# On Alexander horned sphere

As I was drawing pictures for some stuff that should be done a year ago, I found this part would make a cool blog post, so here it is ^^ (well I admit that I mainly just want t show off the picture)

For kids who doesn’t know, let’s first talk a bit about what this ‘sphere’ is:

This is an embedded topological sphere in $\mathbb{R}^3$ which has non-simply connected exterior. Also, Since the surface is compact, through inversion about any point bounded away from infinity by the surface, we obtain a ‘sphere’ that bounds a non-simply connected region inside. This shows that the topology of the complement of a compact surface depends on the embedding, which is not true for embeddings of  compact $1$-dimensional manifolds in $\mathbb{R}^2$. (i.e. all Jordan curves separates the plane into two simply connected open sets, via the Jordan curve theorem)

The construction, as shown in the beautiful 2 page article by Alexander, goes as follows:

Take an ordinary sphere (stage 0), stretch and bend it like a banana so that the two ‘end caps’ are supported on a pair of parallel circles such that one lies vertically on top of the other (state 1). Next, on each cap we develop a banana shape, the banana shape on the two caps link though each other and again has their caps supported on a pair of parallel circles (stage 2).  Continue the process to add successively smaller bananas on the caps produced in the immediate preceding stage.

Claim: The limit is a topological sphere.

To see this, we build homeomorphisms from $S^2$ to each sphere in the intermediate stages. i.e. let

$h_n: S^2 \rightarrow S^2_n$

be a homeomorphism where $S^2_n \subseteq \mathbb{R}^3$ is the embedded sphere at stage $n$.

We may take $h_n$ s.t. $h_n^{-1}$ restricting to the complement of the $(n-1)$th stage caps (denoted by $C_{n-1}$) agree with $h_{n-1}^{-1}$. Hence union the maps $h_n|_{C_{n-1}}$ gives a continuous map on the complement of a Cantor set on the sphere. (Since $C_n$ is increasing and the caps gets smaller) This map can be extended continuously to the whole sphere because any neighborhood of points in the Cantor set contains pre-image of some sufficiently small cap.

The extension $h$ is injective since any two points in the Cantor set will be separated by a pair of disjoint pre-image of small caps. Since the sphere is compact, we conclude $h$ is a homeomorphism. i.e. the limiting surface is a topological sphere.

The exterior of the surface is not simply connected as a loop just outside the ‘equator’ can’t be contracted to a point. In fact, it’s also easy to show that the fundamental group of the exterior is not finately generated.

For some reason, Charles and I wanted to create a diffeotopy of from the standard sphere to an Alexander horned sphere. (with differentiability failing only at time one, and this is necessary since there can be no diffeomorphism from the sphere to the horned sphere, otherwise it would extend to a neighbourhood of the surfaces and hence the whole $\mathbb{R}^3$, but the exteriors of the two are not homeomorphic.)

The above figure is in fact a particular kind of Alexander horned sphere we needed. i.e. it has the property that each cap in the $(n+1)$th stage has diameter less than $1/2$ of that in the $n$th stage, and the distance between the parallel circles is also less than $1/2$ of that in the previous stage. Spheres at each stage is differentiable.

This would allow us to construct a diffeotopy that achieves stage $n$ at time $1-1/2^n$, the diffeotopy is of bounded speed as all horns are half as large as the pervious stage, hence once we get to the first stage with bounded speed, making all points traveling at that maximum speed would get one to the next stage using $1/2$ as much time.

However, we do not know if all horned sphere can be achieved by s diffeotopy from the standard sphere. i.e. does the property of being a ‘diffeotopic sphere’ depend on the embedding in $\mathbb{R}^3$.

Many thanks to Charles Pugh for forcing me to look at this business. It is indeed very fun~