Alexander's trick

Two homeomorphisms of the n-ball which agree on the boundary sphere are isotopic

Alexander's trick, also known as the Alexander trick, is a basic result in geometric topology, named after J. W. Alexander.

Statement

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

More generally, two homeomorphisms of $D^{n}$ {\displaystyle D^{n}} that are isotopic on the boundary are isotopic.

Proof

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Base case: every homeomorphism which fixes the boundary is isotopic to the identity relative to the boundary.

If $f\colon D^{n}\to D^{n}$ {\displaystyle f\colon D^{n}\to D^{n}} satisfies $f(x)=x{\text{ for all }}x\in S^{n-1}$ {\displaystyle 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}}

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

The subtlety is that at $t=0$ {\displaystyle t=0}, $f$ {\displaystyle f} "disappears": the germ at the origin "jumps" from an infinitely stretched version of $f$ {\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 $(x,t)=(0,0)$ {\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 $f,g\colon D^{n}\to D^{n}$ {\displaystyle f,g\colon D^{n}\to D^{n}} are two homeomorphisms that agree on $S^{n-1}$ {\displaystyle S^{n-1}}, then $g^{-1}f$ {\displaystyle g^{-1}f} is the identity on $S^{n-1}$ {\displaystyle S^{n-1}}, so we have an isotopy $J$ {\displaystyle J} from the identity to $g^{-1}f$ {\displaystyle g^{-1}f}. The map $gJ$ {\displaystyle gJ} is then an isotopy from $g$ {\displaystyle g} to $f$ {\displaystyle f}.

Radial extension

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

{\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.