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

TL;DR

This work develops a unified obstruction framework for avoiding prescribed singularities $\eta$ in smooth maps, using stable avoiding ideals and Segre-Stiefel-Whitney classes to obtain universal obstructions and Euler-characteristic formulas for singularity loci. By connecting Thom polynomials, Segre-SW classes, and real Ohmoto–Aluffi-type relations, the authors derive lower bounds for the $\eta$-avoiding numbers, notably for the fold ($A_2$) and Morin ($\Sigma^2$) cases on real projective spaces, and show how these obstructions interplay with Euler characteristics of the singular loci. They present universal formulas (characteristic series) for the Euler characteristics of $\eta$-loci and compute precise parity and non-emptiness results in several configurations, revealing a deeper link between fold and Morin maps. A central conjecture proposes a tight relationship between the $A_2$ and $\Sigma^2$ loci through Segre-SW classes and stable avoiding ideals, suggesting a cobordism-like correspondence between cusp and Morin-type degeneracies and offering practical tools for obstruction-theoretic reasoning in singularity theory. The framework provides concrete computational techniques for Stiefel-Whitney classes and Euler characteristics of a broad class of singularity loci, with implications for immersion-type problems in real algebraic geometry and differential topology.

Abstract

For a singularity type $η$, let the $η$-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 $η$ type singular points. For instance, the case of $η=Σ^1$ is the case of immersions, which has been extensively studied in the case of real projective spaces. In this paper we study the $η$-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 $η(f)$, proving that $η(f)$ cannot be empty. Second, we interpret this obstruction as a non-zero invariant of the singularity locus $η(f)$ 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 ${\rm s}^{\rm sw}_η$ whose lowest degree term is the mod 2 Thom polynomial of $η$. Using these techniques we compute some universal formulas for the Euler characteristic of a singularity locus.

Theorems & Definitions (96)

• Remark 2.1
• Remark 2.2
• Remark 2.3
• Theorem 2.4
• Remark 2.5
• Theorem 2.6
• Theorem 2.7: Ohmoto-Aluffi
• Proposition 2.8
• Theorem 2.9
• proof