Alexander's trick

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

More generally, two homeomorphisms of D n {\displaystyle D^{n}} {\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 f : D n → D n {\displaystyle f\colon D^{n}\to D^{n}} {\displaystyle f\colon D^{n}\to D^{n}} satisfies f ( x ) = x  for all  x ∈ S n − 1 {\displaystyle 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 ) = { t f ( x / t ) , if  0 ≤ ‖ x ‖ < t , x , if  t ≤ ‖ x ‖ ≤ 1. {\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}}} {\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} {\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} {\displaystyle t>0} the transformation J t {\displaystyle J_{t}} {\displaystyle J_{t}} replicates f {\displaystyle f} {\displaystyle f} at a different scale, on the disk of radius t {\displaystyle t} {\displaystyle t}, thus as t → 0 {\displaystyle t\rightarrow 0} {\displaystyle t\rightarrow 0} it is reasonable to expect that J t {\displaystyle J_{t}} {\displaystyle J_{t}} merges to the identity.

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

Radial extension

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

F : D n → D n  with  F ( r x ) = r f ( x )  for all  r ∈ [ 0 , 1 ]  and  x ∈ 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}} {\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

References