Covering maps that are not compositions of covering maps of lesser order

Author:
Jerzy Krzempek

Journal:
Proc. Amer. Math. Soc. **130** (2002), 1867-1873

MSC (2000):
Primary 54C10; Secondary 05C25

Published electronically:
November 15, 2001

MathSciNet review:
1887036

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Abstract | References | Similar Articles | Additional Information

Abstract: In 1995 J.W. Heath asked *which exactly **-to-one maps are compositions of exactly **-to-one maps with **.* This paper deals with compositions of covering maps. Exactly -to-one covering maps on locally arcwise connected continua that are not factorable into covering maps of order are constructed for all 's, and characterized in algebraic terms (fundamental groups). They are not proper compositions of exactly -to-one maps, open maps, or locally one-to-one maps.

**1.**John D. Baildon,*Open simple maps and periodic homeomorphisms*, Proc. Amer. Math. Soc.**39**(1973), 433–436. MR**0324626**, 10.1090/S0002-9939-1973-0324626-2**2.**Karol Borsuk and R. Molski,*On a class of continuous mappings*, Fund. Math.**45**(1957), 84–98. MR**0102063****3.**Jerzy Dydak,*On elementary maps*, Colloq. Math.**31**(1974), 67–69. MR**0370469****4.**L.R. Griffus,*Exactly**-to-**Maps Between Metric Continua,*Ph.D. thesis, Auburn University, 1996.**5.**Jo Heath,*Every exactly 2-to-1 function on the reals has an infinite set of discontinuities*, Proc. Amer. Math. Soc.**98**(1986), no. 2, 369–373. MR**854049**, 10.1090/S0002-9939-1986-0854049-8**6.**Jo W. Heath,*Exactly 𝑘-to-1 maps: from pathological functions with finitely many discontinuities to well-behaved covering maps*, Continua (Cincinnati, OH, 1994) Lecture Notes in Pure and Appl. Math., vol. 170, Dekker, New York, 1995, pp. 89–102. MR**1326835****7.**Jerzy Krzempek,*Compositions of simple maps*, Fund. Math.**162**(1999), no. 2, 149–162. MR**1734914****8.**J. Mioduszewski,*On two-to-one continuous functions*, Rozprawy Mat.**24**(1961), 43. MR**0145490****9.**Derek J. S. Robinson,*A course in the theory of groups*, 2nd ed., Graduate Texts in Mathematics, vol. 80, Springer-Verlag, New York, 1996. MR**1357169****10.**K. Sieklucki,*On superpositions of simple mappings*, Fund. Math.**48**(1959/1960), 217–228. MR**0112123****11.**Edwin H. Spanier,*Algebraic topology*, Springer-Verlag, New York-Berlin, 1981. Corrected reprint. MR**666554**

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Additional Information

**Jerzy Krzempek**

Affiliation:
Institute of Mathematics, Silesian Technical University, Kaszubska 23, PL-44-100 Gliwice, Poland

Email:
krzem@zeus.polsl.gliwice.pl

DOI:
https://doi.org/10.1090/S0002-9939-01-06266-9

Keywords:
Composition,
covering map,
locally one-to-one,
open,
exactly $k$-to-one map,
group action,
fundamental group

Received by editor(s):
November 6, 2000

Received by editor(s) in revised form:
January 9, 2001

Published electronically:
November 15, 2001

Communicated by:
Alan Dow

Article copyright:
© Copyright 2001
American Mathematical Society