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 ${\displaystyle D^{n}}$ which agree on the boundary sphere ${\displaystyle S^{n-1}}$ are isotopic.

More generally, two homeomorphisms of ${\displaystyle D^{n}}$ 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 ${\displaystyle f\colon D^{n}\to D^{n}}$ 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' ${\displaystyle f}$ 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 ${\displaystyle t>0}$ the transformation ${\displaystyle J_{t}}$ replicates ${\displaystyle f}$ at a different scale, on the disk of radius ${\displaystyle t}$, thus as ${\displaystyle t\rightarrow 0}$ it is reasonable to expect that ${\displaystyle J_{t}}$ merges to the identity.

The subtlety is that at ${\displaystyle t=0}$, ${\displaystyle f}$ "disappears": the germ at the origin "jumps" from an infinitely stretched version of ${\displaystyle f}$ 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 ${\displaystyle (x,t)=(0,0)}$. This underlines that the Alexander trick is a PL construction, but not smooth.

General case: isotopic on boundary implies isotopic

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

Radial extension

Some authors use the term Alexander trick for the statement that every homeomorphism of ${\displaystyle S^{n-1}}$ can be extended to a homeomorphism of the entire ball ${\displaystyle D^{n}}$.

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 ${\displaystyle f\colon S^{n-1}\to S^{n-1}}$ 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.

See also

• Clutching construction

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.