Glasnik Matematicki, Vol. 42, No.1 (2007), 89-94.

ON THE FUNDAMENTAL GROUP OF R3 MODULO THE CASE-CHAMBERLIN CONTINUUM

Katsuya Eda, Umed H. Karimov and Dušan Repovš

School of Science and Engineering, Waseda University, Tokyo 169-8555, Japan
e-mail: eda@logic.info.waseda.ac.jp

Institute of Mathematics, Academy of Sciences of Tajikistan, Ul. Ainy 299A, Dushanbe 734063, Tajikistan
e-mail: umed-karimov@mail.ru

Institute of Mathematics, Physics and Mechanics and Faculty of Education, University of Ljubljana, P.O.Box 2964, Ljubljana 1001, Slovenia
e-mail: dusan.repovs@guest.arnes.si

Abstract.   It has been known for a long time that the fundamental group of the quotient of R3 by the Case-Chamberlin continuum is nontrivial. In the present paper we prove that this group is in fact, uncountable.

2000 Mathematics Subject Classification.   54F15, 55Q52, 57M05, 54B15, 54F35, 54G15.

Key words and phrases.   Case-Chamberlin continuum, quotient space, fundamental group, lower central series, weight, commutator.

Full text (PDF) (free access)

DOI: 10.3336/gm.42.1.07

References:

  1. S. Armentrout, unpublished manuscript.

  2. K. Borsuk, On the homotopy types of some decomposition spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 18 (1970), 235-239.
    MathSciNet

  3. J. H. Case and R. E. Chamberlin, Characterization of tree-like continua, Pacific J. Math. 10 (1960), 73-84.
    MathSciNet

  4. W. Hurewicz and H. Wallman, Dimension Theory, Princeton University Press, Princeton, 1941.
    MathSciNet

  5. U. H. Karimov and D. Repovs, On suspensions of noncontractible compacta of trivial shape, Proc. Amer. Math. Soc. 127 (1999), 627-632.
    MathSciNet     CrossRef

  6. U. H. Karimov and D. Repovs, On nonacyclicity of the quotient space of R3 by the solenoid, Topology Appl. 133 (2003), 65-68.
    MathSciNet     CrossRef

  7. J. Krasinkiewicz, On a method of constructing ANR-sets. An application of inverse limits, Fund. Math. 92 (1976), 95-112.
    MathSciNet

  8. R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89, Springer-Verlag, Berlin-New York, 1977.
    MathSciNet

  9. W. Magnus, A. Karras and D. Solitar, Combinatorial Group Theory, Dover Publications, Inc., New York, 1976.
    MathSciNet

  10. N. Shrikhande, Homotopy properties of decomposition spaces, Fund. Math. 116 (1983), 119-124.
    MathSciNet

  11. L. Siebenmann, Chapman's classification of shapes: a proof using collapsing, Manuscripta Math. 16 (1975), 373-384.
    MathSciNet     CrossRef
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