3d analog to the Klein bottle

Question

The Klein bottle is a non-orientable 2-manifold, which can be immersed in 3-space, but requires 4-space to be embedded. A thick version ("glass") version of the Klein bottle is a 3-manifold with boundary, with the same properties. Is there a non-orientable analog of the Klein bottle that is a non-orientable 3-manifold, which can be immersed in 4-space, but requires 5-space for embedding? Does the "glass" version have the same properties?

Answer 1

One can generalize the Klein bottle $K^2$, using its description as the quotient of $S^1 \times [0,1]$ identifying $(x,1) \sim (r(x),0)$ using an orientation reversing homeomorphism $r : S^1 \to S^1$. More generally, you could define $K^n$ to be the quotient of $S^{n-1} \times [0,1]$ identifying $(x,1) \sim (r(x),0)$ again using an orientation reversing homeomorphism. This has all the features that you ask for (with pretty much the same proofs), plus an additional feature you did not mention, namely that $K^n$ is a compact connected manifold.