Algebraic characterization of finite (branched) coverings

M. Mulero

Fundamenta Mathematicae (1998)

  • Volume: 158, Issue: 2, page 165-180
  • ISSN: 0016-2736

Abstract

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Every continuous map X → S defines, by composition, a homomorphism between the corresponding algebras of real-valued continuous functions C(S) → C(X). This paper deals with algebraic properties of the homomorphism C(S) → C(X) in relation to topological properties of the map X → S. The main result of the paper states that a continuous map X → S between topological manifolds is a finite (branched) covering, i.e., an open and closed map whose fibres are finite, if and only if the induced homomorphism C(S) → C(X) is integral and flat.

How to cite

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Mulero, M.. "Algebraic characterization of finite (branched) coverings." Fundamenta Mathematicae 158.2 (1998): 165-180. <http://eudml.org/doc/212309>.

@article{Mulero1998,
abstract = {Every continuous map X → S defines, by composition, a homomorphism between the corresponding algebras of real-valued continuous functions C(S) → C(X). This paper deals with algebraic properties of the homomorphism C(S) → C(X) in relation to topological properties of the map X → S. The main result of the paper states that a continuous map X → S between topological manifolds is a finite (branched) covering, i.e., an open and closed map whose fibres are finite, if and only if the induced homomorphism C(S) → C(X) is integral and flat.},
author = {Mulero, M.},
journal = {Fundamenta Mathematicae},
keywords = {branched covering; open and closed map; ring of continuous functions; flat homomorphism; integral homomorphism},
language = {eng},
number = {2},
pages = {165-180},
title = {Algebraic characterization of finite (branched) coverings},
url = {http://eudml.org/doc/212309},
volume = {158},
year = {1998},
}

TY - JOUR
AU - Mulero, M.
TI - Algebraic characterization of finite (branched) coverings
JO - Fundamenta Mathematicae
PY - 1998
VL - 158
IS - 2
SP - 165
EP - 180
AB - Every continuous map X → S defines, by composition, a homomorphism between the corresponding algebras of real-valued continuous functions C(S) → C(X). This paper deals with algebraic properties of the homomorphism C(S) → C(X) in relation to topological properties of the map X → S. The main result of the paper states that a continuous map X → S between topological manifolds is a finite (branched) covering, i.e., an open and closed map whose fibres are finite, if and only if the induced homomorphism C(S) → C(X) is integral and flat.
LA - eng
KW - branched covering; open and closed map; ring of continuous functions; flat homomorphism; integral homomorphism
UR - http://eudml.org/doc/212309
ER -

References

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  10. [10] L. F. McAuley and E. E. Robinson, Discrete open and closed maps on generalized continua and Newman's property, Canad. J. Math. 36 (1984), 1081-1112. Zbl0552.54005
  11. [11] B. Malgrange, Ideals of Differentiable Functions, Oxford Univ. Press, 1966. 
  12. [12] W. S. Massey, Algebraic Topology: An Introduction, Springer, 1967. 
  13. [13] H. Matsumura, Commutative Ring Theory, Cambridge Univ. Press, 1986. Zbl0603.13001
  14. [14] M. A. Mulero, Algebraic properties of rings of continuous functions, Fund. Math. 149 (1996), 55-66. Zbl0840.54020
  15. [15] M. A. Mulero, Rings of continuous functions and the branch set of a covering, Proc. Amer. Math. Soc. 126 (1998), 2183-2189. Zbl0893.54008
  16. [16] J. C. Tougeron, Idéaux de Fonctions Différentiables, Springer, 1972. Zbl0251.58001

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