Mediterranean Journal of Mathematics

, Volume 10, Issue 1, pp 519–528 | Cite as

On 2-dimensional Nonaspherical Cell-like Peano Continua: A Simplified Approach

  • Katsuya Eda
  • Umed H. Karimov
  • Dušan Repovš


We construct a functor AC(−, −) from the category of path connected spaces X with a base point x to the category of simply connected spaces. The following are the main results of the paper: (i) If X is a Peano continuum then AC(X, x) is a cell-like Peano continuum; (ii) If X is n-dimensional then AC(X, x) is (n + 1)−dimensional; and (iii) For a path connected space X, π 1(X, x) is trivial if and only if π 2(AC(X, x)) is trivial. As a corollary, AC(S 1, x) is a 2-dimensional nonaspherical cell-like Peano continuum.

Mathematics Subject Classification (2010)

Primary 54F15 55N15 Secondary 54G20 57M05 


Noncontractible compactum weak homotopy equivalence trivial shape Peano continuum Snake cone Alternating cone asphericity cell-like space Topologist sine curve 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Barrat M.G., Milnor J.: An example of anomalous singular homology. Proc. Amer. Math. Soc. 13, 293–297 (1962)MathSciNetCrossRefGoogle Scholar
  2. 2.
    K. Borsuk, Theory of Shape, Monografie Math. 59, PWN, Warsaw, 1975.Google Scholar
  3. 3.
    Bredon G.E.: Sheaf Theory, Second Ed., Graduate Texts in Math. 170. Springer, New York (1997)Google Scholar
  4. 4.
    Eda K.: Free σ-products and noncommutatively slender groups. J. Algebra 148, 243–263 (1992)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Eda K., Karimov U.H., Repovš D.: On (co)homology locally connected spaces. Topology Appl. 120, 397–401 (2002)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Eda K., Karimov U.H., Repovš D.: A construction of simply connected noncontractible cell-like two-dimensional continua. Fund. Math. 195, 193–203 (2007)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Eda K., Karimov U.H., Repovš D.: A nonaspherical cell-like 2-dimensional simply connected continuum and related constructions. Topology Appl. 156, 515–521 (2009)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Eda K., Karimov U.H., Repovš D.: On the second homotopy group of SC(Z). Glas. Mat. Ser. III 44(64), 493–498 (2009)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Eda K., Kawamura K.: Homotopy groups and homology groups of the ndimensional Hawaiian earring. Fund. Math. 165, 17–28 (2000)MathSciNetMATHGoogle Scholar
  10. 10.
    Griffiths H.B.: A note on commutators in free products. II, Math. Proc. Cambridge Philos. Soc. 51, 245–251 (1955)MATHCrossRefGoogle Scholar
  11. 11.
    Karimov U.H., Repovš D.: On suspensions of contractible compacta of trivial shape. Proc. Amer. Math. Soc. 127(2), 627–632 (1999)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Lyndon R.C., Schupp P.E.: Combinatorial Group Theory. Princeton Univ. Press, Princeton, N.J., (1971)Google Scholar
  13. 13.
    Spanier E.H.: Algebraic Topology. Springer-Verlag, Berlin (1966)MATHGoogle Scholar
  14. 14.
    Whitehead G.W.: Elements of Homotopy Theory. Springer-Verlag, Berlin (1978)MATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.School of Science and EngineeringWaseda UniversityTokyoJapan
  2. 2.Institute of MathematicsAcademy of Sciences of TajikistanDushanbeTajikistan
  3. 3.Faculty of Mathematics and Physics and Faculty of EducationUniversity of LjubljanaLjubljanaSlovenia

Personalised recommendations