Let us return to this class of homotopies of immersions of the torus in R³. We can easily connect the two objects shown here using a "C" transformation. We take a torus and "self-penetrate" it somewhere, creating a line of double points that forms a circle:
I have used "two colors": gray for the outside of the torus, white for the inside. The self-penetration shown above (which leads to one of the infinite possible immersions of the "standard torus") thus brings a white portion of the surface into view.
Let us observe this object from a point located on the axis of the torus:
At the top, the internal (white) portion of the torus revealed by the self-penetration. We can then perform a "C" transformation and create two cusp points.
At the point indicated by the arrow, we "constrict" a passage. This operation creates two cusp points C1 and C2:
which we can then migrate as shown below:
All that remains is to perform a C⁻¹ transformation (coalescing two cusp points):
We obtain the object:
This immersion of the torus is homotopic to the standard torus.
We see that the operations "C" and its inverse "C⁻¹", which extend the universe of immersions to that of surface foldings in R³, allow us to do interesting things. We can construct the entire set of foldings of classical surfaces (sphere, projective plane, torus, and Klein bottle). How many such classes does this set possess?
We have seen that the sphere and the projective plane belong to the same class (as do the right and left Boy surfaces). How many classes of torus foldings are there? I believe—unless I am mistaken—that this problem is not currently resolved. Can one move from one immersion class of the torus to another using C operations, or not? Intuitively, I would tend to answer no, but this is merely a
conjecture.
A construction cannot prove impossibility, but only illustrate possibility. If someone finds constructions allowing one to jump from one class to another, the theorem would be effectively proven. But the fact that no such constructions have been found does not in itself constitute a proof. The absence of proof is not proof of absence. Saying there are four classes of torus foldings in R³, or that there is only one, are both
conjectures,
mere beliefs, at present.
It so happens that Smale proved the sphere eversion was possible before Phillips provided the first construction. It could just as well have been the other way around. But who would have had the idea to undertake such an endeavor, completely against our geometric intuition?
The C transformation allows transforming a sphere into a cross-cap, then into a Boy surface, via Steiner's Roman surface. See the article. Can it transform a torus into a Klein bottle? Logically, yes, but I do not have a ready answer to this question.
In passing, why speak of the "projective plane"? The shown objects (one-sided) are finite. Souriau's answer:
- On a plane, there is "the line at infinity." We simply glue this plane along the line at infinity.
Which, as one might expect, is a closed curve.
In the Topologicon, there is a small animated drawing, a "foliation," showing how a Möbius strip with three half-twists can transform into a Boy surface. The final image shows this surface minus a disk. It suffices to add this disk to complete the surface. A Boy surface is thus a Möbius strip plus a disk. Exercise: using the tools of the Topologicon, recalculate its Euler-Poincaré characteristic (which equals 1).
Conversely, one could start with the disk and let it grow, self-penetrating, until it reattaches to the three-half-twist Möbius strip—a different construction.
I have recovered these drawings from my 55-page communication presented at the Lacanian psychoanalytic conference in Aix-en-Provence (April 4–5, 1987), dedicated to "Perversion," and included in the proceedings edited by the organizers. I will use this text in a future document titled "JPP chez Lacan."
First image: the disk in the process of contorting.
Next, the beginning of the formation of the self-intersection set:
Next figure: appearance of the triple point:
I stop shading, since the surface is about to become one-sided.
Next, the surface is ready to reattach to itself along its boundary.
Here, we have depicted the three-half-twist Möbius strip, completing the surface:
Next figure: the same Möbius strip.
Then, the complete Boy surface. Regarding the images shown in the Topologicon, one cannot say "we see it from below," as a Boy surface has neither a head nor a tail. Let us say that as it appears, we see its triple point.
Next, its self-intersection set:
You have thus witnessed, before your eyes, the plane folding back onto its "line at infinity." Hence its name: "projective plane," rather odd at first glance. Perhaps this is the first time people have seen infinity so closely.
These images were composed some twenty years ago, and this website or CD finally offers a way to display them. The reader may wonder why they did not appear in "Pour la Science" or "La Recherche." I did send articles to these journals, but the editors did not find the subject interesting.
I hope that with this "geometric toolbox," you will eagerly set about inventing countless new surfaces. Here is one imagined by Madame Ivars. Take a sphere and push in two segments of equal length in diametrically opposite directions until they touch, imagining them welded onto two rods, like this:
When the segments meet, a "surgery" takes place. There is a sheet crossing along the segment, and two conical points at each end. Next, this surface in cross-section.
The same, in perspective:
Among the tools for building surfaces, an "plumbing fitting":
When a fluid enters through an orifice serving as an inlet, it exits circulating on the outside of the outlet pipe. But one can use the outlet as an inlet. The effect is the same. The object is a kind of geometric operator transforming exterior into interior.
By closing this "fitting" onto itself, one obtains a Klein bottle:
But here is another arrangement:
The same, in perspective:
The surface is then two-sided. Which one is it?
Through a regular homotopy, a continuous sequence of immersions, we transform it into this:
You have recognized a box with a handle, which can only be opened from the inside. We can then transform the object. Simply thread a rope through the hole and pull it toward oneself:
The result is:
It is a torus.
Now it's your turn to figure out the next assembly:
made using three of these "fittings."
We will discuss, in another paper, the usefulness of these "fittings" in the case of the sphere eversion.