Alexander's trick

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 f\colon 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{cases} tf(x/t), & \text{if } 0 \leq \|x\| 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 t\rightarrow 0 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 f,g\colon 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\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} 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==