A Lower Bound on the Self-intersections of Fold Singularities

Joshua Drouin; Liam Kahmeyer

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arXiv · 2605.12989

A Lower Bound on the Self-intersections of Fold Singularities

Joshua Drouin·Liam Kahmeyer

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Abstract

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.

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BibTeX RIS

Joshua Drouin, Liam Kahmeyer. 2026-05-13. A Lower Bound on the Self-intersections of Fold Singularities. https://arxiv.org/abs/2605.12989

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