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It is well known that every finitely presented group may be realised as fundamental group of some closed $4$-manifold.

What groups can be obtained as fundamental groups of open subsets of $R^4$? I'm also interested in the same question with $n=3$

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This seems to be open even for $\mathbb{R}^3,$ see this question and many enlightening answers thereto. See also the very nice answer to this MSE question, which gives some obstructions in $\mathbb{R}^3.$

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