This question is a "sequel" to my similar questions about the fundamental group and homology. All of these questions were inspired by seeing a talk, by Tadashi Tokieda, about the interesting physics that appears in toys.
What stories, puzzles, games, paradoxes, toys, etc from everyday life are better understood after learning about cohomology?
I am teaching a short course on cohomology, from chapter three of Hatcher's book. I would like to present a collection of real-life phenomena that are greatly illuminated by actually knowing about cohomology. Ideally, I would refer back to these examples as the course progressed and explain them with the new tools the students learn.
There are some interesting examples, best explained via cohomology, given as answers to my previous questions. These include:
Impossible objects such as the Penrose tribar that exist locally, but not globally. (I would be very very happy to get a much longer, more in-depth reference.)
Here is a non-example:
- The belt trick; this relies on the fundamental group, not on cohomology.
And here is one example firmly on the border:
- Kirchhoff's laws for electrical circuits. Now, these are cohomology made flesh, but they are not quite an "everyday" example...