Does there exist any space with $S^n$ as a covering space, that addmits immersion in $\mathbb{R}^{n+1}$?

Question

Intuitively, it seems the only space covered by $S^n$ that admits immersion into $\mathbb{R}^{n+1}$ is itself. A necessary condition is that the product of the tangent bundle with $\mathbb{R}$ be parallelizable. Is there a rigorous proof of the general case? (or a counter-example) .

Answer 1

Every oriented 3-dimensional manifold $M^3$ admits a (smooth) immersion in $E^4$. This is a corollary of two things:

1. $TM^3$ is trivial, see e.g. here. Hence, $T(M\times (0,1))$ is also trivial.

2. Hirsch-Smale theory which shows that a smooth parallelizable $n$-dimensional manifold admits an immersion in $E^n$. Apply this to $M^3\times (0,1)$.

Now, take any manifold (necessarily orientable!) which is covered by the 3-sphere, say a lens space or the Poincare homology sphere.