This question is a "prequel" to a similar question about homology. Both 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 the fundamental group and/or covering spaces?
To be more precise, I am teaching a short course on the fundamental group and covering spaces, from chapter one of Hatcher's book. I want to motivate the material with everyday objects or experiences.
Here are some examples and then some non-examples, to explain what I am after. First the examples:
$\newcommand{\RR}{\mathbb{R}} \newcommand{\SO}{\operatorname{SO}} \newcommand{\ZZ}{\mathbb{Z}}$
- The plate (or belt) trick; this is a fancy move that a waiter can make with your plate, but it is more likely to appear in a juggling show. It is "explained" by the fact $\pi_1(\SO(3)) = \ZZ/2\ZZ$.
- Tavern puzzles: before trying to solve a tavern puzzle, one should check that the two pieces are topologically unlinked. You can decide this by computing $\pi_1$ of the complement of one of the pieces, and then checking the other piece is trivial.
- The game of skill, the endless chain (also called fast-and-loose), is explained by computing winding number, ie computing in $\pi_1(\RR^2 – 0)$.
- In the woodprint Möbius Strip II the ants illustrate the orientation double-cover (an annulus) of the strip. One could also perform the usual game of cutting the Möbius strip along its core curve to demonstrate a double cover of the circle by the circle.
Noticeably missing are any real life toys/puzzles/games that rely on the idea of homotopy.
Now for the non-examples:
- Impossible objects such as the Penrose tribar that exist locally, but not globally. These can be explained via non-trivial cohomology classes. But homology and cohomology are not discussed in this course. So – no cohomology! You can find many real-life examples of cohomology discussed here.
- Winding number (in the form of linking number) also arises in discussions of DNA replication; see discussions of topoisomerase. However DNA is not an everyday object, so it is not a good example.
- There are no draws in the board game Hex. This is equivalent to the Brouwer fixed-point theorem. This example is not very good, because most people don't know the game.
Best Answer
I recently heard this puzzle from Dror Bar-Natan, and there's a nice solution using the fundamental group.
To solve it, you can first reformulate it as follows: nails correspond to punctures in the plane, and removing a nail corresponds to filling in a puncture. The fundamental group of such a space is freely generated by loops around each puncture, and filling in a puncture corresponds to quotienting by one generator. We'd like a loop that is killed in each of these quotients, and it's easy to write one down inductively using iterated commutators.