Obstructions for Morin and fold maps: Stiefel-Whitney classes and Euler characteristics of singularity loci
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2025
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| author | Fehér, László M. Matszangosz, Ákos K. |
| author_facet | Fehér, László M. Matszangosz, Ákos K. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_07379 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Obstructions for Morin and fold maps: Stiefel-Whitney classes and Euler characteristics of singularity loci Fehér, László M. Matszangosz, Ákos K. Algebraic Geometry Algebraic Topology 32S20, 14E15, 57R45, 58K30, 14P25 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. |
| title | Obstructions for Morin and fold maps: Stiefel-Whitney classes and Euler characteristics of singularity loci |
| topic | Algebraic Geometry Algebraic Topology 32S20, 14E15, 57R45, 58K30, 14P25 |
| url | https://arxiv.org/abs/2502.07379 |