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 Earle-Eells result that $\pi_0 Diff(S^2)$ has precise two components, and in dimension $3$ it is the analogous theorem of Cerf. But I think it's an open problem for n=4.

Cerf (1970) "La stratification naturelle des espaces de fonctions differentiables reelles et le theoreme de la pseudo-isotopie" Inst. Hautes. Etudes Sci Publ. Math 39 5--173

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.