Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In 1995 J.W. Heath asked which exactly $n$-to-one maps are compositions of exactly $k$-to-one maps with $1<k<n$. This paper deals with compositions of covering maps. Exactly $n$-to-one covering maps on locally arcwise connected continua that are not factorable into covering maps of order $\leq n-1$ are constructed for all $n$'s, and characterized in algebraic terms (fundamental groups). They are not proper compositions of exactly $k$-to-one maps, open maps, or locally one-to-one maps.

References [Enhancements On Off] (What's this?)

  • 1. J.D. Baildon, Open simple maps and periodic homeomorphisms, Proc. Amer. Math. Soc. 39 (1973), 433-436. MR 48:2976
  • 2. K. Borsuk, R. Molski, On a class of continuous mappings, Fund. Math. 45 (1957), 84-98. MR 21:858
  • 3. J. Dydak, On elementary maps, Colloq. Math. 31 (1974), 67-69. MR 51:6696
  • 4. L.R. Griffus, Exactly $k$-to-$1$ Maps Between Metric Continua, Ph.D. thesis, Auburn University, 1996.
  • 5. J.W. Heath, Every exactly $2$-to-$1$ function on the reals has an infinite set of discontinuities, Proc. Amer. Math. Soc. 98 (1986), 369-373. MR 87i:54031
  • 6. J.W. Heath, Exactly $k$-to-$1$ maps: from pathological functions with finitely many discontinuities to well-behaved covering maps, in: Continua. With the Houston Problem Book, 89-102, Lecture Notes in Pure and Appl. Math. 170, Dekker, New York, 1995. MR 96d:54015
  • 7. J. Krzempek, Compositions of simple maps, Fund. Math. 162 (1999), 149-162.MR 2001b:54018
  • 8. J. Mioduszewski, On two-to-one continuous functions, Dissertationes Math. (Rozprawy Mat.) 24 (1961). MR 26:3021
  • 9. D.J.S. Robinson, A Course in the Theory of Groups, Springer, New York, 1996. MR 96f:20001
  • 10. K. Sieklucki, On superposition of simple mappings, Fund. Math. 48 (1960), 217-228. MR 22:2981
  • 11. E. Spanier, Algebraic Topology, Springer, New York, 1981. MR 83i:55001

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 54C10, 05C25

Retrieve articles in all journals with MSC (2000): 54C10, 05C25

Additional Information

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

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

American Mathematical Society