Direct stiffness method: Difference between revisions

Revision as of 03:01, 23 January 2014

Alexander's trick, also known as the Alexander trick, is a basic result in geometric topology, named after J. W. Alexander.

Statement

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

Base case: every homeomorphism which fixes the boundary is isotopic to the identity relative to the boundary.

If $f : D^{n} \to D^{n}$ satisfies $f (x) = x for all x \in S^{n - 1}$ , then an isotopy connecting f to the identity is given by

J (x, t) = {\begin{cases} t f (x / t), & if 0 \leq ‖ x ‖ < t, \\ x, & 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 $f, g : D^{n} \to D^{n}$ 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 $g J$ 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 $f : S^{n - 1} \to S^{n - 1}$ be a homeomorphism, then

F : D^{n} \to D^{n} with F (r x) = r f (x) for all r \in [0, 1] 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.

References

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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]].
+==Statement==
+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]].
+More generally, two homeomorphisms of ''D''<sup>''n''</sup> 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 <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
+:<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>
+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".
+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.
+'''General case''': isotopic on boundary implies isotopic
+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>.
+==Radial extension==
+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>.
+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.
+Concretely, let <math>f\colon S^{n-1} \to S^{n-1}</math> be a homeomorphism, then
+:<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>
+defines a homeomorphism of the ball.
+===[[Exotic sphere]]s===
+The failure of smooth radial extension and the success of PL radial extension
+yield [[exotic sphere]]s via [[exotic sphere#Twisted spheres|twisted spheres]].
+==References==
+{{reflist}}
+*{{cite book |last=Hilden |first=V.L. |title=Braids and Coverings |year=1989 |publisher=Cambridge University Press |isbn=0-521-38757-4}}
+[[Category:Geometric topology]]
+[[Category:Homeomorphisms]]
+{{unref|date=December 2007}}

Direct stiffness method: Difference between revisions

Revision as of 03:01, 23 January 2014

Contents

Statement

Proof

Radial extension

Exotic spheres

References

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Direct stiffness method: Difference between revisions

Revision as of 03:01, 23 January 2014

Statement

Proof

Radial extension

Exotic spheres

References

Navigation menu

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