So far, I have discussed what exotic smoothness is and have tried to motivate the basic reason why it might be important for physics. In this post I will talk about the existence of exotic smooth structures and how we see a hint of a deep connection to physics.
Fair warning; to really understand how we know exotic smooth structures exist, one needs quite a bit of difficult mathematics. It’s easy to see why this question would at least be hard to answer; to show two smooth manifolds are exotic, we need to show that we can find a homeomorphism between them, but that there is no diffeomorphism between them. So the brute force method would be to check every function that could possibly exist – obviously too hard! The usual approach one takes to answer such questions is to look for well-defined characteristics that, for instance, all diffeomorphic manifolds have. These are generically called invariants, and usually come with famous names like Euler, Donaldson, Seiberg, and Witten. Proving that such invariants exist (and are actually useful!) is arguably one of the deepest rabbit holes in pure mathematics, so I will not attempt any kind of complete account here. I will give some important results, and try to provide simple arguments for why they work. It will turn out that the most interesting dimension for exotic smoothness is also the dimension we care the most about for physics.
Low Dimensions (<4)
In dimensions under 4, there are no exotic smooth structures. In dimension 1, there are only 2 non-diffeomorphic smooth manifolds, the 1-sphere and the interval , and they are not homeomorphic (since any map cannot be continuous). In dimension 2, you can decompose any (closed) smooth manifold into pieces which looks like spheres, tori, or projective planes, and again we find no exotic smoothness.
Dimension 3 is a little trickier, and in fact has only recently been solved by Perelman in 2003. Prior to that, we had the following conjecture:
Geometrization Conjecture (Thurston): Every smooth, closed 3-manifold can be decomposed into prime manifolds, and every prime, closed 3-manifold can be cut along tori so that the interiors are one of 8 geometries.
The geometries are essentially 3-dimensional generalizations of the spheres and tori from the analogous 2-dimensional result. If the geometrization conjecture is true, then the following classical conjecture of Poincare (1904) would also be true:
Poincare Conjecture: The only simply-connected closed 3-manifold is .
Perelman used Ricci flow to show the existence and uniqueness of these 8 geometric structures, and proved both the previous conjectures. The Ricci flow is worth a very brief aside (actually worth more then I’m giving it). It tells us that Ricci curvature on a manifold is given by the flow of the metric:
Perelman showed that no matter what metric you start with, you flow to one of the 8 possible geometries. The Ricci flow is currently an area of active research, considering higher dimensional Ricci flows and how these flows tend towards singularities, for instance.
A significant result in math circles, Perelman was awarded a Fields medal (which he did not accept) and $1 million from the Clay Mathematics institute (also not accepted). For our purposes, this story ends with: there is no exotic smoothness in dimension 3!
High Dimensions (>4)
It turns out that for high dimensional manifolds, there is only one story to tell: the h-cobordism theorem. The h stands for homotopy, which is the process of continuously (not smoothly!) deforming one object into another. The classic example is homotoping a coffee mug into a donut (gif from Wikipedia, of course…):
The h-cobordism theorem essentially says that if you can do this between two manifold of dimension , then the two manifolds are diffeomorphic. The technique this time is to decompose the manifold into pieces (called handles), and make sure these handles can be untwisted in such a way to make a homotopy. The reason they always can is something called The Whitney Trick, which essentially relies on having “enough space” to move the handles around. Now, this trick only works in dimensions 5 or greater, so the h-cobordism theorem fails to tell us anything about dimension 4. Seeing a trend yet?
One thing that’s different about the higher dimensional case is that there actually are exotic smooth structures. In fact, the first example of exotic smoothness (in any dimension) were the exotic 7-spheres found by John Milnor (1957). There are many other specific examples of exotic smoothness in higher dimensions, but since the universe is (apparently) dimension 4, we will turn there now.
So the problem is that decomposition techniques generally fail in dimension 4, due to the added complexity but failure of the Whitney disk trick. Now, the topological version of the -cobordism theorem works; meaning that two manifolds that are homotopic in dimension four are also homeomorphic. Of course, that doesn’t help us very much because we are at least in the category of continuity; want we want is the difference between continuous and smooth. Well, in this short introduction I won’t give any more details (perhaps in another post), but it turns out that in dimension four the thing you really need is the intersection form. This is (roughly) a matrix which describes how two-dimensional surfaces intersect in a four-dimensional manifold. By a complete classification of these forms (done by Freedman and Donaldson), you can do things like try and decompose the manifold while preserving the intersection form. This leads to some contradictions, the most interesting of which leads to the existence of exotic . These would be smooth 4-manifolds which are homeomorphic to our usual , but which are not diffeomorphic to the usual . Things are even worse (or better!) – there are infinitely-many exotic !
So the situation is this; in terms of exotic smoothness, dimension 4 is special. This presents a major motivation for studying exotic smoothness in the context of physics. We have already discussed that since exotic smooth structures are not smoothly equivalent, we would not expect any results which relied on calculus (like physics!) to be the same on both of them. Of course, this would not matter if we were studying the physics of space alone – since it is 3-dimensional, there is no exotic smoothness. But as soon as we move to the dimension in which all our fundamental theories are based, exotic smoothness suddenly becomes non-trivial.
This is either a very significant observation, or it is not! The next post will discuss how we might try to study exotic smoothness in physics, from both model-building and observational standpoints.