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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.

Short version:

What stories, puzzles, games, paradoxes, toys, etc from everyday life are better understood after learning about cohomology?

Long version:

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:

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:

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    $\begingroup$ For the Penrose triangle, how about Chris Mortensen, Peeking at the Impossible, Notre Dame J. Formal Logic Volume 38, Number 4 (1997), 527-534. doi.org/10.1305/ndjfl/1039540768 ? $\endgroup$ – David Roberts May 10 at 7:26
  • $\begingroup$ Ah, very good. I'll add a link to one of Penrose's articles. $\endgroup$ – Sam Nead May 10 at 7:36
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A very similar question was already asked on this site: Teaching homology via everyday examples. One of the answers mentioned the book Robert Ghrist, Elementary applied topology which has a lot of everyday examples of cohomology (and many other everyday examples).

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  • $\begingroup$ Yes - I asked that question and I also linked to it in the first line of this question. :) $\endgroup$ – Sam Nead May 11 at 11:40
  • $\begingroup$ @Sam Nead: anyway, I found this book only recently, and I added the reference to my old answer and to this one. $\endgroup$ – Alexandre Eremenko May 11 at 12:24
  • $\begingroup$ Thank you for adding the reference! $\endgroup$ – Sam Nead May 11 at 15:17

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