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

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". In the original 2-page paper, J. W. Alexander explains that for each $t>0$ the transformation $J_{t}$ replicates $f$ at a different scale, on the disk of radius $t$ , thus as $t\rightarrow 0$ it is reasonable to expect that $J_{t}$ merges to the identity.

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

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, Vagn Lundsgaard (1989). Braids and coverings: selected topics. London Mathematical Society Student Texts. Vol. 18. Cambridge: Cambridge University Press. doi:10.1017/CBO9780511613098. ISBN 0-521-38757-4. MR 1247697.
Alexander, J. W. (1923). "On the deformation of an n-cell". Proceedings of the National Academy of Sciences of the United States of America. 9 (12): 406–407. Bibcode:1923PNAS....9..406A. doi:10.1073/pnas.9.12.406. PMC 1085470. PMID 16586918.

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