Realizations of Countable Groups as Fundamental Groups of Compacta

Details Realizations of Countable Groups as Fundamental Groups of Compacta Realizations of Countable Groups as Fundamental Groups of Compacta

2008-10-20

arxiv.org/abs/0810.3412v2

Mathematics

Geometric Topology

14 pages, 7 figures; Added extended Abstract and Introduction

Scientific paper

It is an open question (Pawlikowski) whether every finitely generated group can be realized as a fundamental group of a compact metric space. In this paper we prove that any countable group can be realized as the fundamental group of a compact subspace of four dimensional Euclidean space. According to theorems of Shelah (see also Pawlikowski) such space can not be locally path connected if the group is not finitely generated. This constructions complements realization of groups in the context of compact Hausdorff spaces, that was studied by Keesling and Rudyak, and Przezdziecki .

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