# Teaching homology via everyday examples

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...
• This is not directly related to the question but the nlab maintains an excellent list for cohomology: ncatlab.org/davidcorfield/show/intuitions+about+cohomology – Callan McGill Jan 1 '14 at 16:32
• The striking example is -- just ordinary addition: 5+6=11, 2+3=05 - the first digit in sum (i.e. 0 or 1) as a function of summands is a 2-cocycle for the group Z/10Z. See mathoverflow.net/questions/60108/… – Alexander Chervov Jan 3 '14 at 12:11
• I'm not convinced kneading dough is continuous. See comments on mathoverflow.net/questions/251921/… on why it is suspicious that stirring coffee is continuous. In particular, as Dejan Govc points out, any self-homeomorphism of the ball of dough would need to map the boundary to the boundary, which is not what happens in kneading. – Jim Conant Jan 24 '17 at 19:21
• @JimConant - In the continuous model of matter, the baker's transformation (aka kneading) is not injective at the boundary of the dough. Note that Govc's objection, "...no two points of the coffee get glued together...", is using the atomic model to reject the continuous model. I find that very unsatisfying - continuous dynamics make predictions about (large-scale) reality which we can check. (Just one example - the paper "Topological optimisation of rod-stirring devices".) – Sam Nead Jan 25 '17 at 13:06
• @SamNead thanks for that comment! It is fascinating to me that a model of matter where points can be identified actually describes real world situations so well. – Jim Conant Jan 25 '17 at 19:01

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.

• Excellent! $\phantom{fgdfgd}$ – Steven Gubkin Jan 1 '14 at 19:29
• Lovely example! If by some chance you happen to remember the reference, please update your answer. I'm probably not the only one who wants to read it... – Vidit Nanda Jan 3 '14 at 4:20
• I will if I find it. Actually I am even not sure that this was Kirillov. But this was one of those remarkable Russian lectures for high school children published by the Moscow Center. – Alexandre Eremenko Jan 3 '14 at 4:32
• If there there is no global currency then moving your money around a cycle in the appropriate direction gives you greater and greater profits! I think some people actually are rich from currency exchange... – Steven Gubkin Jan 3 '14 at 16:10
• Arbitrage, no? __ – Paul Reynolds Jan 3 '14 at 16:56

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.)

• I am somewhat skeptical. I would really like to see the scatter plots of y versus x for various species. Do you have a link to the original paper? – Steven Gubkin Jan 1 '14 at 20:04
• @StevenGubkin: The book shows scatterplots for the sunfish (landsburg.org/img_0442.jpg) and water strider (landsburg.org/img_0443.jpg), and references by Braemer ("A critical review of the sun-azimuth hyptothesis") and Birukow ("Innate types of chronometry in insect orientation"). I have not looked at either paper. – Steven Landsburg Jan 1 '14 at 21:46
• That really is pretty convincing! – Steven Gubkin Jan 1 '14 at 21:53

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.

• The Youtube video is very nice. The version of this "puzzle" with two metronomes pretty much exactly fits my requirements. I'll have to think about which initial phases converge fast enough to make a good demonstration. – Sam Nead Jan 1 '14 at 15:44
• And, of course, for three metronomes, we are looking for a retraction from the three-sphere to the trefoil knot. :) – Sam Nead Jan 1 '14 at 20:57
• youtube.com/watch?v=aSNrKS-sCE0 is relevant as well. – Steven Gubkin Jan 2 '14 at 3:18
• Ok - I borrowed a pair of metronomes and made a set-up similar to the one Steven Strogatz uses in his TED talk. And it doesn't work. You can actually see this happening in his video - the metronomes go into sync (mostly) and then barrel right through it. I had some trials where the metronomes seemed to settle into a phase difference of 180... I messed around a fair bit with the "rocking base" and the settings of the metronomes, but my take-away lesson is that you need a much better setup to make a good demo. – Sam Nead Jan 3 '14 at 16:03
• The water bottles do seem pretty dinky. How about hanging the platform at each corner? That seems like it would work a lot better. – Steven Gubkin Jan 3 '14 at 16:07

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.

• I find it telling that just prior to this posting, two out of three top rated answers were from the realm of biological sciences and another interesting example involved statistics, while none of the answers involved physics. Clearly, physics is no longer viewed as the most important area for applying mathematics to "real life". – Victor Protsak Jan 2 '14 at 8:57
• To be fair, I mentioned that my example worked for pendulums too which is "physics". I agree that your example is probably more fundamental because it actually uses the homology groups directly. – Steven Gubkin Jan 2 '14 at 16:27
• This is discussed in more detail in Quantum Field Theory III: Gauge Theory by Eberhard Zeidler (Springer, 4/2011), pg. 1009, 1020, 1027. – Tom Copeland Jan 24 '17 at 18:53

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 at http://www.math.wayne.edu/~isaksen/Expository/carrying.pdf

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.

• One of my favorite games to show students of all ages. I have never met an elementary schooler who can do it. Some bright middle schoolers can. High school and up it pretty much divides into who thinks about it, and who just wants to have fun getting more and more tangled. Really a marvelous puzzle, and (in my opinion) the basic topological insight needed before the more rigid geometry of tavern puzzles. – Steven Gubkin Jan 7 '14 at 3:35
• Unfortunately the image link is dead. Can someone who knows this exercise find another image, and/or put a sufficient description into the text? – Peter LeFanu Lumsdaine May 10 '19 at 9:37
• @PeterLeFanuLumsdaine, done. – Michael May 10 '19 at 16:04

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.

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.
• The first two are nice puzzles, but they are pretty far from everyday experience. The last one (sandwich cutting, cake cutting, etc) is definitely an everyday activity! Can this be turned into a puzzle or game or story in some way, making it more "alive"? – Sam Nead Jan 1 '14 at 15:54

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).

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.

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.

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.

• Given a directed graph. View it as a graph, or $1$-dimensional simplicial complex. For each triple of vertices that exhibits transitivity, glue a triangle. Take $H^1$ of this two-dimensional simplicial complex. – Will Sawin Jan 1 '14 at 17:03

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.

• I think this interview between John D. Cook and Robert Ghrist would also be of interest to get a quickly overview of applied topology. – Shamisen Jan 6 '14 at 14:38

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.

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.

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