Isotopic diffeomorphisms of the sphere

Question

Assume that $f:\mathbb{S}^n\to\mathbb{S}^n$ is a diffeomorphism and assume that there is an orientation preserving diffeomorphism $F:\mathbb{R}^{n+1}\to\mathbb{R}^{n+1}$ such that $F|_{\mathbb{S}^n}=f$. As far as I know it is well known that this implies that $f$ is isotopic to the identity $\operatorname{id}:\mathbb{S}^n\to\mathbb{S}^n$ though diffeomorphisms of $\mathbb{S}^n$.

While it can easily be proved that $f$ is isotopic to $\operatorname{id}:\mathbb{S}^n\to\mathbb{S}^n$ through embeddings of $\mathbb{S}^n$ to $\mathbb{R}^{n+1}$, I do not see how to turn these embedings into isotopy of diffeomorphisms of the sphere. Can someone explain it to me or give a reference?

I also know that in general if $\mathbb{S}^n$ is replaced by other manifolds, then the answer is negative, see Easiest example where pseudo-isotopy fails to be the same as isotopy?

Answer 1

This is Cerf's pseudoisotopy-implies-isotopy theorem.

Cerf's result is true in high dimensions, while it's independently known in a low-dimensional range. In dimension $2$ it goes back to the Earle–Eells result that $\pi_0\, \hbox{Diff}(S^2)$ has precisely two components, and in dimension $3$ it is the analogous theorem of Cerf. But I think it's an open problem for $n=4$.

Jean Cerf (1970), La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie, Inst. Hautes. Etudes Sci Publ. Math 39:5–173.