Every countable group is the fundamental group of some compact subspace of R^4

TL;DR

This paper demonstrates that any countable group can be realized as the fundamental group of a compact, path-connected subspace within four-dimensional Euclidean space, simplifying previous constructions.

Contribution

It provides a simpler construction method for embedding any countable group as a fundamental group in R^4, improving upon recent complex approaches.

Findings

Constructed a compact, path-connected subspace of R^4 with any given countable fundamental group.

Simplified the existing methods for realizing countable groups as fundamental groups in Euclidean space.

Established a new, more accessible approach to topological group realization in four dimensions.

Abstract

For every countable group G we construct a compact path connected subspace K of R^4 whose fundamental group is isomorphic to G. Our construction is much simpler than the one found recently by Virk.