Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-19T04:10:50.375Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

1. Cernavskiï, A. V., Finite-to-one open mappings on manifolds, Mat. Sb. (N.S.) 65 (107) (1964), 357–369. (Russian)Google Scholar
2. Church, P. T., Differentiate open maps on manifolds, Trans. Amer. Math. Soc. 109 (1963), 87–100.Google Scholar
3. Church, P. T. and Hemmingsen, E., Light open maps on n-manifolds, Duke Math. J. 27 (1960), 527–536.Google Scholar
4. Church, P. T. and Hemmingsen, E., Light open mappings on n-manifolds. II, Duke Math. J. 28 (1961), 607–623.Google Scholar
5. Church, P. T. and Hemmingsen, E., Light open mappings on n-manifolds. III, Duke Math. J. 30 (1963), 379–389.Google Scholar
6. Cronin, J. and McAuley, L. F., Whyburn's conjecture for some differentiable mappings, Proc. Nat. Acad. Sci. U.S.A. 56 (1966), 405–412.Google Scholar
7. Hurewicz, W. and Wallman, H., Dimension theory, Princeton Mathematical Series, Vol. 4 (Princeton Univ. Press, Princeton, N. J., 1941).Google Scholar
8. McAuley, L. F., Conditions under which light open mappings are homeomorphisms, Duke Math. J. 33 (1966), 445–452.Google Scholar
9. McAuley, L. F., Concerning a conjecture of Whyburn on light open mappings, Bull. Amer. Math. Soc. 71 (1965), 671–674.Google Scholar
10. Radó, T. and Reichelderfer, P. V., Continuous transformations in analysis;, With an introduction to algebraic topology; Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berucksichtigung der Anwendungsgebiete, Bd. 75 (Springer-Verlag, Berlin-Göttingen-Heidelberg, 1955).Google Scholar
11. Stoïlow, S., Sur les transformations continues et la topologie des fonctionsanalytiques, Ann. Sci. École Norm. Sup. III 45 (1928), 347–382.Google Scholar
12. Titus, C. J. and Young, G. S., A Jacobian condition for inferiority, Michigan Math. J. 1 (1952), 89–94.Google Scholar
13. Väisälä, J., Discrete open mappings on manifolds, Ann. Acad. Sci. Fenn. Ser.A I No. 392 (1966), 10 pp.Google Scholar
14. Whyburn, G. T., Topological analysis (Princeton Univ. Press, Princeton, N. J., 1958).Google Scholar