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[edit]

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 Dn that are isotopic on the boundary are isotopic.

Proof[edit]

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 \mbox{ 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), & \mbox{if } 0 \leq \|x\| < t, \\ x, & \mbox{if } t \leq \|x\| \leq 1. \end{cases}

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".

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[edit]

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 \mbox{ with } F(rx) = rf(x) \mbox{ for all } r \in [0,1] \mbox{ and } x \in S^{n-1}

defines a homeomorphism of the ball.

Exotic spheres[edit]

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

References[edit]

  • Hilden, V.L. (1989). Braids and Coverings. Cambridge University Press. ISBN 0-521-38757-4.