O. Saeki, CONSTRUCTING GENERIC SMOOTH MAPS OF A MANIFOLD INTO A SURFACE WITH PRESCRIBED SINGULAR LOCI, Annales de l'Institut Fourier, 45(4), 1995, pp. 1135
It is known that the singular set S(f) of a generic smooth map f : M -
-> N of an n-dimensional manifold into a surface is a closed 1-dimensi
onal submanifold of M and that it has a natural stratification induced
by the absolute index. In this paper, we give a complete characteriza
tion of those 1-dimensional (stratified) submanifolds which arise as t
he singular set of a generic map in terms of the homology class they r
epresent.