SCHWEITZER 2004 Problem 9

Let $F$ be a smooth (i.e. $C^{\infty}$ ) closed surface. Call a continuous map $f\colon F\rightarrow \mathbb{R}^2$ an almost-immersion if there exists a smooth closed embedded curve $\gamma$ (possibly disconnected) in $F$ such that $f$ is smooth and of maximal rank (i.e., rank 2) on $F\backslash \gamma$ and each point $p\in\gamma$ admits local coordinate charts $(x,y)$ and $(u,v)$ about $p$ and $f(p)$ , respectively, such taht the coordinates of $p$ and $f(p)$ are zero and the map $f$ is given by $(x,y)\rightarrow (u,v), u=|x|, v=y$ . Determine the genera of those smooth, closed, connected, orientable surfaces $F$ that admit an almost-immersion in the plane with the curve $\gamma$ having a given positive number $n$ of connected components.