Compact, simply connected $C^1$ manifold bounded between two spheres

Question

Say that we have a compact, simply connected $C^1$ manifold $M$ of dimension $n-1$ embedded in $\mathbb{R}^n$ and $c_1, c_2 > 0$ such that for $v\in M$, $c_1\leq \|v\|\leq c_2$. It seems like we should be able to say that $M$ is $C^1$-diffeomorphic to $S^{n-1}$, but I'm not sure how to formalize this. I know that if I can show that $M$ is homotopy equivalent to $S^{n-1}$, then it follows from the generalized Poincaré conjecture, but it's not even clear to me that a homotopy between $M$ and $S^{n-1}$ exists. Does anybody have any ideas, or possibly a counterexample? Thanks!

Answer 1

This isn't true. Pick an embedding of $S^2 \times S^2 \hookrightarrow \mathbb{R}^5$, pick a point $p$ not on the image and translate the origin to that point. Pick a radius smaller than the distance from the image (due to compactness, the distance is positive) and pick a radius greater than the distance from the image (which is finite also due to compactness). Then the embedding of $S^2 \times S^2$ is between those two spheres and is not diffeomorphic to $S^4$.