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Transactions of the American Mathematical Society

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


Author: J. R. Quine
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
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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:

$\displaystyle \chi (M) - 2\chi ({M^ - }) + \sum \mu ({p_k}) = (\deg f)\chi (N)$

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.

References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1978-0474378-X
Keywords: Singularities, cusp points, folds, nanifold, Euler characteristic, excellent map, Riemann-Hurwitz formula
Article copyright: © Copyright 1978 American Mathematical Society

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