From the point of view of topology, holes are one of the key properties of the surface. If you put a loop of thread on the surface of a ball or cylinder, you can tighten it without a knot, and this space is called simply connected. With a bagel, this will not work: the hole will prevent it. It is impossible to turn figures of different linear connectivity one into another without gaps or gluings. Topological figures for which this is possible are connected by homeomorphic transformations, as when playing with a piece of plasticine. The cup and bagel are homeomorphic, the bagel and cylinder are not. But why is man homeomorphic?

Traditional Chinese medicine teaches that a person has seven holes: the entrance and exit of the gastrointestinal tract, the ears, nostrils and exits of the external genital organs. Modern anatomy thinks differently - for example, the external genitalia in men end with testicles, therefore, from the point of view of topology, they do not form holes. This is a dead end, a “hollow”, which can be eliminated by homeomorphic transformations without any gluing and tearing. Enlightenments of the female genital organs end with fallopian tubes that open into the body cavity. She also does not communicate with the external environment, making this "hole" just a "hollow."

This also applies to ears, the openings of which are closed by an airtight (normal) eardrum. But with the remaining openings, the situation is more complicated: in addition to the “entrance” and “exit” of the gastrointestinal tract, gaps starting in the nostrils approach it in the region of the nasopharynx. We still have four holes connected to each other - a difficult case. The editors of “PM” had to involve a mathematician-topologist to find out: a person is homeomorphic to a spinner. More precisely, the triple torus.

## Andrei Konyaev, Candidate of Physics and Mathematics, Associate Professor of the Department of Differential Geometry and Applications of Mechanics and Mathematics, Moscow State University

“In topology, it is not always easy to say which simplest figure a surface can be reduced to: a sphere, a torus, etc. There is no general rule in this regard, it all depends on the concrete surface and how it is defined. If we describe it with the formula (as a sphere: x2 + y2 + z2 = 1), then this task is usually very difficult. If the surface is defined by an atlas, that is, by a set of the individual figures that make it up (maps) and the rules for gluing them together, then you can find the original figure quickly enough. ”

The article “Homeomorphic little men” was published in the journal Popular Mechanics (No. 3, March 2018).