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Finite-to-One Open Mappings

Published online by Cambridge University Press:  20 November 2018

Edwin Duda  and
W. Hugh Haynsworth
Edwin Duda
Affiliation:
University of Miami, Coral Gables, Florida
W. Hugh Haynsworth
Affiliation:
The College of Charleston Charleston, South Carolina
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The class of finite-to-one open mappings on manifolds contains some important subclasses. Any non-constant analytic function from a bounded region in its domain of definition is finite-to-one. Church [2] showed that any light strongly open Cn map f: Rn → Rn is discrete. A number of papers concerning discrete open mappings on manifolds have been published; see [1-6; 8-9; 11-14].

A result of Černavskiĭ [1] (see also [13]) shows that for any discrete strongly open mapping f : Mn → Nn of an n-manifold into an n-manifold, the branch set of f has dimension less than n – 1. If f is also a closed map, then N(f) is finite and the set of points x for which N(x, f) = N(f) is an open dense connected subset of Mn.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 1 , February 1971 , pp. 77 - 83
Copyright
Copyright © Canadian Mathematical Society 1971

References

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