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 | References | Similar Articles | Additional Information

Abstract: We prove the following important generalization of a famous theorem by Newman: If is a closed manifold, there is an such that if is a closed manifold and is a finite-to-one open surjective mapping which is not a homeomorphism, then there is at least one such that diam .

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.

**[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)**

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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