What are the orientable prime compact three-manifolds that can be embedded in $\mathbb{R}^4$?

Question

I am a physicist working in quantum field theory and it happens that I stumble on the problem of the title: What are the orientable compact prime three-manifolds that can be embedded in $\mathbb{R}^4$. My starting point is an orientable compact three-manifold embedded in $\mathbb{R}^4$ that I can decompose in connected sums. I am intuitibly assuming that this decomposition can be done inside $\mathbb{R}^4$and that the resulting prime manifolds are also embedded in $\mathbb{R}^4$. Any information on the higher dimensional problem, that is, prime n-manifolds embedded in $\mathbb{R}^{n+1}$ would also be much appreciated.

Answer 1

Actually, regarding connected sums, it is much more subtle than you think. On one hand, if $M_1, M_2$ are 3-dimensional manifolds each of which embeds in $\mathbb R^4$, then their connected sum also embeds. But the converse is false. For instance, lens spaces $L(p,q), p>1$, do not embed smoothly in $\mathbb R^4$, while for any coprime integers $p, q$, with $p$ odd, the connected sum of the lens spaces $L_{p,q} \# (- L_{p,q})$ embeds smoothly in $\mathbb R^4$. (Here the negative sign means the opposite orientation.) See my answer here and reference therein, as well as

Donald, Andrew, Embedding Seifert manifolds in $S^{4}$, Trans. Am. Math. Soc. 367, No. 1, 559-595 (2015). ZBL1419.57046.

All in all, the problem of smooth embeddings of closed 3-manifolds in $\mathbb R^4$ is wide-open. It is Problem 3.20 on Kirby's famous list of problems in 3-dimensional topology. Even in the case of Seifert manifolds, which is arguably, the nicest class of 3-dimensional manifolds, only partial answers are known. (And, with one exceptions, all Seifert manifolds are prime.)