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.