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

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- by J. R. Quine PDF
- Trans. Amer. Math. Soc.
**236**(1978), 307-314 Request permission

## 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.

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## Additional Information

- © Copyright 1978 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**236**(1978), 307-314 - MSC: Primary 58C25
- DOI: https://doi.org/10.1090/S0002-9947-1978-0474378-X
- MathSciNet review: 0474378