Fundamental group torsion for subsets of Euclidean 3-space
The corresponding problem in has a negative answer. The corresponding problem in for has a positive answer. If the subset of has a regular neighbourhood with a smooth boundary, the answer is negative. Similarly the homology of the subset is known to have no torsion via an Alexander duality argument. So any torsion in the fundamental group must be in the commutator subgroup.
[E] Eda, K. Fundamental group of subsets of the plane. Topology and its Applications Volume 84, Issues 1-3, 24 April 1998, Pages 283-306
* indicates original appearance(s) of problem.