Advertisement

Mediterranean Journal of Mathematics

, Volume 10, Issue 3, pp 1573–1589 | Cite as

Realizations of Countable Groups as Fundamental Groups of Compacta

  • Žiga Virk
Article

Abstract

It has been an open question for a long time whether every countable group can be realized as a fundamental group of a compact metric space. Such realizations are not hard to obtain for compact or metric spaces but the combination of both properties turn out to be quite restrictive for the fundamental group. The problem has been studied by many topologists (including Cannon and Conner) but the solution has not been found. In this paper we prove that any countable group can be realized as the fundamental group of a compact subspace of \({\mathbb{R}^4}\). According to the theorem of Shelah [10] such space can not be locally path connected if the group is not finitely generated. The theorem is proved by an explicit construction of an appropriate space X G for every countable group G.

Mathematics Subject Classification (2010)

55Q05 

Keywords

Fundamental group compacta Peano space the realization theorem homotopical closeness 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    N. Brodsky, J. Dydak, B. Labuz and A. Mitra, Covering maps for locally path connected spaces, arXiv: 0801.4967v3.Google Scholar
  2. 2.
    Cannon J.W., Conner G.R.: The combinatorial structure of the Hawaiian earring group. Topology Appl. 106, 225–271 (2000)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Cannon J.W., Conner G.R.: On the fundamental groups of onedimensional spaces. Topology Appl. 153, 2648–2672 (2006)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    G. R. Conner and D. Fearnley, Fundamental groups of spaces which are not locally path connected, preprint (1998).Google Scholar
  5. 5.
    J. Dydak, and Ž. Virk, An alternate proof that the fundamental group of a Peano continuum is finitely presented if the group is countable, Glas. Mat. Ser. III, 46 (2011).Google Scholar
  6. 6.
    S. Ferry, Homotopy, simple homotopy and compacta. Topology 19 (1980), no. 2, 101-110Google Scholar
  7. 7.
    J. E. Keesling and Y. B. Rudyak, On fundamental groups of compact Hausdorff spaces, Proc. Amer. Math. Soc. 135 (2007), no. 8, 2629-2631.Google Scholar
  8. 8.
    J. Pawlikowski, The fundamental group of a compact metric space, Proc. Amer. Math. Soc. 126, no. 10, (1998), 3083-3087.Google Scholar
  9. 9.
    A. Przeździecki, Measurable cardinals and fundamental groups of compact spaces, Fund. Math. 192 (2006), no. 1, 87-92.Google Scholar
  10. 10.
    S. Shelah, Can the Fundamental (Homotopy) Group of a Space be the Rationals?, Proc. Amer. Math. Soc. 103, no. 2, (1988), 627-632 .Google Scholar
  11. 11.
    Virk Ž.: Small loop space. Topology Appl. 157, 451–455 (2010)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.Faculty of Mathematics and PhysicsUniversity of LjubljanaLjubljana 100Slovenia

Personalised recommendations