# Alexander's trick

Alexander's trick, also known as the Alexander trick, is a basic result in geometric topology, named after J. W. Alexander.

## Statement

More generally, two homeomorphisms of Dn that are isotopic on the boundary are isotopic.

## Proof

Base case: every homeomorphism which fixes the boundary is isotopic to the identity relative to the boundary.

$J(x,t)={\begin{cases}tf(x/t),&{\mbox{if }}0\leq \|x\| Visually, the homeomorphism is 'straightened out' from the boundary, 'squeezing' $f$ down to the origin. William Thurston calls this "combing all the tangles to one point".

The subtlety is that at $t=0$ , $f$ "disappears": the germ at the origin "jumps" from an infinitely stretched version of $f$ to the identity. Each of the steps in the homotopy could be smoothed (smooth the transition), but the homotopy (the overall map) has a singularity at $(x,t)=(0,0)$ . This underlines that the Alexander trick is a PL construction, but not smooth.

General case: isotopic on boundary implies isotopic

Some authors use the term Alexander trick for the statement that every homeomorphism of $S^{n-1}$ can be extended to a homeomorphism of the entire ball $D^{n}$ .
$F\colon D^{n}\to D^{n}{\mbox{ with }}F(rx)=rf(x){\mbox{ for all }}r\in [0,1]{\mbox{ and }}x\in S^{n-1}$ 