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Realizations of Countable Groups as Fundamental Groups of Compacta

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.

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Correspondence to Žiga Virk.

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Virk, Ž. Realizations of Countable Groups as Fundamental Groups of Compacta. Mediterr. J. Math. 10, 1573–1589 (2013). https://doi.org/10.1007/s00009-013-0274-0

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  • DOI: https://doi.org/10.1007/s00009-013-0274-0

Mathematics Subject Classification (2010)

  • 55Q05

Keywords

  • Fundamental group
  • compacta
  • Peano space
  • the realization theorem
  • homotopical closeness