Rational irreducible plane continua without the fixedpoint property
Authors:
Charles L. Hagopian and Roman Manka
Journal:
Proc. Amer. Math. Soc. 133 (2005), 617625
MSC (2000):
Primary 54F15, 54H25
Published electronically:
August 20, 2004
MathSciNet review:
2093087
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We define rational irreducible continua in the plane that admit fixedpointfree maps with the condition that all of their tranches have the fixedpoint 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 fixedpoint property.
 [A]
Marwan
M. Awartani, The fixed remainder property for self homeomorphisms
of Elsa continua, Proceedings of the 1986 Topology Conference
(Lafayette, LA, 1986), 1986, pp. 225–238. MR 945500
(89g:54073)
 [B]
Harold
Bell, On fixed point properties of plane
continua, Trans. Amer. Math. Soc. 128 (1967), 539–548. MR 0214036
(35 #4888), http://dx.doi.org/10.1090/S00029947196702140362
 [Bi]
R.
H. Bing, The elusive fixed point property, Amer. Math. Monthly
76 (1969), 119–132. MR 0236908
(38 #5201)
 [D]
Eldon
Dyer, Irreducibility of the sum of the elements of a continuous
collection of continua, Duke Math. J. 20 (1953),
589–592. MR 0058198
(15,335f)
 [H]
C. L. Hagopian, Irreducible continua without the fixedpoint property, Bull. Pol. Acad. Sci. Math. 51 (2003), 121127.
 [HM]
Charles
L. Hagopian and Roman
Mańka, Simple spirals on double Warsaw circles,
Topology Appl. 128 (2003), no. 23, 93–101. MR 1956606
(2004c:54029), http://dx.doi.org/10.1016/S01668641(02)001050
 [I]
S.
D. Iliadis, Positions of continua on the plane, and fixed
points, Vestnik Moskov. Univ. Ser. I Mat. Meh. 25
(1970), no. 4, 66–70 (Russian, with English summary). MR 0287522
(44 #4726)
 [KW]
Victor
Klee and Stan
Wagon, Old and new unsolved problems in plane geometry and number
theory, The Dolciani Mathematical Expositions, vol. 11,
Mathematical Association of America, Washington, DC, 1991. MR 1133201
(92k:00014)
 [Ku1]
C. Kuratowski, Théorie des continus irréductibles entre deux points II, Fund. Math. 10 (1927), 225276.
 [Ku2]
K.
Kuratowski, Topology. Vol. II, New edition, revised and
augmented. Translated from the French by A. Kirkor, Academic Press, New
YorkLondon; Państwowe Wydawnictwo Naukowe Polish Scientific
Publishers, Warsaw, 1968. MR 0259835
(41 #4467)
 [L]
Wayne
Lewis, Continuum theory problems, Proceedings of the 1983
topology conference (Houston, Tex., 1983), 1983, pp. 361–394. MR 765091
(86a:54038)
 [M1]
Roman
Mańka, On irreducibility and indecomposability of
continua, Fund. Math. 129 (1988), no. 2,
121–131. MR
959436 (89g:54079)
 [M2]
Roman
Mańka, On spirals and fixed point property, Fund. Math.
144 (1994), no. 1, 1–9. MR 1271474
(95c:54061)
 [Mr]
R. L. Moore, Concerning upper semicontinuous collections of continua, Trans. Amer. Math. Soc. 27 (1925), 416428.
 [N]
Sam
B. Nadler Jr., Continua which are a onetoone continuous image of
[0,∞), Fund. Math. 75 (1972), no. 2,
123–133. (errata insert). MR 0317301
(47 #5848)
 [S]
K.
Sieklucki, On a class of plane acyclic continua with the fixed
point property., Fund. Math. 63 (1968),
257–278. MR 0240794
(39 #2139)
 [A]
 M. M. Awartani, The fixed remainder property for selfhomeomorphisms of Elsa continua, Topology Proc. 11 (1986), 225238. MR 89g:54073
 [B]
 H. Bell, On fixed point properties of plane continua, Trans. Amer. Math. Soc. 128 (1967), 539548. MR 35:4888
 [Bi]
 R. H. Bing, The elusive fixed point property, Amer. Math. Monthly 76 (1969), 119132. MR 38:5201
 [D]
 E. Dyer, Irreducibility of the sum of the elements of a continuous collection of continua, Duke Math. J. 20 (1953), 589592. MR 15:335f
 [H]
 C. L. Hagopian, Irreducible continua without the fixedpoint property, Bull. Pol. Acad. Sci. Math. 51 (2003), 121127.
 [HM]
 C. L. Hagopian and R. Manka, Simple spirals on double Warsaw circles, Topology and its Appl. 128 (2003), 93101. 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, 6670. MR 44:4726
 [KW]
 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), 225276.
 [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), 361394. MR 86a:54038
 [M1]
 R. Manka, On irreducibility and indecomposability of continua, Fund. Math. 129 (1988), 121131. MR 89g:54079
 [M2]
 , On spirals and the fixed point property, Fund. Math. 144 (1994), 19. MR 95c:54061
 [Mr]
 R. L. Moore, Concerning upper semicontinuous collections of continua, Trans. Amer. Math. Soc. 27 (1925), 416428.
 [N]
 S. B. Nadler, Continua which are a onetoone continuous image of , Fund. Math. 75 (1972), 123133. MR 47:5848
 [S]
 K. Sieklucki, On a class of plane acyclic continua with the fixed point property, Fund. Math. 63 (1968), 257278. 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 958196051
Email:
hagopian@csus.edu
Roman Manka
Affiliation:
Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00956 Warsaw, Poland
Email:
manka@impan.gov.pl
DOI:
http://dx.doi.org/10.1090/S0002993904075434
PII:
S 00029939(04)075434
Keywords:
Fixedpoint 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
