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

For a singularity type $\eta$, let the $\eta$-avoiding number of an $n$-dimensional manifold $M$ be the lowest $k$ for which there is a map $M\to\mathbb{R}^{n+k}$ without $\eta$ type singular points. For instance, the case of $\eta=\Sigma^1$ is the case of immersions, which has been extensively studied in the case of real projective spaces. In this paper we study the $\eta$-avoiding number for other singularity types. Our results come in two levels: first we give an abstract reasoning that a non-zero cohomology class is supported on the singularity locus $\eta(f)$, proving that $\eta(f)$ cannot be empty. Second, we interpret this obstruction as a non-zero invariant of the singularity locus $\eta(f)$ for generic $f$. The main technique that we employ is Sullivan's Stiefel-Whitney classes, which are mod 2, real analogues of the Chern-Schwartz-MacPherson (CSM) classes. We introduce the Segre-Stiefel-Whitney classes of a singularity ${\rm s}^{\rm sw}_\eta$ whose lowest degree term is the mod 2 Thom polynomial of $\eta$. Using these techniques we compute some universal formulas for the Euler characteristic of a singularity locus.