Alexander's trick

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Alexander's trick, also known as the Alexander trick, is a result in topology, a discipline within mathematics. It is named after J. W. Alexander. It states that two homeomorphisms of the n-dimensional ball Dⁿ which agree on the boundary sphere S^n-1, are isotopic, or more generally, that two homeomorphisms of Dⁿ that are isotopic on the boundary, are isotopic.

[edit] Proof

The key is that every homeomorphism which fixes the boundary is isotopic to the identity relative to the boundary. In fact, if f : Dⁿ → Dⁿ satisfies f(x) = x for all x ∈ S^n-1, then an isotopy connecting f to the identity is given by

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

Now if f,g: Dⁿ → Dⁿ are two homeomorphisms that agree on S^n-1, then g^-1f is the identity on S^n-1, so we have an isotopy J from the identity to g^-1f. The map gJ is then an isotopy from g to f.

[edit] Homeomorphisms of spheres

There is a lovely corollary to the trick: Let f be any homeomorphism of the n-1 sphere. Now take a pair of n balls and glue their boundaries via f. The resulting space, called a twisted sphere by Milnor, is homeomorphic to the n sphere. This is a small step in his discovery of twisted spheres which are homeomorphic to the n sphere but not diffeomorphic.

[edit] Note

It seems that 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 Dⁿ. Indeed, let f : S^n-1 → S^n-1 be a homeomorphism, then