Cantor Set in Alexander Horned Sphere Construction

Question

I have seen it said in several different places that in the standard construction of the Alexander horned sphere, given by successive embeddings of a sphere with $2^n$ handles, either limited or intersected, a Cantor set shows up as a place where certain aspects of the construction have to be modified, or where otherwise something interesting happens (I know I should give a more precise description, but it is such a standard construction that that would seem somewhat superfluous). In Hatcher, this Cantor set is said to correspond to the intersection of all the handles. However, I am unable to see what this set looks like, and I definitely don't understand what makes this set special. If anyone could help explain this phenomena I would be very grateful.

Answer 1

I'll start with a cool video of an Alexander Horned Sphere being constructed for YouTube:

A YouTube construction of Alexander Horned Sphere

We can see an obvious connection to the Cantor set by using an address system to identify a 1-1 connection between torus's at different levels of the construction and the different levels of Cantor's construction.

Reading Allen Hatcher's Algebraic Topology we find the material related to the question on pages 170-171.

The key point behind Hatcher's construction is that the handles (which form the starting point for the defined homeomorphisms $h_n:B_{n-1}\to B_n$) are initially left intact. The $h_n$ remove a section from each handle of the $B_{n-1}$ ball, and build two new interlocking handles in its place. This is different from the construction in the video, which is purely constructive.

Next a map $f$ is defined as $f:B_0\to \mathbb{R^3}$, where $f=\lim_\limits{n\to\infty}{h_n h_{n-1} \dots h_1 h_0}$. If we define an additional function $f_N=h_N\dots h_0$ we can see what happens next better.

We are looking at the points of $f_N(B_0)$ that are not in $f(B_0)$ for some $N$. These are the points that never stabilize as part of a handle. If we start with $f_1(B_0)=B_1$, remembering that the next level of handles are still intact, we have a certain set of points. At the next level, $f_2(B_0)=h_2(f_1(B_0))=B_2$, (and so we have $B_n\subset B_{n-1}$) we remove the sections from the two handles, and add four intact new interlocking handles, which is giving us a 'growing' Cantor set related to the missing points.

I'm not sure about the next bit - the image of $f_N(B_0)-B_0$, i.e. $f(f_N(B_0)-f(B_0))$, appears to be an arbitrary null point, because the points are just deleted. And anyway I don't see what the 'intersection of all the handles' means, because they don't intersect, and if you re-complete the missing section, the other handles get obliterated.