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 $S^$ 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 $f(x) = x \text x \in S^$, then an isotopy connecting ''f'' to the identity is given by

: $J(x,t) = \begin tf(x/t), & \text 0 \leq \, x\, < t, \\ x, & \text t \leq \, x\, \leq 1. \end$

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 $S^$, then $g^f$ is the identity on $S^$, so we have an isotopy $J$ from the identity to $g^f$. 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 $S^$ 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 $f\colon S^ \to S^$ be a homeomorphism, then
: F\colon D^n \to D^n \text F(rx) = rf(x) \text r \in ,1\text x \in S^ 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

*
* {{cite journal, first=J. W., last= Alexander, authorlink=James Waddell Alexander II, title=On the deformation of an ''n''-cell, journal= Proceedings of the National Academy of Sciences of the United States of America , volume=9, issue=12 , year=1923, pages= 406–407, doi=10.1073/pnas.9.12.406, pmid= 16586918, pmc= 1085470, bibcode=1923PNAS....9..406A, doi-access=free

Geometric topology
Homeomorphisms