A Lower Bound on the Self-intersections of Fold Singularities
Fuente:
arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866910215472939008 |
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| author | Drouin, Joshua Kahmeyer, Liam |
| author_facet | Drouin, Joshua Kahmeyer, Liam |
| contents | For an oriented surface $S$, the singular set of a fold map $f:S\rightarrow \mathbb{R}^2$ is a collection of smooth curves, also known as fold singularities. We construct a sharp lower bound on the number of self-intersections of such fold singularities. This is done by first establishing a sharp lower bound on the number of self-intersections of the boundary of a surface immersed in $\mathbb{R}^2$. We then construct a sharp lower bound for the number of self-intersections of the singular set of a simple stable fold map of a surface to $\mathbb{R}^2$ by viewing the connected components of the singular set as the boundary components of smaller surface components, and invoking the previously constructed lower bound for the number of self-intersections of an immersed boundary. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_12989 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Lower Bound on the Self-intersections of Fold Singularities Drouin, Joshua Kahmeyer, Liam Geometric Topology 57R45, 58K30, 57R42, 57K20 For an oriented surface $S$, the singular set of a fold map $f:S\rightarrow \mathbb{R}^2$ is a collection of smooth curves, also known as fold singularities. We construct a sharp lower bound on the number of self-intersections of such fold singularities. This is done by first establishing a sharp lower bound on the number of self-intersections of the boundary of a surface immersed in $\mathbb{R}^2$. We then construct a sharp lower bound for the number of self-intersections of the singular set of a simple stable fold map of a surface to $\mathbb{R}^2$ by viewing the connected components of the singular set as the boundary components of smaller surface components, and invoking the previously constructed lower bound for the number of self-intersections of an immersed boundary. |
| title | A Lower Bound on the Self-intersections of Fold Singularities |
| topic | Geometric Topology 57R45, 58K30, 57R42, 57K20 |
| url | https://arxiv.org/abs/2605.12989 |