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The geometry of open continuous mappings having two valences between Riemann surfaces

Published online by Cambridge University Press:  24 October 2008

Abdallah Lyzzaik
Abdallah Lyzzaik
Affiliation:
Department of Mathematics, American University of Beirut, Beirut, Lebanon
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Abstract

An open continuous function from an open Riemann surface with finite genus and finite number of boundary components into a closed Riemann surface is termed a (p, q)-map, 0 < q < p, if it has a finite number of branch points and assumes every point in the image surface either p or q times, counting multiplicity, with possibly a finite number of exceptions.

The object of this paper is to prove that the geometry of any (p, q)-map resembles that of a (p, q)-map whose q-set (the set of image points of f that are taken on exactly q times, counting multiplicity), constitutes a finite set of Jordan arcs or curves (loops). This leads to interesting geometrie results regarding (p, q)-maps without exceptional points. Further, it yields that every (p, q)-map is homotopic to a simplicial (p, q)-map having the same covering properties.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 120 , Issue 2 , August 1996 , pp. 309 - 329
Copyright
Copyright © Cambridge Philosophical Society 1996

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References

REFERENCES

[1]Agoston, M. K.. Algebraic topology: a first course (Marcel Dekker, Inc., 1976).Google Scholar
[2]Ahlfors, L. V. and Sario, L.. Riemann surfaces (Princeton Univ. Press, 1960).CrossRefGoogle Scholar
[3]Keldyš, L. V.. Topological imbeddings in Euclidean space. Proc. Steklov Inst. Math. 8 (1966).Google Scholar
[4]Lyzzaik, A. K.. Covering properties of open continuous mappings having two valences between Riemann surfaces. Math. Proc. Cambridge Phil. Soc. 118 (1995), 321–340.Google Scholar
[5]Lyzzaik, A. K. and Stephenson, K.. The structure of open continuous mappings having two valences. Trans. Amer. Math. Soc. 327, (1991), 525–566.CrossRefGoogle Scholar
[6]Massey, W. S.. Algebraic topology: an introduction (Springer-Verlag, 1977).Google Scholar
[7]Pommerenke, Chr.. Univalent functions (Vandenhoeck & Ruprecht, 1975).Google Scholar
[8]Stoïlow, S.. Principes topologiques de la théorie des fonctions analytiques (Gauthier-Villars, 1938).Google Scholar