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 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 satisfies, then an isotopy connecting f to the identity is given by 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". In the original 2-page paper, J. W. Alexander explains that for each t>0 the transformation J_t replicates f at a different scale, on the disk of radius t, thus as it is reasonable to expect that J_t merges to the identity. 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 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 be a homeomorphism, then

Exotic spheres

The failure of smooth radial extension and the success of PL radial extension yield exotic spheres via twisted spheres.