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| '''Alexander's trick''', also known as the '''Alexander trick''', is a basic result in [[geometric topology]], named after [[James_Waddell_Alexander_II|J. W. Alexander]].
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| ==Statement==
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| Two [[homeomorphism]]s of the ''n''-[[dimension]]al [[ball (mathematics)|ball]] <math>D^n</math> which agree on the [[Boundary (topology)|boundary]] [[sphere]] <math>S^{n-1}</math> are [[homotopy#Isotopy|isotopic]].
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| More generally, two homeomorphisms of ''D''<sup>''n''</sup> that are isotopic on the boundary are isotopic.
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| ==Proof==
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| '''Base case''': every homeomorphism which fixes the boundary is isotopic to the identity relative to the boundary.
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| If <math>f\colon D^n \to D^n</math> satisfies <math>f(x) = x \mbox{ for all } x \in S^{n-1}</math>, then an isotopy connecting ''f'' to the identity is given by
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| :<math> J(x,t) = \begin{cases} tf(x/t), & \mbox{if } 0 \leq \|x\| < t, \\ x, & \mbox{if } t \leq \|x\| \leq 1. \end{cases} </math>
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| Visually, the homeomorphism is 'straightened out' from the boundary, 'squeezing' <math>f</math> down to the origin. [[William Thurston]] calls this "combing all the tangles to one point".
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| The subtlety is that at <math>t=0</math>, <math>f</math> "disappears": the [[Germ (mathematics)|germ]] at the origin "jumps" from an infinitely stretched version of <math>f</math> 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 <math>(x,t)=(0,0)</math>. This underlines that the Alexander trick is a [[Piecewise linear manifold|PL]] construction, but not smooth.
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| '''General case''': isotopic on boundary implies isotopic
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| If <math>f,g\colon D^n \to D^n</math> are two homeomorphisms that agree on <math>S^{n-1}</math>, then <math>g^{-1}f</math> is the identity on <math>S^{n-1}</math>, so we have an isotopy <math>J</math> from the identity to <math>g^{-1}f</math>. The map <math>gJ</math> is then an isotopy from <math>g</math> to <math>f</math>.
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| ==Radial extension==
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| Some authors use the term ''Alexander trick'' for the statement that every [[homeomorphism]] of <math>S^{n-1}</math> can be extended to a homeomorphism of the entire ball <math>D^n</math>.
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| However, this is much easier to prove than the result discussed above: it is called radial extension (or coning) and is also true [[piecewise linear homeomorphism|piecewise-linearly]], but not smoothly.
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| Concretely, let <math>f\colon S^{n-1} \to S^{n-1}</math> be a homeomorphism, then
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| :<math> 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}</math>
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| defines a homeomorphism of the ball.
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| ===[[Exotic sphere]]s===
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| The failure of smooth radial extension and the success of PL radial extension
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| yield [[exotic sphere]]s via [[exotic sphere#Twisted spheres|twisted spheres]].
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| ==References==
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| {{reflist}}
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| *{{cite book |last=Hilden |first=V.L. |title=Braids and Coverings |year=1989 |publisher=Cambridge University Press |isbn=0-521-38757-4}}
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| [[Category:Geometric topology]]
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| [[Category:Homeomorphisms]]
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| {{unref|date=December 2007}}
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Andrew Simcox is the title his mothers and fathers gave him and he totally enjoys this title. He works as a bookkeeper. My wife and I live in Kentucky. To climb is something I truly appreciate performing.
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