Which closed 3-manifolds can be embedded in $R^4$? I wonder which closed orientable 3-manifolds can be embedded in $\mathbb R^4$ and which in $\mathbb R^5$. Is there a way to determine whether given closed 3-manifold, obtained, say by Dehn surgery on knot, can be embedded into $R^4$ ?
Is the answer known for spherical 3-manifolds (finite fundamental group) ?
I am mainly interested in topological properties of manifolds. The known answers for PL or smooth embedding are of value as well for me. I appreciate any kind of answer. I am not interested in such peculiarities as exotic $R^4$.
 A: There is an analogy to surfaces in a sense.  For 3-manifolds that fibre over surfaces there is a complete answer.  For a variety of Seifert-fibred manifolds there are complete answers -- but not all.  For example, Seifert-fibred homology spheres are still problematic.   The preprint that Ian linked to in his comments has much more results of this kind in it.
At present, in summary:

*

*We likely do not have a complete set of invariants that obstruct embedding into $\mathbb R^4$.


*We appear to be far from knowing all the "natural" constructions of embeddings of 3-manifolds into $\mathbb R^4$ for the manifolds that are known to embed.
It is quite possible there are elements of formal logic obstructing both 1 and 2.
For example, if a compact boundaryless connected 3-manifold embeds in $S^4$ it separates it into two components.  It is possible that one or even both of these components has a fundamental group with an unsolvable word problem.  This would restrict the kinds of techniques one could use for creating obstructions in (1).
edit: I see Agol and Freedman's paper on this topic as connected to this last concern.  2-manifolds in $S^3$ have the Fox re-embedding theorem.  So you could hope for some nice re-embedding theorems for $3$-manifolds in $S^4$.  You shouldn't expect too nice a re-embedding theorem in $S^4$, since the tool that makes Fox's theorem work is Dehn's lemma, and the analogies to Dehn's lemma in 4-manifold theory are generally not true.
A: Results on the subject seem spotty, but the last of them seems to be:  
MR3271270 Reviewed 
Donald, Andrew(4-GLAS-SMS)
Embedding Seifert manifolds in S4. (English summary) 
Trans. Amer. Math. Soc. 367 (2015), no. 1, 559–595. 

