Realizations of Countable Groups as Fundamental Groups of Compacta

TL;DR

This paper demonstrates that any countable group can be realized as the fundamental group of a compact subspace in four-dimensional Euclidean space, expanding understanding of group realizations in topology.

Contribution

It proves that all countable groups can be realized as fundamental groups of compact subspaces in four-dimensional Euclidean space, addressing an open question.

Findings

Any countable group can be realized as a fundamental group of a compact subspace in 4D Euclidean space.

Such spaces are generally not locally path connected if the group is not finitely generated.

This work complements previous studies on group realizations in compact Hausdorff spaces.

Abstract

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 .