4 corrected previous oversight

Here's a problem I've found entertaining.

Is it possible to find a subset of 3-dimensional Euclidean space such that its homology groups (integer coefficients) or one of its fundamental groups contains an element of finite order?

Context: The analogous question has a negative answer in dimension 2. This is a theorem of Eda's (1998). In dimension 4 and higher, the answer is positive as the real projective plane embeds. If the subset of 3-space has a regular neighbourhood with a smooth boundary, a little 3-manifold theory says the fundamental group and homology groups are torsion-free.

edit: Due to Richard Kent's comment and the ensuing discussion, torsion in the homology has been ruled out . So any torsion in provided the fundamental group must belong to subset of $\mathbb R^3$ is compact and has the commutator subgrouphomotopy-type of a CW-complex (more precisely, if Cech and singular cohomologies agree).

3 update description because of observations in latter discussion

Here's a problem I've found entertaining.

Is it possible to find a subset of 3-dimensional Euclidean space such that its homology groups (integer coefficients) or one of its fundamental groups contains an element of finite order?

Context: The analogous question has a negative answer in dimension 2. This is a theorem of Eda's (1998). In dimension 4 and higher, the answer is positive as the real projective plane embeds. If the subset of 3-space has a regular neighbourhood with a smooth boundary, a little 3-manifold theory says the fundamental group and homology groups are torsion-free.

edit: Due to Richard Kent's comment and the ensuing discussion, torsion in the homology has been ruled out. So any torsion in the fundamental group must belong to the commutator subgroup.

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