Alexander\'s trick - Citizendia

Alexander's trick, also known as the Alexander trick, is a basic result in geometric topology, named after J. W. Alexander. In Mathematics, geometric topology is the study of Manifolds and their Embeddings Low-dimensional topology, concerning questions of dimensions James Waddell Alexander II ( September 19, 1888 – September 23, 1971) was an important Topologist of the pre-WWII era and part of

1 Statement
2 Proof
3 Radial extension
- 3.1 Exotic spheres

Statement

Two homeomorphisms of the n-dimensional ball $D n$ which agree on the boundary sphere $S n - 1$ , are isotopic. Topological equivalence redirects here see also Topological equivalence (dynamical systems. In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it In Mathematics, a ball is the inside of a Sphere; both concepts apply not only in the three-dimensional space but also for lower and higher dimensions and for metric For a different notion of boundary related to Manifolds see that article "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe In Topology, two continuous functions from one Topological space to another are called homotopic ( Greek homos = identical

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 $f\colon D^n \to D^n$ 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, you straighten it out from the boundary, squeezing $f$ down to the origin. William Thurston calls this "combing all the tangles to one point". William Paul Thurston (born October 30, 1946) is an American Mathematician.

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. In Mathematics, a germ of ( continuous, differentiable or analytic) functions is an Equivalence class of (continuous differentiable 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. Piecewise linear may refer to Piecewise linear function Piecewise linear manifold

General case: isotopic on boundary implies isotopic

Now 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 $g J$ 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 $D n$ . Topological equivalence redirects here see also Topological equivalence (dynamical systems.

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. Piecewise linear may refer to Piecewise linear function Piecewise linear manifold

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

The failure of smooth radial extension and the success of PL radial extension yield exotic spheres via twisted spheres. In Mathematics, an exotic sphere is a Differentiable manifold that is Homeomorphic to the standard Euclidean n - Sphere, but not In Mathematics, an exotic sphere is a Differentiable manifold that is Homeomorphic to the standard Euclidean n - Sphere, but not In Mathematics, an exotic sphere is a Differentiable manifold that is Homeomorphic to the standard Euclidean n - Sphere, but not

Contents

Statement

Proof

Radial extension

Exotic spheres