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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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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