Definitions

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

$J\left(x,t\right) = begin\left\{cases\right\} tf\left(x/t\right), & mbox\left\{if \right\} 0 leq ||x|| < t, x, & mbox\left\{if \right\} t leq ||x|| leq 1. end\left\{cases\right\}$

Visually, you straighten it 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 $\left(x,t\right)=\left(0,0\right)$. This underlines that the Alexander trick is a PL construction, but not smooth.

General case: isotopic on boundary implies isotopic

Now if $f,gcolon D^n to D^n$ are two homeomorphisms that agree on $S^\left\{n-1\right\}$, then $g^\left\{-1\right\}f$ is the identity on $S^\left\{n-1\right\}$, so we have an isotopy $J$ from the identity to $g^\left\{-1\right\}f$. The map $gJ$ is then an isotopy from $g$ to $f$.

Some authors use the term Alexander trick for the statement that every homeomorphism of $S^\left\{n-1\right\}$ 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 $fcolon S^\left\{n-1\right\} to S^\left\{n-1\right\}$ be a homeomorphism, then

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

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