Boundary-fixing homeomorphism group of disk is transitive on interior points

Statement

Suppose $D$ is a closed disk in $\mathbb {R} ^{n}$, and let $\partial D$ denote the boundary of $D$ ($\partial D$ is thus a sphere in $\mathbb {R} ^{n}$). Then, for any points $a,b$ in the interior of $D$ (i.e., $a,b$ in $D\setminus \partial D$), there exists a self-homeomorphism $\sigma :D\to D$ such that $\sigma (a)=b$ and $\sigma (x)=x$ for all $x\in \partial D$.

Related facts

Applications

• Connected manifold implies homogeneous: Given any two points in a connected manifold, there is a homeomorphism from the manifold to itself that sends the first point to the other.
• Euclidean implies compactly homogeneous: Given any two points $a,b\in \mathbb {R} ^{n}$, there exists a compact subset $K$ of $\mathbb {R} ^{n}$ and a homeomorphism of $\mathbb {R} ^{n}$ sending $a$ to $b$ that is the identity map outside $K$.

Proof

The reflection construction

The idea here is to consider, for every $x\in \partial D$, the line segments joining $a$ to $x$ and the line segment joining $b$ to $x$. The first line segment is reflected onto the second: the reflection occurs in such a way that ratios of lengths are preserved. In particular, $x$ gets sent to $x$ and $a$ gets sent to $b$.

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The translation construction

This construction is based on the idea that we have a homeomorphism between the interior of $D$ and $\mathbb {R} ^{n}$ that works radially: it uses a homeomorphism between $[0,1)$ and $[0,\infty )$ and scales every point along the radial line from the center of the sphere, via this homeomorphism.

The idea is to take the unique homeomorphism of the interior of $D$ that corresponds to a translation map in $\mathbb {R} ^{n}$ between the points of $\mathbb {R} ^{n}$ corresponding to $a$ and $b$ under this homeomorphism. It turns out that this homeomorphism of the interior of $D$ extends to a homeomorphism of $D$ that preserves all the boundary points of $D$.

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