Given any finitely-presented group $G$, there are a few equivalent techniques for constructing smooth/PL 4-manifolds $M$ such that $\pi_1 M$ is isomorphic to $G$.  For most constructions of these 4-manifolds, they embed naturally in $S^5$ (as the boundary of regular neighbourhoods of $2$-complexes in $S^5$.)  

Question: Are there are any smooth/PL 4-dimensional submanifolds $M$ of $S^4$ such that $\pi_1 M$ has an unsolvable word problem?  $M$ would of course have to be a smooth $4$-manifold with non-empty boundary.

I'm aware there are several constructions and obstructions to $2$-complexes embedding in $S^4$.  Moreover, I've heard some of the construction techniques fall into the tame topological world and may not be smoothable.  The condition given by Kranjc (that $H^2$ of the 2-complex is cyclic) is generally a non-computable condition for a group with non-solvable word problem. Although, perhaps there are many groups with non-solvable word problem and $H^2$ trivial.  The closest to references on the subject that I know:

M. Kranjc, "Embedding a 2-complex K in R^4 when H^2(K) is a cyclic group," Pac. J. Math. 150 (1991), 329-339. 

A. Shapriro, "Obstructions to the imbedding of a complex in Euclidean space, I. The first obstruction," Ann. of Math., 66 No. 2 (1957), 256--269.