variétés.

It is known that the singular set $S\left(f\right)$ of a generic smooth map $f:M\to N$ of an $n$-dimensional manifold into a surface is a closed 1-dimensional submanifold of $M$ and that it has a natural stratification induced by the absolute index. In this paper, we give a complete characterization of those 1-dimensional (stratified) submanifolds which arise as the singular set of a generic map in terms of the homology class they represent.

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