Algebraic topology for everyone


I have long thought that much of modern mathematics is accessible, in some form, to anyone who is interested, regardless of prior mathematical background. While mathematics is typically taught in such a way that requires the student to have learned much previous material, at least some form of many subjects can be done with, at least, substantially less, and possibly none at all. Naturally, the progression in place in the undergraduate curriculum is quite good at training mathematicians. However, it’s rather bad for inspiring the wider population. Few students who are not math major take any upper-level math classes in college; in many cases, this is not due to lack of interest, but because the classes require substantial previous knowledge that few non-majors have.

As a result, most of the great achievements of mathematicians are hidden from almost the entire world. These are spectacular results, on par with the great works of literature and music, or the great works of architecture. Understanding a great work of music in depth requires lots of musical background knowledge, but becoming acquainted with a piece requires almost none. I would like to suggest that this is true of at least large chunks of mathematics as well.

I’d also like to suggest that the lack of approachability to mathematics is harmful for us. Other disciplines have figured this out. It’s easy to find literature classes, which are open to everyone, that discuss some of the greatest novels ever written. Anyone who is interested can read and learn about the best works of literature! Similarly, musicians perform the great works of music; if one is interested in learning more, there are generally program notes to read or lectures to listen to, aimed at a general audience of nonmusicians, which describe the music and its historical context in greater detail. If one is interested in learning about art, one could go to an art gallery to see great paintings and learn a bit about them; if ey is still curious, ey can take an art history class. Even in physics, whose depth of concepts and background knowledge is roughly comparable to ours, opportunities about for getting a glimpse into high-level research in quantum mechanics and cosmology (but maybe not string theory).

As a result, the community has an opportunity to become excited about these subjects, even if most people can’t really understand them properly. Naturally, that’s a healthy state of affairs: if more people are curious about a subject and have the opportunity to learn a bit, some will want to learn a lot. This chain spurs on more great discoveries and great research. And, lest I appear too naïve, I must mention that this helps these departments to receive funding.

Of course, it would be remiss of me to neglect the role that popular math books play in popularizing modern mathematics. Some of them have been inspirational to me: reading An Imaginary Tale by Paul Nahin when I was fourteen or fifteen forced me to learn complex analysis as rapidly as possible. (And yes, I do mean to say I was forced. I probably haven’t been as drawn and obsessed with anything in my entire life as I was with complex analysis when I was in high school.) More recent popular math books discuss various other fascinating topics in mathematics, including, rather remarkably, the monster group! (I haven’t read Mark Ronan’s Symmetry and the Monster yet, but even I am deeply skeptical of the possibility of presenting the monster group to a general audience.)

Recently, I have been entertaining the idea of running lectures presenting various topics in modern mathematics to anyone who is interested, regardless of background. On Sunday, Nathan and I endeavored to teach a class on algebraic topology to students in grades 7–12 with no prerequisites. As far as modern mathematics goes, topology is possibly the easiest topic to present to a general audience, but we have to start somewhere. Still, this was a rather ambitious project. SUMaC’s program II is on algebraic topology as well; this is for a self-selected group of math nerds entering their junior and senior years of high school, and they get 4 weeks at it. I told some SUMaC students that I was planning to present SUMaC program II in two hours with no background necessary; they thought I was insane. We wouldn’t have time for all of it, of course, but we’d do a good chunk. The primary goal of the class was to have them understand why the sphere and the torus are not homeomorphic.

Nathan taught the first leg, starting with topological spaces (only as subsets of Euclidean spaces) and homeomorphisms between them. Then we showed a video of a coffee cup turning into a torus and back. Nathan then told them that the point of topology is to understand invariants to determine when two topological spaces are homeomorphic, or, more accurately, when they aren’t. (Of course, one can argue this, but deeper issues, such as functoriality, were beyond the scope of this class.)

After that, it was my turn, and I discussed the first of the two invariants we introduced: the Euler characteristic. From the vantage point of an introduction to topology, I don’t see a lot of value in triangulations specifically, so I told them about polygonizing surfaces. We handed out ping pong balls and chalk and had them draw polygonizations of spheres and count vertices, edges, and faces. About half of the students got this wrong, so I fixed the ones that were impossible. Some of them had seen V-E+F=2 before, so that was quickly recognized once I fixed the wrong ones. I suppose, however, that none of them had seen a proof, or anything approaching one, of why this is the case. So, I argued using common refinements that we should expect V-E+F not to depend on the choice of polygonization. I didn’t mention this in the class, but proving this properly is actually surprisingly difficult. (For example, how can we really be certain that we can make a common refinement for two choices of polygonization?) A slightly fictionalized account of suggested proofs and history of this problem can be found in Imre Lakatos’s Proofs and Refutations.

We then passed out clay to mold into tori and felt pens to polygonize them. They already knew from the previous argument that we would look at V-E+F, but they didn’t know what answer to expect this time. Only one student managed to do this correctly. But, it’s hard to count right on a torus. So, I showed them the identification space of the torus and how to count much more easily using it. Since the Euler characteristic of the torus is 0, we had proven that the torus is not homeomorphic to a sphere. But, to whet their appetites for what was to come, I pointed out that the Euler characteristic of a circle is also 0, so we can’t yet guarantee that a circle and a torus are not homeomorphic.

As an aside, I’d like to mention that I still don’t really feel that I understand the Euler characteristic. I understand why everything works out (co)homologically, in terms of the alternating sum of Betti numbers, but relating this to counting vertices, edges, faces, and so on is still a bit confusing to me. The best I’ve come up with so far, admittedly without trying too hard, is the idea of a minimal model for a polygonization. For example, since H^0 and H^2 of a sphere are 1-dimensional, and the other cohomologies are trivial, we should expect a minimal model of one vertex and one face. And we do. Similarly, with a torus, we have a minimal model with one vertex, two edges, and one face, which looks like the identification space of the torus. Since I’m (relatively) happy with common refinements, this convinces me that the Euler characteristics of the sphere and the torus are what we expect. And we can do something different for any orientable surface, since I know its fundamental group and hence can draw its identification space. But, once we start getting torsion in the (co)homology, we can’t possibly have a minimal model, so I start to be confused about how V-E+F relates to the (co)homological Euler characteristic. Any suggestions?

After that, Nathan was up again, and this time he talked about groups. He introduced the group axioms and used as examples the integers, Z/12, Z/2, the trivial group, and the dihedral group D_8 (or D_4, if you’re one of those people who thinks that we should only write D_8 if we also write S_24). We handed out post-its to help them do computations in D_8.

Then it was my turn again. I started by talking about homotopies of based loops on a space, and I showed them that the homotopy classes form a group, called the fundamental group. I then convinced them on an intuitive level that the fundamental group of an annulus is isomorphic to the integers. Then, I proved using the universal cover (but I didn’t call it that) that the fundamental group of a circle is isomorphic to the integers as well.

By then, we were nearly out of time, so Nathan wrapped up fairly quickly by working out the fundamental groups of the sphere, the torus, and a wedge of two circles. That was the end of the class.

Many of the students stayed engaged throughout the entire class. That was pretty inspiring: two hours of math in one sitting is probably more than they’re used to, and the concepts are much deeper than those they see in school. This was especially nice to see given that many of the students were in grades 7 and 8. (Maybe that shouldn’t be too surprising; they might not have been taught that math is hard and boring by that age!)

The most shocking thing that happened was that one student asked a question about Dehn invariants! That was totally out of the blue, and it was simply a stroke of good fortune that I had read about them only a few months ago. He told us after the class was over that he was going to college to be a math major next year, and that he had learned about Hilbert’s third problem for some sort of high school research project.

I’m now more confident in my assertion that it’s possible to teach certain topics in modern mathematics to a general audience with no background in mathematics. Granted, basic algebraic topology is probably about the easiest topic possible since one can do so much just by drawing pictures. Still, I believe that one can do much more.

My next project is to teach Galois theory from no background, in two hours. Does that tip me beyond the bounds of sanity?

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About Simon

Hi. I'm Simon Rubinstein-Salzedo. I'm a mathematics postdoc at Dartmouth College. I'm also a musician; I play piano and cello, and I also sometimes compose music and study musicology. I also like to play chess and write calligraphy. This blog is a catalogue of some of my thoughts. I write them down so that I understand them better. But sometimes other people find them interesting as well, so I happily share them with my small corner of the world.
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5 Responses to Algebraic topology for everyone

  1. michiexile says:

    Your minimal models could be reinterpreted as smallest (in number of non-degenerate simplices) simplicial sets with the corresponding right homeomorphism type of their geometric realization. This is one of the way you can get to a vertex+face model for the sphere, and vertex+2edges+face model for the torus.
    Given this, it’s a mere matter of figuring out how to build a simplicial set for a torsion space. Notice that there is a way to glue up the Möbius strip with a single vertex, single edge and a single face. Since this is a manifold-with-boundary, it’s not quite as helpful as we might think; but gluing in two of those along the edge gives us a simplicial set with a single vertex, a single edge and two faces; and with torsion homology.
    I suspect that torsion won’t quite be visible, in minimal models, in the degree it happens in, but rather in an adjacent degree.
    All that said – for Galois theory – can I come play?

    • Simon says:

      After I posted this and was trying to fall asleep yesterday, I realized that relating V-E+F to the (co)homological Euler characteristic is not hard. The fact that they’re the same is just that the Euler characteristic of a complex is equal to the Euler characteristic of its (co)homology, assuming the complex is finite. So, if we’re happy with “singular=simplicial” (co)homology, then we’re done.

  2. pastwatcher says:

    I am in agreement that math should be more accessible. I’ve always figured that we probably drive away half of our potential mathematicians by boring or confusing them with the standard primary school curriculum before revealing anything really pretty (geometry sort of counts here, I suppose).
    I don’t see how you could motivate Galois theory with no background, or even display it at work. It’s hard enough to motivate the highly accessible questions of number theory; some people like it, but many don’t. Would it help to add in geometry, ruler-and-compass and such? But that’s just me being overly visual of course. Still, you have to start by introducing irrational and/or imaginary numbers…lattices???
    Also, I admit that I believe the contrapositive of that which makes me “one of those people”. We should say S_4 for that group what moves 4 things in 24 ways, thus D_4 for that group what moves ’em in 8 ways, yo.

  3. hajari_ says:

    “they might not have been taught that math is hard and boring by that age!”
    I think that’s exactly the problem! It’s very sad.
    I went on a maths camp a few years ago where we had lectures on topology (mostly knot theory), number theory, and something else I don’t know the name for… looking at deterministic finite automata and the sets of strings the machines accept, and then some other stuff expanding on that. None of that really required prior knowledge.
    Some of the other groups also did game theory, which I know nothing about but I imagine you wouldn’t need much background for it.

  4. Zach Landes says:

    This sounds like a great project – you briefly mentioned it to me before, but now that I have the full picture…I want you to show *me* this stuff, too. I know absolutely nothing about topology. And I recall V-E+F=2 being used in your proof of the house & utilities problem, so I’d love to learn the derivation of that law.

    I have, myself, considered starting a math website that purports to teach mathematics intuitively. I figure once something “clicks” for me I ought to be able to explain it to others. And in so doing, learn it better. There is nothing more valuable for reinforcing a concept than being forced to explain it to others. Particularly others who are trying to learn.

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