Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $D^2$.

share|improve this question

1 Answer 1

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.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.