Alexander’s horned sphere is a wild construction that has become one of the symbols of topology.

Today I want to talk to you about the most beautiful of the sciences – topology. We will start with a very trivial reasoning (behind the simplicity of the formulation of which lies a whole abyss), and end with the construction and study of an amazing object – Alexander’s horned sphere. So let’s go!

Jordan’s theorem

Here I will show only the simplest formulation of the theorem:

It is clear from the figure that if there is a closed curve that does not have self-intersections, then it divides the plane into two regions, and for any two points in one of the regions there is a piecewise broken line that connects them without intersecting the closed curve. It will not work to connect two points from different areas with such a line.

A more general statement, called the Schoenflies theorem, states that the two regions into which a plane is divided by a closed curve are arranged in the same way.

Sameness of areas

Here some excursus into topology is necessary. One can speak about the “identity” of two topological objects when each of them can be obtained from the other by means of a homeomorphism – a continuous reversible one-to-one transformation. For example, consider several triangles on a plane:

All of them are homeomorphic, i.e. topologically equivalent: from each of these triangles, another can be obtained by stretching or squeezing the sides. It is also obvious that any closed figure without self-intersections, having more or no corners at all (for example, a circle), can be obtained by continuous reversible transformations from the triangles in the figure above.

Distances, angles do not interest us. By the way, similarity is just a special case of homeomorphism.

One of the main tasks of topology is the classification of homeomorphic objects. Somewhere this can be done simply, as in the figure above: explicitly present a sequence of actions that do not violate the rules of the game.

Topological and homotopy equivalents

However, in most cases, the direct construction of a homeomorphism is difficult, especially when it comes to spaces of higher dimensions. Therefore, topologists have developed the so-called. topological invariants are some characteristics of objects that do not change under homeomorphisms. A characteristic of a space that is preserved under a homeomorphism. In other words, to prove that objects (spaces) are homeomorphic means to present a certain value (invariant) that is the same for them. The most obvious topological invariant is the connection.

A connected space cannot be represented as the union of two or more non-intersecting non-empty open subsets. The set A is connected, B is not connected Connectedness is a generalization of the concept of linear connectivity. In a path-connected space, any two points can be connected by a continuous curve

A path-connected space is said to be simply connected if every loop on it contracts to a point. Without going into strict formulations, I will show what this means in the figure:

Any loop inside curve C shrinks to one point, and in the outer part there is a loop that does not shrink (if you “girdle” area C)

Simply connected is no longer a topological invariant, but a weaker one, the so-called. homotopic. If the spaces X and Y are homeomorphic, then they are homotopy equivalent; the converse is not true in general.

For example, a circle is homotopically equivalent to a donut (solid torus), but not homeomorphic to it

For example, the Euler characteristic (homotopy invariant) of a circle and a solid torus coincide, but they cannot be obtained from each other by continuous deformations (they are not homeomorphic)

For example, Euler characteristic (homotopy invariant) the circle and the solid torus coincide, but they cannot be obtained from each other by continuous deformations (they are not homeomorphic)

It is the homotopy equivalence that is meant by the phrase “they are arranged in the same way.”

So, the Schoenflies theorem on the plane states that the two regions into which any closed curve divides the plane are arranged in the same way, regardless of the shape of the curve.

IN space the exterior of each closed surface (in fact, a sphere) does not have non-contractible loops (after all, if you put a loop on a sphere, then thanks to the third dimension, it can be easily removed). It would seem that this is true for any sphere … however, problems immediately arose with the proof for at least three dimensions. Something told the mathematicians that the generalization of the theorem was wrong.

James Alexander and his “wild” sphere

James Alexander, an American mathematician known for his contributions to topology and differential geometry, proposed a counterexample to Schoenflies’ conjecture in 1924.

By the way, James was married to the Russian Natalya Levitskaya.  They had two children.

By the way, James was married to the Russian Natalya Levitskaya. They had two children.

Let us go through all the stages of constructing this counterexample. To begin with, let’s take a disk and “pull out” two horns from it:

Now let’s look at the surfaces of these two disks and pull out a couple more horns from them, crossing them with a “lock”:

Further actions will be repeated indefinitely. Thus, in the first step we will get 2 horns, then 4, 8, and so on. So we built it.

Now let’s see, firstly, why this thing is called “Alexander’s horned sphere”, and secondly, why it is a counterexample to Schoenflies’ conjecture. If everything is clear with “hornedness” (just an epithet), then the term “sphere” requires clarification. The key here is to understand that none of the “horns” touch each other.

Yes, they can approach to a distance, obviously less, of any given value, but still! This means that we can turn everything back – i.e. our deformation is a homeomorphism (no gluings, no breaks), which means we have before us a special, but still a sphere! Since we have a sphere, then it has the property of simply connectedness:

And what about the appearance of Alexander’s horned sphere? Can we present a loop that cannot be contracted to a point? In other words, can we prove that the interior and exterior of a sphere are not homotopically equivalent? Turns out it’s easy and simple!

The figure above shows examples of three loops that cannot be pulled together. Sooner or later, when trying to remove this loop from the Alexander sphere, we will stumble upon one of its horns, because we need to get as close to them as we like, closer than any preset value! And there the jungle is something like this:

Consequently, the exterior of the Alexander sphere and its interior are arranged otherwisethan the exterior and interior of a closed region on the plane, which refutes the Schoenflies conjecture! Thank you for your attention!

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