Fundamental group of a smooth closed 3-manifold

Question

I wanna find an example of a smooth, without boundary, 3-manifold M, such that $\pi_1(M)$ have non trivial torsion, and M can be smoothly embedded in $R^4$. Any ideas?

Answer 1

You can find a very detailed discussion of the problem of smooth embeddability of compact 3-manifold in $R^4$ in

Budney, Ryan; Burton, Benjamin A., Embeddings of 3-manifolds in $S^4$ from the point of view of the 11-tetrahedron census, ZBL07589799.

(A free arXiv version is here.)

Of interest to you:

(a) The manifold $S^3/Q_8$ (where $Q_8$ is the "quaternion group" embedded in $S^3$ as a subgroup of unit quaternions) embeds in $R^4$: Take any smooth embedding $RP^2\to R^4$. Then take the boundary of a tubular neighborhood of the image. It will be diffeomorphic to $S^3/Q_8$. The manifold $S^3/Q_8$ has fundamental group of order 8.

(b) 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 $R^4$. The fundamental group of this manifold is infinite but contains elements of order $p$. The "minus" sign means "opposite orientation."

See the discussion following Theorem 2.15 in the paper.