Direct stiffness method

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

$J(x,t)=\{\begin{array}{ll}tf(x/t),& \text{if }0\le \Vert x\Vert <t,\\ x,& \text{if }t\le \Vert x\Vert \le 1.\end{array}$

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 $f,g:D^n\to D^n$ 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 $f:S^{n-1}\to S^{n-1}$ be a homeomorphism, then

$F:D^n\to D^n\text{ with }F(rx)=rf(x)\text{ for all }r\in [0,1]\text{ and }x\in S^{n-1}$

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

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