Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
MathSciNet review: 0474378
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 58C25

Retrieve articles in all journals with MSC: 58C25


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1978-0474378-X
PII: S 0002-9947(1978)0474378-X
Keywords: Singularities, cusp points, folds, nanifold, Euler characteristic, excellent map, Riemann-Hurwitz formula
Article copyright: © Copyright 1978 American Mathematical Society