As determined here, there exist spaces covered by $S^n$ that can be immersed in $R^{n+1}$. However, the existence of an embedding is a stronger condition. I suspect it to be impossible.
Is there some sort of "trick"/easy proof?
What I've calculated so far:
For any space $M$ covered by the sphere, we have $H_{0}\cong H_{n}\cong\mathbb{Z}$.
All other homology groups are are torsion groups $T_m$, with $k*g=0,\forall g\in T_{m}$, where $k$ is the degree of the map from $S^n$ to $M$.
If $M$ is embeddable, then there exists $N$ such that $\partial N=M$.
With use of the Alexander duality theorem, we can determine the relationships between the homology groups of $M$ and those of $N$:
$T_{k}(M)\cong T_{k}(N)\times T_{n-k-1}(N)$, where $T_k$ represents the torsion-subgroup of $H_k$ .
Also, $H_k(N)=T_K(N)$ for $k>0$.
Additional : If $M$ admits an embedding, then it must admit an immersion. $M$ admits an immersion if and only if $M\times (0,1)$ is parallelizable.
