# Teaching the fundamental group via everyday examples

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.
• Issues like orientability have nice explanations using the fundamental group. Say, take an embedding of an $I$-bundle over $S^1$ in $\mathbb R^3$. If the base circle (0-section) is unknotted you could explain the isotopy-classes of such embeddings using various fundamental group computations, so people can see whether or not a twice-twisted annulus is isotopic to a 0-twisted annulus, or a -2-twisted annulus. Oct 21, 2014 at 20:26
• If you try making or solving a crossword puzzle on the surface of a cube (rather than a planar set), you will notice that the orientation of letters becomes wrong when you "write around a corner". People naturally try to keep the letters locally oriented although global orientation is not possible. Global orientation is possible if you solve the puzzle on the covering space of the surface with corners removed. (In general, when I write on something that has corners I imagine writing on the covering space.) Oct 21, 2014 at 20:46
• Should this be Community Wiki? If applications of Brouwer fixed-point theorem are allowed, you can use the one about placing a map on a table, or others from this similar thread: mathoverflow.net/questions/19272/… Oct 21, 2014 at 20:47
• Phone/tablet games that require the player to manipulate an array of colored/labeled items by "pulling" them past each other appear to be a good way to motivate homotopy. Oct 21, 2014 at 21:32
• No-draw for Hex is only equivalent to the Brouwer fixed-point theorem in a weak sense. Proving the no-draw theorem is as easy without the Brouwer theorem as it is with the Brouwer theorem, even if the latter is assumed as known. Oct 22, 2014 at 15:18

I recently heard this puzzle from Dror Bar-Natan, and there's a nice solution using the fundamental group.

There are $n$ nails arranged in a line on a wall. Find a way of hanging a picture from these nails so that if any 1 nail is removed, then the picture will fall.

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.

My favorite example is this. You want to hang a picture (to which a piece of rope is attached in the usual way) on a wall. There are two nails drawn in the wall, close to each other. One has to hang it with the following conditions:

a) The picture must be hanging (does not fall).

b) When one nail is removed (any one), the picture falls.

When you solve this, generalize to n nails.

I asked this question many people from 10 years olds to participants of a conference in low dimensional topology. The average time required to find a solution is about the same, and it is strictly greater than $0$. Rope and nails or some substitute were always present when this question was asked).

I use this example in my teaching in two ways. When I explain the correct statement of Cauchy's theorem in Complex Variable, and the difference between the fundamental group and the first homology group, and when I teach what is a non-Abelian group.

My second favorite example is the belt trick which is associated to the name of Dirac, and which is mentioned in the question. Sometimes I do it without a belt, using just my hand holding a pencil.

EDIT. Sorry, it is a duplicate:-) But I have a reference: http://www.math.purdue.edu/~eremenko/problems.html I did not invent this problem myself:-( It was one German student who asked me this question when we were hiking in the mountains about 15 years ago. I don't remember his name:-(

• By the way, the paper "Picture-Hanging Puzzles" by Demaine et al provides a reference for the possible origins of this problem (Spivak in Quantum 1997). Oct 22, 2014 at 17:04
• @Sam Nead: Thanks! I downloaded Demaine. What is "Quantum"? Is this the English translation of the Russian Kvant journal? Or something else? MSN does not list the journal with such name. Oct 22, 2014 at 18:36