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

Source abstract

Let M and N be smooth compact oriented connected 2-mani-folds. Suppose f : M → N f:M \to N is smooth and every point p ∈ M 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 + {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 , … , p n {p_1}, \ldots ,{p_n} be the cusp points and μ ( p k ) \mu ({p_k}) the local degree at the cusp point p k {p_k} . We prove the following: $$χ ( M ) − 2 χ ( M − ) + ∑ μ ( p k ) = ( deg ⁡ f ) χ ( N ) \chi (M) - 2\chi ({M^ - }) + \sum \mu ({p_k}) = (\deg f)\chi (N)$$ where χ \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.