A compact $n-1$ manifold can be embedded in $\mathbb{R}^n$ iff it can be embedded in $S^n$

Question

I would like to ask how to show that a compact $n-1$ manifold embedded in $S^n$ can be embedded in $\mathbb{R}^n$? By Alexander duality one can show that some space can not be embedded in $\mathbb{R}^n$. But I'm not sure how to decide if it can be embedded in $\mathbb{R}^n$.

Answer 1

If you can embed it in $S^n$ then the embedding must not be surjective since then it would be a homeomorphism, but such a thing is impossible since they are manifolds of different dimensions. So remove some point not in the image and you have a space homeomorphic to $\mathbb{R}^n$ in which your manifold embeds into. Note this never used compactness.