Sphere inversion and Klein bottle immersion

The Sphere Inversion

December 7, 2004

page 1

**Introduction. **

We will consider closed surfaces, such as the sphere, the torus and some others. These are surfaces in the sense that an average person understands, that is, they are two-dimensional objects that are represented in a three-dimensional Euclidean space, R3, which is our mental space of representation. These surfaces can be represented in several ways. If they do not intersect themselves, we will say that they are embedded (in R3). If they do intersect, we will then speak of immersions and this intersection will then be represented by the presence of a self-intersection set (self-intersection).

In our embeddings, we will assume that the tangent plane varies continuously and that the surface is an example of regularity, like for example the tip of a cone. Our surfaces will be regular.

In the case of immersions, we will require that along the lines of self-intersection, the two tangent planes of the intersecting sheets are distinct.

The world of geometry, as conceived by the mathematician, is quite different from the physical world. The fact that surfaces can self-intersect does not bother him at all. The physical world does not allow such things. But it becomes possible in the metaphysical world. Thus, in the Bible it is written that when the dead are resurrected, they will be in the form of "glorious bodies". They will then be able to pass through anything and in principle be able to intersect themselves. Thus, when the time of the Last Judgment comes, if you walk in Rome in the form of a glorious body, and you are lost and looking for the Piazza Navona, you might be tempted to ask a resurrected mortal, who has the same appearance as you, for directions. Suppose the person you question is heading in the opposite direction relative to this square. In the ordinary physical space, he would have to turn around to point his finger in that direction. But if he is walking in the form of a glorious body, this rotation will no longer be necessary. He can point his index finger at his navel and intersect himself. When his hand reappears emerging from his back, he will only have to say to you "it's that way". By inserting his arm through his stomach, he will have created in his body a self-intersection set consisting of two circles, which will disappear when he returns to his normal configuration.

If a human being closes his mouth, puts a clothespin on his nostrils to block them, and we ignore his other natural openings, his body envelope then adopts the topology of the sphere S2. Imagine a resurrected being in the form of a glorious body whose natural openings are thus blocked. We know that he can intersect himself, that is, his body envelope can go from a situation of embedding to a situation of immersion. One of the metaphysical problems that then arose was whether a resurrected person in the form of a glorious body could turn himself inside out without making folds.

A small remark in passing. Magicians know how to use "magic circles" that can interpenetrate "magically". One could imagine representing surfaces using a sort of "magic grid" such that the two sheets, represented here in black and pink, can intersect each other without difficulty.

The Magic Grid

Anyway, one must agree that there is often not much difference between mathematics and magic. I created a comic book twenty years ago: the Topologicon. It is now out of print and unavailable, except as a collectible item. On one of the pages, you could see this:

It is a pity that Belin editions decided to abandon this collection. It must be said that with a production cost of barely more than one euro, selling the albums for 13 euros (plus shipping), sold by mail order, not only left a profit margin of 12 euros, that is, a profit exceeding 92 percent of the selling price, but it did not correspond to a very obvious business strategy, especially for black and white.

Consider a sphere S2 embedded in R3. We assume that its outer surface is gray and its interior is a pinkish color. We can press two antipodal points, which we arbitrarily call "north pole" and "south pole", until they touch at a point. You can do this, for example, with a donut. When it comes to a mathematical donut (we don't know if donuts are resurrected or not in the form of glorious bodies), the two polar regions, after having touched at a point, can self-intersect along a self-intersection curve that takes the shape of a circle. Anticipating, we will say that this surface has undergone a catastrophe of the type Do.

One might then be tempted to try to turn the donut, the sphere, by continuing the operation. But then a bulge will form, which will degenerate into an ugly fold, or more precisely, a wrinkled surface (figure d).

At the end of the 1950s, the serious question of whether one could turn a metaphysical donut without making folds remained unresolved. To be honest, everyone thought it was strictly impossible. But in 1957, a mathematician, Stephen Smale (who received the Fields Medal but for an entirely different work) proved that the different immersions of the sphere S2 in R3 formed a single set and that it was always possible to find a sequence of continuous deformations of immersions (also called regular homotopy) allowing to go from one situation to another. The corollary was that one should be able to pass, using a continuous sequence of immersions, from the standard embedding of the sphere S2 to the antipodal embedding. In simpler terms: one should be able to turn a sphere without making folds, provided one allows it to turn itself.

Smale's advisor was Raoul Bott. He asked his student how one should proceed and Smale replied that he had no idea, but that his theorem was completely unassailable. Smale could not visualize in space but he didn't care (as is the case with many geometers). And, if we are completely honest, after proving his theorem, he completely disregarded the way one could actually perform the operation and quickly moved on to another subject, leaving his mathematician colleagues in the greatest confusion. I find it not very nice to create problems like that and then leave people to find the solution ten years later.

It must be said that it is quite difficult to imagine immersions in one's head. Yet we know of surfaces that can only be represented in R3 in this way. The Klein bottle, for example.

Klein Bottle

It is represented here with a mesh-coordinate system consisting of two sets of closed curves, like the torus. Thus, one can mesh a Klein bottle without creating a mesh singularity. But as can be seen, this surface necessarily self-intersects along a closed curve, a circle. Therefore, one cannot embed a...