Extending a homeomorphism of the open disk to the boundary.

Question

Let $D^2 = \{x \in \mathbb{R}^2 : ||x||\leq 1\}$ denote the closed disk and $int(D^2)$ denote its interior. If I have a homeomorphism $\ f: int(D^2) \rightarrow int(D^2)$ it is clear that it is not necessarily extendable to the whole disk. As an intuitive example, take a homeomorphism that rotates the disk infinitely fast, the closer you get to the boundary. What is however, if $\ f$ is isotopic to the identity? Can it be extended to $D^2$? Many thanks!

Answer 1

The map $f\colon (r,\theta)\mapsto (r,\theta+\frac1{1-r})$ that rotates faster as you approach the boundary is isotopic to the identiy, via $f_t\colon(r,\theta)\mapsto (r,\theta+\frac t{1-r})$, $0\le t\le 1$.