I'll assume that the subset is compact.

Then, if you use Cech cohomology, [Alexander duality][1] turns this into a question about the complement, which is a 3-manifold.

So, I answer with another question:  Can a (wild) open submanifold of the 3-sphere have torsion in its homology? (My guess is no. But then I'm not RH Bing.)


  [1]: http://en.wikipedia.org/wiki/Alexander_duality