Obstructions for Morin and fold maps: Stiefel–Whitney classes and Euler characteristics of singularity loci

For a singularity type , let the ‐avoiding number of an ‐dimensional manifold be the lowest for which there is a map without type singular points. For instance, the case of is the case of immersions, which has been extensively studied in the case of real projective spaces. In this paper, we study the ‐avoiding number for other singularity types. Our results come in two levels: first we give an abstract reasoning that a nonzero cohomology class is supported on the singularity locus , proving that cannot be empty. Second, we interpret this obstruction as a nonzero invariant of the singularity locus for generic . The main technique that we employ is Sullivan's Stiefel–Whitney classes, which are mod 2, real analogues of the Chern–Schwartz–MacPherson classes. We introduce the Segre–Stiefel–Whitney classes of a singularity whose lowest degree term is the mod 2 Thom polynomial of . Using these techniques, we compute some universal formulae for the Euler characteristic of a singularity locus.