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On Newman's theorem for finite-to-one open mappings on manifolds


Authors: L. F. McAuley and Eric E. Robinson
Journal: Proc. Amer. Math. Soc. 87 (1983), 561-566
MSC: Primary 57N15; Secondary 54C10, 55M25, 55M35, 57S17
DOI: https://doi.org/10.1090/S0002-9939-1983-0684659-X
MathSciNet review: 684659
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Abstract: We prove the following important generalization of a famous theorem by Newman: If $ (M,d)$ is a closed manifold, there is an $ \varepsilon > 0$ such that if $ Y$ is a closed manifold and $ f:M \twoheadrightarrow Y$ is a finite-to-one open surjective mapping which is not a homeomorphism, then there is at least one $ y \in Y$ such that diam $ {f^{ - 1}}(y) \geqslant \varepsilon $.

A version of the above theorem was first proved by Černavskii using rather complicated covering arguments. Our proof by comparison is much simpler and uses modern homology theory.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1983-0684659-X
Keywords: Manifold, continuous decomposition, proper open map, degree, group action
Article copyright: © Copyright 1983 American Mathematical Society