I have just completed one of the more difficult assignments of my mathematical life: to give a popular presentation of the work of John Milnor immediately following the formal announcement that he was the winner of this year’s Abel Prize. Of course, in one way the task is very straightforward, since Milnor is a mathematical giant and has a large number of fascinating theorems to his name. However, these theorems are not in my field, the talk was supposed to last fifteen minutes, and my immediate audience included not necessarily mathematical journalists who were supposed to understand what I was saying. If you go to the Abel Prize website, you will find a webcast of the whole announcement, including my talk (which includes a telephone interview with Milnor himself), and also a link to a written version of the talk, in which I go into more detail. But if you are a mathematician, then be warned that even the more detailed version is more about the background to Milnor’s results than to the results themselves. And since I was obliged to prepare the talk in secret, I cannot rule out that some of what I have said is wrong, or gives the wrong emphasis.
Why did they ask somebody in a different field? There are two answers to this. First of all, I was asked whether I could do the popular presentation before the winner was even decided: in other words, it was not supposed to be a talk given by an expert in the winner’s area. (One might speculate about whether my name came up because there were people being considered whose work was closer to mine. I have no idea about this: as you can imagine the members of the committee were extremely tight-lipped about their deliberations.) Secondly, one can even see my lack of expertise as an advantage in some ways, since it made it less likely that I would talk way over the heads of the non-mathematicians.
As I started to prepare the talk, I realized I had a problem. Although Milnor is famous for many results, the one for which he is best known is his construction of a seven-dimensional “exotic sphere”. This means a differentiable 7-manifold that is homeomorphic to a 7-sphere but not diffeomorphic to a 7-sphere. My problem with this result was that it felt false: given a homeomorphism from the 7-sphere to another differentiable manifold, surely one can just “iron out the kinks” or “smooth off the corners”, or however one wants to put it. This was not the only false-seeming result I had to contend with. Another was Milnor’s famous counterexample to the Hauptvermutung: he found two different triangulations of a triangulable space (that is, a topological space that is homeomorphic to a simplicial complex) that have no common refinement. Later it was shown by Casson and Sullivan that you could do this for manifolds as well. And Freedman discovered a 4-manifold that cannot be triangulated at all. My intuition tells me that you can just scatter a large number of points fairly randomly into a manifold and then join them up to form a triangulation. Certainly, this works in two dimensions, and similar arguments show that any two triangulations have a common refinement.
So why aren’t these amazing results of Milnor and others obviously false? Given that I couldn’t immediately answer this question to my satisfaction (and to some extent still can’t) I had an obvious impulse: ask a question on Mathoverflow. But I had to be very careful not to give the slightest clue about the Abel Prize, so instead of asking about exotic spheres, I asked about the existence of differentiable structures on manifolds. (The result that not every topological manifold can be given a differentiable structure is due to Michel Kervaire. To throw people off the scent still further, I asked specifically about 4-manifolds.) I had some interesting responses, which you can look at here. From that I learned that the manifolds that have these strange properties are not in themselves pathological objects . One might have thought that they were wild and fractal and that that was why you couldn’t smooth them out, but actually they are much nicer: for example (and this is also clear from reading Milnor’s original papers, even if you don’t follow the details) you can make them by taking two very nice manifolds and gluing them together in not quite the obvious way, but still a nice way. In fact, Smale showed that every exotic sphere can be obtained by taking two hemispheres and gluing them along the equator using a diffemorphism. Topological spheres produced in this way are called twisted spheres.
The best explanation I have at the moment for why you can’t just iron out the non-differentiability is that the nature of the singularities is much more complicated than it is in any situation that one can visualize. (I myself have trouble visualizing non-differentiable maps beyond maps from to ) In a higher dimension, you can find that the set of singularities forms a submanifold that you can’t just iron away: you can push the crease from one place to another, but you still have a crease. If something like that is correct, then you would expect that the existence or otherwise of exotic differentiable structures would depend very much on subtle topological properties of the spheres, and this I know to be true: the sequence of numbers of differentiable structures on spheres of various different dimensions starts 1, 1, 1, ?, 1, 1, 28, 2, 8, 6, 992, 1, 3, 2, 16256, 2, 16, 16, 523264, 24 (the question mark being because nobody knows whether there are exotic 4-spheres, or whether, if so, the number of exotic 4-spheres is finite), and these incredible numbers, worked out by Kervaire and Milnor, are, I read, related to homotopy groups of spheres. And I presume something similar is true for triangulations: that there are local difficulties to triangulating a manifold that cause every triangulation to pick out some privileged direction, or orientation, or something topological at any rate, in such a way that one cannot make a continuous choice. If anybody can enlighten me further, I’d be very interested and grateful …