If you roll out a lump of clay into a snake, then tie a string loosely around it, can you contort the ends of the snake, without actually pressing them together, so that you cannot get the string off?

You can stretch the clay into a Medusa’s head of snakes, and tangle them as you like, again without letting them touch. If you are allowed to rest the string on the surface of the clay, then you can get it off: you just slide it around and over what was an end of the original snake.

If however no point of the string may touch the snake, then you can trap the string so that it cannot be removed. You let the clay start as an amphisbaena, with heads at both ends; you let each head be a hydra’s head, sprouting two new ones from the stump when cut off; and you cut off all heads as they appear, ad infinitum. The result is a solid binary tree, as discussed in “The Tree of Life.” With the branches of the tree intertwined just so, you get what is called Alexander’s Horned Sphere.

You may object that you cannot actually do something ad infinitum. A person called James R. Meyer has this objection. His website is devoted to showing what’s wrong with mathematics and philosophy. I too sometimes take issue with contemporary professional philosophy in my blog. I like to think that I am saved from being a crank by knowing, as a mathematician, that the truth cannot be given uniquely to me. Meyer himself has a page about cranks, taking issue with how others define them. I found first his page about the horned sphere. The page shows no sign of understanding what the horned sphere is for.

The horned sphere is a topological sphere. It arises as a counterexample to a formerly conjectured three-dimensional version of the Jordan–Schönflies Theorem.

I read about that theorem as a child, or at least about the simpler form, the Jordan Curve Theorem: a simple closed curve divides the plane into an inside and an outside. I did not understand what there was to be excited about. I suppose now the point is the following. A simple closed curve is a continuous function f from the unit interval [0, 1] into ℝ2 that repeats a value precisely at 0 and 1, so that, f being the function,

f(t) = f(u) ⇔ t = u ∨ {t,u} = {0, 1}.

A simple example is g, given by

g(t) = (sin (2πt), cos (2πt)),

tracing out the unit circle; but f might not be given by a formula. How are you going to define a function h on ℝ2 so that h(x,y) = 1, if (x,y) is “inside” the curve given by f, and otherwise h(x,y) = 0? I just did define h, but only by begging the question of what inside means.

If g is as above, and f is some simple closed curve, then fg−1 is a homeomorphism between the two curves, just for being continuous in both directions. According to the Jordan–Schönflies Theorem, the homeomorphism extends to a homeomorphism from the whole plane ℝ2 to itself.

Passing to three dimensions, one may think that if there is a homeomorphism from a sphere, considered as a surface in ℝ3, to some other surface in space, then that homeomorphism should extend to a homeomorphism from the whole space ℝ3 to itself.

The horned sphere shows that one would be wrong. Here I want to work out some details of the proof. It may serve as another example of my recent theme, that mathematics is the science whose findings are proved by deduction. Topology in particular can seem to be a counterexample, since it seems to rely on physical intuition, albeit an intuition that tolerates supposed absurdities like completed infinite processes.

The horned sphere has been the inspiration of some sculptures pictured in Claire Ferguson, Helaman Ferguson: Mathematics in Stone and Bronze (Erie, Penn.: Meridian Creative Group, 1994). A friend recently gave me the book, and the book is a reason for this post.

J. W. Alexander described his construction in “An example of a simply connected surface bounding a region which is not simply connected” (Proc. N. A. S. 10, 1924). A simply connected space is one in which

  • you can carry a string from any point to any other like Ariadne, and

  • if you carry the string back to where you started from, then you can hold the two ends and draw the whole string to yourself.

In more technical language, the loop of string must be null-homotopic, meaning there is a continuous function f from the square [0, 1] × [0, 1] into the space in question such that the function t ↦ f(0, t) is the original loop, and t ↦ f(1, t) is constant.

Alexander describes his construction with words and a drawing:

The surface Σ is the limiting surface approached by the sequence Σ1, Σ2, Σ3, .. It will be seen without difficulty that the interior of the limiting surface Σ is simply connected, and that the surface itself is of genus zero and without singularities, though a hasty glance at the surface might lead one to doubt this last statement. The exterior R of Σ is not simply connected, however, for a simple closed curve in R differing but little from the boundary of one of the cells γi cannot be deformed to a point within R.

The surface Σ is the surface of our clay. To say that it is of genus zero means it has zero holes; a torus has genus one. I myself do see without difficulty that Σ will have no holes. I am not sure what Alexander means by singularities. That the exterior of Σ is not simply connected is not clear without a proof. Alexander himself confesses, at the end of his short article,

This example shows that a proof of the generalized Schönfliess theorem announced by me two years ago, but never published, is erroneous.

If he was wrong then, why is he not wrong now?

The relevant Wikipedia article, “Alexander Horned Sphere,” lists a reference on my shelves, Spivak’s Comprehensive Introduction to Differential Geometry, Volume One (2nd ed., Houston: Publish or Perish, 1979). Defining Alexander’s Σ using a drawing like his, an exercise asks the reader to show what I said was clear, that Σ has no holes. (Spivak calls the surface S and asks, “Show that S is homeomorphic to S2,” the latter being the surface, which is two-dimensional, of a sphere.) The exercise then asserts, without explicitly asking for a proof, that the S together with the the region outside it fails to be a manifold-with-boundary, though the student has shown that it would have to be one, if S were differentiable.

I pass to another of the Wikipedia references, Hocking and Young, Topology (Reading, Mass.: Addison-Wesley, 1961), which says on page 175 (where the circumflex on ĥ is a tilde in the original),

Let S be a simple closed surface in [Euclidean space] E3, that is, S is a homeomorph of S2, and let h be a homeomorphism of S onto the unit sphere S2 in E3. Is there an extension ĥ of h such that ĥ is a homeomorphism of E3 onto itself? … Alexander … gave a famous example, the Alexander horned sphere, showing that the answer must be “no” in the general case. This example is pictured [below]. We can see from the picture alone that it is quite obvious that the complement of the horned sphere is not simply connected. Since the complement of S2 in E3 is simply connected, it follows that no homeomorphism of E3 onto itself will throw the horned sphere onto S2. Note that there is a Cantor set of “bad” points on the horned sphere.

My sense is that in topology nothing is obvious. In Real Analysis II in graduate school, the professor made an “obvious” topological assertion that I questioned and later disproved with the help of a book I found: Steen and Seebach, Counterexamples in Topology. I am going to have to work out a proof of what Hocking and Young think is obvious. I am helped by yet another reference in the Wikipedia article, though a reference to which the link was dead, till I revived it: Allen Hatcher, Algebraic Topology.

We started with a lump of clay and rolled it out into a snake. Alternatively, we pulled two horns from it. Now continuing, from each horn we pull two more horns, ad infinitum.

More precisely, letting the initial lump of clay be B0, we pull two horns out of it to get B1. In general, Bn will have 2n horns, and when we make each horn into two, we get Bn+1. Each Bn is closed and bounded, hence compact.

Perhaps ordinary clay is incompressible, so that our original lump retains its volume throughout the reshaping. We prefer the volume to grow, as if we add horns at each step. Let us say that Bn has 2n horns, one for each binary sequence σ of length n. Let the corresponding horn itself be called Hσ. To obtain Bn+1, to each Hσ we attach the horns Hσ0 and Hσ1. Now we have a strictly increasing chain:

B0B1B2 ⊂ …

We may think of Bn as a balloon filled with air; when we lower the pressure outside, the balloon expands from Bn into Bn+1.

Since it depends on physical intuition, this description is perhaps not “universally valid.” We could write down equations in three variables, defining the surfaces of the Bn. Then inequalities would define the solids themselves.

There is a homeomorphism hn from Bn to Bn+1. We “could” define it precisely, but we don’t want to bother. Still there are conditions it must satisfy. For each n, we have that the composite function

hn∘ … ∘h0

is a homeomorphism from B0 to Bn+1. We want the sequence

(h0, h1h0, h2h1h0, …)

of these functions to converge uniformly to an injective function h. This is not automatic; we have to choose the hn right. The uniform convergence will imply that h itself is continuous. Every closed subset of B0 is bounded and therefore compact, so its image under h is compact and therefore closed. Thus the inverse of h is also continuous, so h is a homeomorphism onto its image.

We call that image B. Now this is homeomorphic to the original B0. Why then have we bothered to create B? We want the complement of B in space not to be homeomorphic to the complement of B0. We achieve this by attaching in two steps to Hσ the horns Hσ0 and Hσ1, for each binary sequence σ of length n, for each n:

  1. Attach a “handle” to Hσ.

  2. Cut out part of the handle, leaving the horns Hσ0 and Hσ1.

The cut-out part will include the handles that will be attached to the horns Hσ0 and Hσ1 in the next step. Thus Bn gets 2n handles attached, resulting in Xn. These form a decreasing chain of compact sets. In particular, we now have

B0B1B2 ⊂ … ⊂ B ⊂ … ⊂ X2X1X0.

It is not automatic that B is the intersection of the Xn, but we can have ensured that this will be so.








The key move really needs three dimensions, and the pictures just above don’t handle this. For each n, for each σ of length n, the handles attached respectively to the horns Hσ0 and Hσ1 should be interlocking. This is to ensure that

  • the complement of Xn+1 is not simply connected,

  • every loop in the complement of Xn that is null-homotopic in the complement of Xn+1 was already null-homotopic in the complement of Xn.

If one accepts this, then the complement of B also fails to be simply connected; for, by the compactness of X0, any loop in the complement of B that is null-homotopic must already have been null-homotopic in the complement of one of the Xn, and we know that that complement has loops that are not null-homotopic. Indeed, suppose we are given a homotopy of loops in the complement of B, namely a continuous function f from the square [0, 1] × [0, 1] into the complement of B such that f(0,0) = f(0,1) and f(1,0) = f(1,1). Let K be the image of the square under f. Then KB is empty, but is the intersection of the descending chain of closed subsets KXn of the compact set X0. Therefore, by compactness, some set in the chain must be empty. Thus f is a homotopy in the complement of some Xn+1.

2 Trackbacks

  1. By Why It Works « Polytropy on September 26, 2020 at 10:49 am

    […] « Knottedness […]

  2. By Multiplicity of Mathematics « Polytropy on October 8, 2020 at 5:54 am

    […] “Knottedness”: Topologically speaking, there is a sphere whose outside is not that of a sphere. The example is Alexander’s Horned Sphere, but it cannot actually be physically constructed. […]

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