What is the most suggestive piece of mathematics you know? Aside from the feeling of suggestive similarities between Gödel’s theorem and the Uncertainty Principle, the answer to this for me has always been the subject of exotic manifolds.
Exotic manifolds are spaces that are homeomorphic but not diffeomorphic. This is pretty strange because it means that the usual constructions we use in setting up the differential geometry needed for Einstein’s General Relativity don’t go through like it says they should in the receipe book. A key step in defining a manifold is to require charts and because of the curvature of space and time we give up the idea of having a single coordinate system cover the whole space. Instead we ask only that we can put down a collection of coordinate systems that overlap correctly – think of a map book covering a city. You want to be able to move smoothly from one page to another without things going crazy. The transition functions that take you from one chart to the next should be smooth.
Exotic spaces have smooth structures on them that are inequivalent. Intuitively I think of this as saying that you can put down different systems of smooth coordinates which you cannot map into each other with diffeomorphisms.
This is pretty radical because one of the most fundamental ideas in relativity is the idea that physics in all coordinate systems is the same. But do such inequivalent differentiable structures actually exist? Well it turns out that exotic manifolds don’t exist in one, two or three dimensions: all differentiable structures are equivalent. If two spaces are homeomorphic then they are diffeomorphic. But in higher dimensions they do indeed exist.
Exotic manifolds were first found by Milnor in 1956 by studying the 7-sphere. You might think that this isn’t important for physics but Schleich and Witt showed that these inequivalent differentiable structures do contribute to the usual path integral formulation of quantum gravity.
You may argue that while surprising, none of this is living up to the billing of being super suggestive! So to get to the action lets ask how the existence of exotica depends on the dimensionality of spacetime.
Results for dimensions higher than three depend on what spaces one considers, so what about R^n = R x R x … x R, the usual n-dimensional Euclidean space we know and love? In addition to the result referred to above that there are no exotica for n < 4, it is known that there are also no exotica for n > 4. So far so good. But what about n = 4?
Hold on to your dentures because it turns out that in R^4 there are an uncountably infinite number of inequivalent differentiable structures. Pow.
I think any physicist who reads this probably has the same question jump into their mind as I did when I first came across this result nearly twenty years ago. Does this have anything to do with the dimensionality of spacetime?
One could imagine that a full theory of quantum gravity might involve not only the infamous sum of topologies (known to be impossible to even write down in four dimensions) but also a “sum” over all dimensions. If each inequivalent differentiable structure contributes to the sum – and that is exactly what the results of Schleich and Witt appear to demonstrate – then it seems plausible that at “low-energies” such a sum might be dominated by R^4, providing a dynamical origin for the dimensionality of spacetime.
In a sense this would be the opposite of the usual situation in gauge theories where one has to reduce the number of solutions integrated over because many are simply gauge transformations of each other. Instead here one would have to allow for the increased multiplicity of exotic structures.
Needless to say this is nothing more than suggestive, but it is one of those very cute, shiny results that you hope is somehow relevant because it seems so awesome.
More recent work on exotic manifolds in physics, which I confess I have not looked at at all, can be found here, here in this book. And of course you can always count on John Baez to have covered any interesting topic in mathematic physics on his “blog-before-blogs-existed”, which seems like an appropriate place to mention the Usenet forums for anyone too young to know about the early days of the internet when the web was something spiders made and Netscape hadn’t been born yet.