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Let $D^2 \subset \mathbb{R}^2$ the unit disk and $f: D^2 \rightarrow D^2$ a homeomorphism with the property that $f$ restricted to the boundary $\partial D^2$ is the identity. Then $f$ is ambient isotopic to the identity.

I know the Annulus Theorem and I can use it to ambient isotope $f$ to the identity on any circle inside $D^2$, but I have no clue how to extend it such that it turns f to the identity on an open set around this circle or even construct the isotopy that works for the whole $D^2$.

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up vote 2 down vote accepted

Subdivide the disk $D^2$ into an annulus and a smaller disk $D_r^2$ of radius $r$. Now let $f_r$ be the identity on the annulus. On the smaller disk take $f_r$ to be a rescaled $f$.

Now $f_0$ is the identity and $f_1$ is f and $f_r$ is an isotopy.

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