Teaching homology via everyday examples

Question

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

To be more precise, I am teaching a short course on homology, from chapter two of Hatcher's book. Before diving into the details of Delta-complexes, good pairs, long exact sequences, degrees, and so on, I would like to present a collection of real-life phenomena that are greatly illuminated by actually knowing homology theory. Ideally, I would refer back to these examples as the course progressed and explain them with the new tools the students learn.

Here is what I have thought of so far.

• Tavern puzzles: before trying to solve a tavern puzzle, it is always a good idea to check that the two pieces are topologically unlinked. You can approximate this by computing the linking number, via the degree of the map from the torus to the two-sphere.
• Cowlicks: after getting a very short haircut, there is a place on your scalp (typically near the crown) where the hair is standing up. This is "explained" by the hairy ball theorem: a tangent vector field on the two-sphere must vanish somewhere. "Explained" is in quotes because your hair is a vector field on a disk, not on a sphere.
• When you knead dough, and then push it back into its original shape, there is always a bit of dough that has returned to its original position. This is the Brouwer fixed-point theorem. This example is a bit tricky, because it is impossible to see the fixed bit of dough. (Now, when you are hiking, you can clearly see that there is a point of the map lying directly above the point it represents. However, this is better explained by the contraction mapping principle.)
• There are no draws in the board game Hex. This is equivalent to the Brouwer fixed-point theorem. This isn't a perfect example, because most people don't know the game.

Noticeably missing are puzzles, etc that rely on homological algebra (diagram chasing, long exact sequences, etc).

$\newcommand{\SO}{\operatorname{SO}} \newcommand{\ZZ}{\mathbb{Z}}$ EDIT: Let me also give some non-examples, to clarify what I am asking.

• Linking number also arises in discussions of DNA replication; see discussions of topoisomerase. However DNA is not an everyday object, so doesn't work as an example.
• 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 knowing $\pi_1(\SO(3)) = \ZZ/2\ZZ$. However, this is a fact about the fundamental group, not about homology.
• Impossible objects such as the Penrose tribar that exist locally, but not globally. These can be explained via non-trivial cohomology classes. But, cohomology is in the next class, so if/when I teach that...

Answer 1

Here is an example of economic interpretation of cohomology from some Russian popular lecture that I read (unfortunately, I do not remember the source; I only remember it was in Russian).

Consider a geographic map with some countries in it. By some stretch of imagination, assume that the countries make an open covering:-) Now each country has a currency, and on the "common boundary" (the intersection of two open countries), an exchange rate between the two currencies exists. The question is whether a universal currency can be established, so that each country's currency can have an exchange rate with respect to this universal currency, so that the existing exchange rates between pairs of currencies are compatible with it.

EDIT. I found a book which has this example (this is not the Russian book I mentioned in the first paragraph, but another book, which is full of such examples): Robert Ghrist, Elementary applied topology. This example in in Chapter 6, section 6. The book is freely readable online.

Answer 2

Take an organism that ordinarily migrates south, put it in an artificial environment, and reset its circadian clock by $x$ hours (by gradually shifting the hours of daylight). Release it into the wild, and instead of migrating south, it will migrate in a new direction $y$. Thinking of $x$ and $y$ as elements of $S^1$ and $y$ as a function of $x$, this gives a map $S^1\rightarrow S^1$. Apparently the winding number of this map is characteristic of the species (e.g. $0$ for the pond skater, $1$ for the sunfish). (Source: Arthur Winfree's Geometry of Biological Time.)

Answer 3

A cool example I learned from Jim Fowler:

Yeast colonies have a life cycle which repeats on a regular interval. We can associate to each yeast colony a phase $\theta \in S^1$. If yeast colonies are combined, over time they settle down to the same phase.

Consider the function $f:S^1\times S^1 \to S^1$ which takes as input the phase of two yeast colonies and outputs the phase of the yeast colony obtained by putting them together and letting them reach a new phase.

It would appear that $f(\theta,\theta) = \theta$, and $f(\alpha,\beta) = f(\beta,\alpha)$, so $f$ factors through the mobius strip.

So really what we want is a retract of the mobius strip onto its boundary circle! At this point algebraic topology can step in to show there is no such function...

What went wrong? There is an unstable equilibrium when the two phases are opposite, so the map is not even well defined.

The exact same story can be told with two metronomes synchronizing on a common countertop. See https://www.youtube.com/watch?v=kqFc4wriBvE for example.

Answer 4

Kirchhoff's rules (or laws) of electrotechnics can be restated in homological terms. In more detail, a (branched) electrical circuit is represented by a graph, or a simplicial 1-complex, $\Gamma,$ so that there is a constant electrical current $I_i$ flowing through each edge. After arbitrarily orienting the edges, currents can be assigned numerical values (positive or negative, depending on whether the direction of the current agrees with the chosen orientation) and the full set of currents in the circuit is specified by a $1$-chain $I=\{I_i\}$. Kirchhoff's first rule states that

The algebraic sum of currents at every node is equal to $0.$

This translates into the condition that the chain $I$ is a $1$-cycle, $\partial I=0.$ In particular, current distributions that obey the first rule form a real vector space of dimension $\operatorname{rk}H_1(\Gamma,\mathbb{R}).$

Kirchhoff's second rule involves more ingredients, namely, the resistances $R_i$ of the edges, voltage drops $R_i I_i$ and the electromotive forces $\mathcal{E}_i$ (emf's). It can be rephrased as follows:

For every loop in $\Gamma$, the sum of the quantities $I_iR_i-\mathcal{E}_i$ over the edges in the loop is equal to $0.$

Here it is assumed that the edges forming the loop have been consistently oriented.

It is commonly stated in physics and electrotechnics textbooks that the equations given by the second rule are not independent and one needs to chose a suitable minimal set. Indeed, there are $\operatorname{rk}H_1(\Gamma,\mathbb{R})$ linearly independent conditions. Combining this with the first law allows one to uniquely "solve for" the currents in the circuit given the resistances of the edges and the emf's.

Answer 5

This isn't classical homology, but "addition with carrying", which is much more of an "everyday life" item than most of the cited examples, can be explained in terms of group cohomology. The carry operation is, in essence, a 1-cocycle. See for example the entertaining article

A Cohomological Viewpoint on Elementary School Arithmetic, Daniel C. Isaksen, The American Mathematical Monthly, Vol. 109, No. 9. (Nov., 2002), pp. 796-805.

It's available on JStor, or directly from the Wayback Machine.

EDIT: Isaksen got the idea from James Dolan, who explained it in a series of $\tt sci.math$ posts in January 1994.

Answer 6

Tie two students' hands like this, and ask them to untangle from each other.

If they actually try jumping/diving between the partners' hands, as younger kids do, explain that climbing over each other wouldn't work because (1st student + rope) combo, if considered as a continuous object, forms a generator of $H_1(R^3-($2nd student + rope combo$))$, also if considered as a continuous object. If (1st student + rope) would be able to untangle himself from the other one that would imply that the generator of $H_1$ is trivial.

After that point out that the gap between one of the wrists and the rope makes (2nd student + rope) topologically different from a circle, and therefore (1st student + rope) doesn't actually represent a generator of what topologically is a disjoint union of the 2nd student and his rope.

If they are unable to untangle from each other after all those explanations they are not ready to attend an algebraic topology class.

Answer 7

One good way to understand Arrow's paradox (not sure if the 'everyday life' condition applies here!) is via homology. See for example

Baryshnikov: Unifying Impossibility Theorems: A Topological Approach.

Basically, voter's preferences, as well as the aggregate preference, are represented as points in a topological space. The choice rule is a map between these spaces. The conditions of Arrow's theorem force the induced map on homology to be that of a dictator.

Answer 8

Some more fun examples along the same lines:

• There are two antipodal points on the surface of the earth with the same temperature and humidity. This is an application of the Borsuk–Ulam theorem to the function $S^2 \to \mathbb{R}^2$ given by $$x \mapsto (\text{temperature at}\ x, \text{humidity at}\ x)$$ Of couse this assumes that temperature and humidity are continuous.
• There is a place in the world where the wind isn't blowing. This is another application of the hairy ball theorem assuming wind can be modelled as a continuous tangent vector field.
• The ham sandwich theorem is a nice result which follows quickly from the Borsuk–Ulam theorem: it implies that it's possible to slice a ham sandwich (with a plane) in such a way that the two slices of bread and the the slice of ham are each cut exactly in half.

Answer 9

This may not be a helpful answer to the precise question, but in the 1970s I got disillusioned with homology as a first course in algebraic topology and started teaching knot theory; the examples are immediate, and one gets links with group theory, and with some homological type ideas, via the free differential calculus. Students could quickly see what the subject was about, and it showed that you were getting some answers but incomplete ones. Also there are lots of nice computational examples to do.

It also fits with my view developed over the years that algebraic topology should be and can be more nonabelian, to reflect the geometry. For example, one would really like to write the boundary of the standard diagram of the Klein Bottle as $a+b-a+b$, not just $2b$.

This knotty activity led to me giving general lectures on "How mathematics gets into knots", and into all sorts of things, including making a mathematical exhibition and promoting the work of the sculptor John Robinson.

Others took over the course and even got a nice book out of it ("Knots and surfaces" by Gilbert and Porter).

Answer 10

When I first learned about fundamental groups at Canada/USA Mathcamp during the summer after 11th grade, our teacher suggested that we had already known about monodromy for years. He was explaining covering spaces, and he asked if we had done the following when we were 5 years old. Often children will find a themselves in a building that's not simply-connected and then wonder, after going around a loop, whether they're in the same place or an identical copy of where they were before. Or I've experienced games in which children go around a column and then feel that they have to go back around the column the same number of times in the opposite direction in order to be back where they were before.

Answer 11

I am not sure the best way to formalize it, but "Rock-paper-scissors" always felt like it should have one dimensional first cohomology group. Maybe just that there is a cycle in the directed graph of "stronger than". I guess this generalizes: want to measure how far away a directed graph is from being a poset? The first cohomology group will tell you how many "inconsistencies" must be resolved.

Answer 12

How about stuff on planar graphs (Euler characteristic)? This is certainly just elementary combinatorics, on the other hand it has a pretty explanation in terms of Betti numbers.

Answer 13

Check out the web page (and links therefrom) of Stanford's Applied and Computational Algebraic Topology group. I'm not sure how much of this counts as "everyday", but it certainly involves concrete questions about data analysis, etc that are pretty easy to explain in intuitive terms.

This survey article by Gunnar Carlson is a good place to start.

Answer 14

One of my favorite examples illustrating the power of (co)homology is Sperner Lemma; see the very nice short paper of D.I.A. Cohen On the Sperner Lemma, J. Combin. Theory, 2(1967), 585-587, or these notes.

Answer 15

The following may not quite count as "everyday life," but in the 2013 MIT Mystery Hunt, one of the most elaborate team puzzle-solving contests in the world, one of the puzzles involved solving a crossword on the surface of a CW complex. As explained in the solution, to figure out which edges were identified, you had to complete the grid; this caused the dotted lines on the surface to join up into loops, and you had to figure out which loops were trivial. No algebraic topology is really needed to solve the puzzle, but it is needed to prove the correctness of the intended solution.

Answer 16

Allen Hatcher's book on topology has a "motivating" discussion of cohomology. He says it's a map.

the countries represent elements of the homology. latitude and longitude are functions from the topological space to the reals. The level sets are then also represented by curves on our map (such as isotherms or isobars). and these will be elements of the cohomology