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)



Rational irreducible plane continua without the fixed-point property

Authors: Charles L. Hagopian and Roman Manka
Journal: Proc. Amer. Math. Soc. 133 (2005), 617-625
MSC (2000): Primary 54F15, 54H25
Published electronically: August 20, 2004
MathSciNet review: 2093087
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We define rational irreducible continua in the plane that admit fixed-point-free maps with the condition that all of their tranches have the fixed-point property. This answers in the affirmative a question of Hagopian. The construction is based on a special class of spirals that limit on a double Warsaw circle. The closure of each of these spirals has the fixed-point property.

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

  • [A] M. M. Awartani, The fixed remainder property for self-homeomorphisms of Elsa continua, Topology Proc. 11 (1986), 225-238. MR 89g:54073
  • [B] H. Bell, On fixed point properties of plane continua, Trans. Amer. Math. Soc. 128 (1967), 539-548. MR 35:4888
  • [Bi] R. H. Bing, The elusive fixed point property, Amer. Math. Monthly 76 (1969), 119-132. MR 38:5201
  • [D] E. Dyer, Irreducibility of the sum of the elements of a continuous collection of continua, Duke Math. J. 20 (1953), 589-592. MR 15:335f
  • [H] C. L. Hagopian, Irreducible continua without the fixed-point property, Bull. Pol. Acad. Sci. Math. 51 (2003), 121-127.
  • [H-M] C. L. Hagopian and R. Manka, Simple spirals on double Warsaw circles, Topology and its Appl. 128 (2003), 93-101. MR 2004c:54029
  • [I] S. Iliadis, Positions of continua on the plane and fixed points, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 1970, no. 4, 66-70. MR 44:4726
  • [K-W] V. Klee and S. Wagon, Old and New Unsolved Problems in Plane Geometry and Number Theory, Dolciani Mathematical Expositions, vol. 11, Math. Assoc. Amer., Washington, DC, 1991. MR 92k:00014
  • [Ku1] C. Kuratowski, Théorie des continus irréductibles entre deux points II, Fund. Math. 10 (1927), 225-276.
  • [Ku2] -, Topology, Vol. 2, 3rd ed., Monografie Mat., Tom 21, PWN, Warsaw, 1961; English transl., Academic Press, New York; PWN, Warsaw, 1968. MR 41:4467
  • [L] I. W. Lewis, Continuum theory problems, Topology Proc. 8 (1983), 361-394. MR 86a:54038
  • [M1] R. Manka, On irreducibility and indecomposability of continua, Fund. Math. 129 (1988), 121-131. MR 89g:54079
  • [M2] -, On spirals and the fixed point property, Fund. Math. 144 (1994), 1-9. MR 95c:54061
  • [Mr] R. L. Moore, Concerning upper semi-continuous collections of continua, Trans. Amer. Math. Soc. 27 (1925), 416-428.
  • [N] S. B. Nadler, Continua which are a one-to-one continuous image of $[0,\infty )$, Fund. Math. 75 (1972), 123-133. MR 47:5848
  • [S] K. Sieklucki, On a class of plane acyclic continua with the fixed point property, Fund. Math. 63 (1968), 257-278. MR 39:2139

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 54F15, 54H25

Retrieve articles in all journals with MSC (2000): 54F15, 54H25

Additional Information

Charles L. Hagopian
Affiliation: Department of Mathematics, California State University, Sacramento, Sacramento, California 95819-6051

Roman Manka
Affiliation: Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-956 Warsaw, Poland

Keywords: Fixed-point property, rational continua, irreducible continua of type $\lambda $, spiral, double Warsaw circle, plane continua, retractions
Received by editor(s): March 13, 2003
Received by editor(s) in revised form: October 17, 2003
Published electronically: August 20, 2004
Additional Notes: The authors wish to thank Mark Marsh for helpful conversations about the topics of this paper
Communicated by: Alan Dow
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society