A Lower Bound on the Self-intersections of Fold Singularities

For an oriented surface $S$, the singular set of a fold map $f:S\to {\mathbb{R}}^2$ 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.