Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
MathSciNet review: 684659
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

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

  • [C] A. V. Černavskii, Finite-to-one open mappings of manifolds, Mat. Sb. (1964), 357-369. MR 0172256 (30:2476)
  • [D] A. Dress, Newman's theorem on transformation groups, Topology 8 (1969), 203-207. MR 0238353 (38:6629)
  • [M] Deane Montgomery, Remark on continuous collections (to appear). MR 695277 (85d:54046)
  • [N] M. H. A. Newman, A theorem on periodic transformations of spaces, Quart. J. Math. Oxford Ser. 2 (1931), 1-9.
  • [R] E. E. Robinson, A characterization of certain branched coverings as group actions, Fund. Math. 103 (1979), 43-45. MR 535834 (80g:57056)
  • [S] P. A. Smith, Transformations of finite period. III. Newman's Theorem, Ann. of Math. (2) 42 (1941), 446-458. MR 0004128 (2:324c)
  • [V] J. Väisälä, Discrete open mappings on manifolds, Ann. Acad. Sci. Fenn. Ser. A I Math. (1966).
  • [W] G. T. Whyburn, Analytic topology, Amer. Math. Soc. Colloq. Publ., vol. 28, Amer. Math. Soc., Providence, R. I., 1942. MR 0007095 (4:86b)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 57N15, 54C10, 55M25, 55M35, 57S17

Retrieve articles in all journals with MSC: 57N15, 54C10, 55M25, 55M35, 57S17

Additional Information

Keywords: Manifold, continuous decomposition, proper open map, degree, group action
Article copyright: © Copyright 1983 American Mathematical Society

American Mathematical Society