To topologists, mugs and doughnuts are the same. You may be wondering: how? “Are they mad?” I hear you cry. “I can drink coffee out of a mug but I certainly cannot drink coffee out of a doughnut.” By considering the number of holes in objects, we will explore whether humans are also doughnuts—or, if not, we will figure out which object humans are the same as.
With numerous words in mathematics, the mathematical use/meaning does not align with our everyday usage of the word; examples include ring and hole. Here, the topological definition of a hole does not align with the everyday use of the word. For example, if you had dug a grave, you would probably say “there is a hole in the ground,” but to a topologist, this does not count as a satisfactory hole in the Earth’s surface.
A topologist’s view
First, let’s figure out why topologists can’t tell the difference between a mug and a doughnut. Topology is concerned with properties of geometric objects; these objects can exist in any number of dimensions. The more precise version of our statement ‘to topologists, mugs and doughnuts are the same’ is ‘a mug and a doughnut are topologically equivalent’. (If we wanted to take this one step further, topologists would use the mathematical term solid torus rather than doughnut.) One ingredient of topological equivalence is homotopy equivalence. This is when one object can be ‘deformed’ into the other, by a series of stretching, squashing, twisting and bending. Things we are not allowed to do to deform our object include cutting it, opening/closing holes, and gluing parts together.
Homotopy equivalence is not exactly the same as topological equivalence, but it is easier to work with. One of the reasons why mugs and doughnuts are homotopy equivalent is that they each have a single one-dimensional hole. The centre hole of the doughnut aligns with the hole made by the handle of a mug. To see this more clearly, imagine that you have a stretchy and inflatable mug—If you find this difficult to imagine, have a look at the cartoon at the start of this article to help you along.
Equivalence: homotopy v topological
Homotopy equivalence is when one object can be continuously deformed into another by a series of stretching, squashing, twisting and bending. Topological equivalence is a stronger condition than homotopy equivalence: it means that there is a continuous bijective map between the objects. If two objects, say, $A$ and $B$, are topologically equivalent, then each point on $A$ corresponds to exactly one point on $B$ and vice versa.
Any pair of objects that are topologically equivalent are homotopy equivalent, but it doesn’t go both ways: homotopy equivalence does not imply topological equivalence. For example, a solid torus (doughnut) and a circle are homotopy equivalent, as we can retract the solid torus to the circle. However, this retraction causes multiple points of the solid torus to map to a single point on the circle. This means that we cannot `undo’ the retraction and return to the solid torus, and so we don’t have topological equivalence.
For simplicity, we will call the part of the mug that your drink goes in the container hole.
If we keep inflating the mug, the container hole will pop out, and we’ll be left with a cylinder with a handle attached. Then, we can squish and stretch the mug, redistributing the air, until the cylinder and handle have the same thickness and it looks like a doughnut, and voila! Doughnuts and mugs are the same (or at least, homotopy equivalent).
Previously, we mentioned that digging a hole in the Earth would not be a satisfactory hole for a topologist, so let’s figure out why this is. As we did for the mug, imagine a stretchy and inflatable sphere representing the surface of the Earth. On the Earth/sphere, a ‘garden hole’ would be a small crater on the surface. After inflating the Earth slightly—and some squishing—we would return to our spherical Earth. This shows that the Earth’s surface along with garden holes is homotopy equivalent to the sphere, not the torus (Chalkdust does not endorse this as an excuse for digging holes in your neighbour’s garden).
Potatommy
Humans are three-dimensional objects, and we want to find out what they are homotopy equivalent to, so first we need a model of a body. We will begin with a simple model of a human whose only functions are eating and breathing. We will proceed to build our model as you would a Mr Potato Head, with some small (perhaps, more realistic) adaptations. We will refine this model multiple times, becoming more realistic at each iteration.
In our simplest model, we will draw our digestive system as a straight-through road, open at both ends. After some stretching and squishing, we obtain a potato-doughnut. However, we also need to breathe, so let’s factor in the respiratory system. Because humans can breathe through their noses, we will begin by modelling our respiratory system as a tube from the nose to the lung, as in the image. We will assume the nostril is always open but the tube and lungs form a closed path.
Potatommy’s digestive system is equivalent to single hole…
… and Potatommy’s respiratory system is equivalent to another single hole
In this model, our lungs act in the same way as the container hole in a mug. Then, as we did before, with our inflating, squishing and stretching abilities, we can ‘pull’ them about so that we just have a single, simple ‘breathing hole’.
However, this is an incredibly simplified and inaccurate model of the respiratory system. Of course, we can choose to breathe either through our nose or mouth. Let’s explore how, if at all, this complicates things. As we did previously, let’s draw this inside our potato, and see what we can squish and stretch it into.
On the other hand, when we modelled the digestive tract, we assumed that both ends were open. Obviously, humans do not have their mouths open all the time. The mouth closing gives us two separate container-style holes which we can pull out and then we obtain a sphere!
Another thing to take into account is that the respiratory system and digestive system are connected. For example, the epiglottis (the flappy bit in your throat that stops you from inhaling food) blocks the trachea and oesophagus from being open at the same time. However, this only happens when swallowing. First we will look at the ‘resting’ state of a human.
As before, the lungs can be squished into the tube connecting to the nose and mouth. This leaves us with a single hole, but now it has three exits (the nose, the mouth and the anus), rather than the two exits we are used to seeing.
Potatommy’s nasal and mouth passage
This changes the homotopy of our model in comparison to previous versions. For a moment, let’s see how this further changes if we look at a ‘swallowing’ potato head.
When we swallow, breathing temporarily stops, as the epiglottis covers the trachea. This closes the path from the outside world to the lungs, so we don’t accidentally inhale our food. This closing of the trachea by the epiglottis changes our potato’s homotopy quite dramatically. The lungs being cut off adds an empty sphere inside our potato and we still have the two holes we had previously: one from nose to mouth, and the other from mouth to anus. This now gets us to a point where it is difficult to count the holes and spaces we obtain inside the body.
It is also important to note that the body is much more complex than the ‘through-roads’ we have modelled. If we begin thinking about even just one part of the body in more detail, there are so many extra things that we should take into account, such as membranes and pores. In our everyday definition of a hole, we would say that pores are holes because sweat leaves them, and we may sometimes describe pores as blocked. There are millions of pores on your body and we could not feasibly see these on a model of a human, especially if we are limited to this page size and only 72 pages of the magazine. You can probably imagine some of the difficulties in this. Just going from an open mouth to a closed one completely changed our homotopy equivalence, never mind thinking about—well, let’s not go there.
Betti numbers
You might have noticed that, although most of the holes we’ve considered so far were one-dimensional, there’s a sneaky two-dimensional ‘cavity’ that represents the lungs when our potato head is swallowing. In fact, we can have holes in all sorts of different dimensions.
For example, a one dimensional hole is equivalent to circle. A hollow doughnut (AKA a rubber ring) has two one-dimensional holes and one two-dimensional hole. Because one-dimensional holes are circles, to count them, we can count how many different circular cuts we can make to the object without creating two distinct pieces. For two-dimensional holes, we can think of the number of cavities inside the object, or the number of punctures necessary for the air inside our object to escape.
One dimensional holes on a hollow torus
To classify the number of holes an object has, topologists use Betti numbers. The $k$th Betti number, $B_k$, tells us the number of $k$-dimensional holes on a topological surface. So for our hollow torus, $B_1=2$ and $B_2=1$. We can see that $B_1=2$ on the image above, and $B_2=1$ as there is only one cavity inside the doughnut, and a puncture anywhere on the surface will cause the air to escape.
You may be surprised that the two-dimensional hole of a torus is not the hole in the centre of the doughnut that we would (in everyday use) call the doughnut hole. There is another name for classifying these holes in a surface: genus.
As another example, let’s calculate the Betti numbers $B_1$ and $B_2$ of a hollow double torus. (Picture one of those two-person rubber rings you get on the really scary water slides.) We can see that the genus of the double torus is $2$ as there are two ‘doughnut holes’. It is also relatively easy for us to determine that $B_2=1$ for the double torus, as a puncture anywhere on its surface will make the air inside it escape. However, $B_1$ is a bit harder to find, as we need to find the maximum number of circular cuts we can make to our surface without making two distinct pieces. To do this, we can think of all the different types of closed loops our surface can have.
Cutting these four one-dimensional holes on a double torus would give a single piece…
…but cutting either of these would lead to having fewer than four cuts or more than one piece.
There are six different loops/circles on our double torus. The maximum number of these we can cut and still only have one piece is four. These are the two blue and two green loops in the top image above. Cutting other combinations of loops either results in fewer than four cuts or two distinct pieces.
An easier way to find $B_1$, in this case, is to know that for a closed orientable surface (such as a double torus), $B_1=2g$ where $g$ is the genus of a surface. So we quickly obtain that our genus-2 surface, the double torus, has $B_1=4$.
Topologically equivalent spaces have the same Betti numbers. So, in an attempt to see whether humans and doughnuts are ‘the same’, we can compare the Betti numbers of both objects. If they are the same, then we have some progress to showing they are topologically equivalent—but we don’t get all the way. This is because of the important distinction that, although topologically equivalent objects always have the same Betti numbers, it doesn’t work in the other direction. In other words, having the same Betti numbers is a necessary but not sufficient condition of topological equivalence.
Finally, let’s return to our ‘resting potato head’. If we make the slight change of closing the potato heads mouth, we no longer have three exits to our hole. With some slight squishing and stretching, we obtain our potato head doughnut, or equivalently, a mug. Just promise me you won’t start drinking coffee out of humans…
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