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

Two homeomorphisms of the n-dimensional ball $D^{n}$ which agree on the boundary sphere $S^{{n-1}}$ are isotopic.

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.

If $f\colon D^{n}\to D^{n}$ satisfies ${\displaystyle f(x)=x{\text{ for all }}x\in S^{n-1}}$, then an isotopy connecting f to the identity is given by

${\displaystyle J(x,t)={\begin{cases}tf(x/t),&{\text{if }}0\leq \|x\|<t,\\x,&{\text{if }}t\leq \|x\|\leq 1.\end{cases}}}$

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

If $f,g\colon D^{n}\to D^{n}$ are two homeomorphisms that agree on $S^{{n-1}}$, then $g^{{-1}}f$ is the identity on $S^{{n-1}}$, so we have an isotopy $J$ from the identity to $g^{{-1}}f$. The map $gJ$ is then an isotopy from $g$ to $f$.

Radial extension

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 $D^{n}$.

However, this is much easier to prove than the result discussed above: it is called radial extension (or coning) and is also true piecewise-linearly, but not smoothly.

Concretely, let $f\colon S^{{n-1}}\to S^{{n-1}}$ be a homeomorphism, then

${\displaystyle F\colon D^{n}\to D^{n}{\text{ with }}F(rx)=rf(x){\text{ for all }}r\in [0,1]{\text{ and }}x\in S^{n-1}}$ defines a homeomorphism of the ball.

Exotic spheres

The failure of smooth radial extension and the success of PL radial extension yield exotic spheres via twisted spheres.

References

• Hansen, V.L. (1989). Braids and Coverings. Cambridge University Press. ISBN 0-521-38757-4.