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EVERY COUNTABLE GROUP IS THE FUNDAMENTAL GROUP OF SOME COMPACT SUBSPACE OF $\mathbb{R}^{4}$

Part of: Homotopy groups

Published online by Cambridge University Press:  17 April 2015

ADAM J. PRZEŹDZIECKI
ADAM J. PRZEŹDZIECKI*
Affiliation:
Warsaw University of Life Sciences – SGGW, Warsaw, Poland email adamp@mimuw.edu.pl
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Abstract

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For every countable group $G$ we construct a compact path connected subspace $K$ of $\mathbb{R}^{4}$ such that ${\it\pi}_{1}(K)\cong G$. Our construction is much simpler than the one found recently by Virk.

Keywords

fundamental groupcompacta

MSC classification

Primary: 55Q05: Homotopy groups, general; sets of homotopy classes
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 92 , Issue 1 , August 2015 , pp. 145 - 148
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

References

Pawlikowski, J., ‘The fundamental group of a compact metric space’, Proc. Amer. Math. Soc. 126 (1998), 3083–3087.CrossRefGoogle Scholar
Virk, Ž., ‘Realizations of countable groups as fundamental groups of compacta’, Mediterr. J. Math. 10 (2013), 1573–1589.CrossRefGoogle Scholar