Are any pair of homotopic diffeomorphisms isotopic? Let $f_0, f_1:M\to N$ be two $C^k$ diffeomorphisms homotopic via a continuous homotopy $H_t$. We can approximate the homotopy by a smooth homotopy $\tilde{H}_t$ rel $\{0,1\}$. 
If $M$ is compact, then since $\text{Diff}^k(M,N)$ is open in $C_S^k(M,N)$, it follows that $\tilde{H}_t \in \text{Diff}^k(M,N)$ if $t$ is sufficiently close to $0$ or $1$ since $\tilde{H}_0 = f_0, \tilde{H}_1 = f_1$. I think compactness of $M$ is needed in this last step, since then the weak and strong topologies agree. 
Is it possible to choose the homotopy $\tilde{H}_t$ so that $\tilde{H}_t \in \text{Diff}^k(M,N)$ for all $t \in [0,1]$? Does the answer depend on whether or not $M$ is compact?
 A: Here are few things to know:


*

*If two homeomorphisms (diffeomorphisms) of closed 2-dimensional manifolds are homotopic then they are isotopic (smoothly isotopic); this is due to D.B.A.Epstein, 1966. 

*If you do not assume that the surface is closed then the following is a counter example: $f(z)=1/\bar{z}$, the inversion of the punctured complex plane ${\mathbb C}^*$. It is homotopic but not isotopic to the identity map.

*For closed aspherical 3-dimensional manifolds, homotopic homeomorphisms (diffeomorphisms) are isotopic (smoothly isotopic). A reference for this is a bit painful to trace, since it is scattered between several papers (Haken case is due to Waldhausen, I think; hyperbolic case is due to Gabai, etc.), the hardest part of the proof is Perelman's Geometrization Theorem.    

*For general closed 3-dimensional manifolds, homotopy does not imply isotopy, see  J. L. Friedman, D.M. Witt, Homotopy is not isotopy for homeomorphisms of 3-manifolds,  Topology 25 (1986), no. 1, 35–44. (The same in the smooth category.)  

*Probably the most famous example is in dimension 6, due to Milnor: There are diffeomorphisms of 6-dimensional sphere which are continuously isotopic but not smoothly isotopic.  (Gluing two copies of the 7-ball via such diffeomorphisms results in exotic 7-spheres.)  
