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Alexander's Trick




: F:D^{n+1} o D^{n+1} \mbox{ with } F(rx) = rf(x) \mbox{ for all } r \in {Link without Title} \mbox{ and } x \in S^n
defines a homeomorphism of the ball.

A similar result, which is likewise named ''Alexander's trick'', states two homeomorphisms on ''D''''n''+1 which agree on the boundary, are Isotopic . This follows from the fact that every homeomorphism which fixes the boundary is isotopic to the identity relative to the boundary. In fact, if ''f'' : ''D''''n''+1 → ''D''''n''+1 satisfies ''f''(''x'') = ''x'' for all ''x'' ∈ ''S''''n'', then an isotopy connecting ''f'' to the identity is given by