A global theorem for singularities of maps between oriented $2$-manifolds

Abstract:

Let M and N be smooth compact oriented connected 2-mani-folds. Suppose $f:M \to N$ is smooth and every point $p \in M$ is either a fold point, cusp point, or regular point of f i.e., f is excellent in the sense of Whitney. Let ${M^ + }$ be the closure of the set of regular points at which f preserves orientation and M the closure of the set of regular points at which f reverses orientation. Let ${p_1}, \ldots ,{p_n}$ be the cusp points and $\mu ({p_k})$ the local degree at the cusp point ${p_k}$. We prove the following: \[ \chi (M) - 2\chi ({M^ - }) + \sum \mu ({p_k}) = (\deg f)\chi (N)\] where $\chi$ is the Euler characteristic and deg is the topological degree. We show that it is a generalization of the Riemann-Hurwitz formula of complex analysis and give some examples.