# Spacetime has an infinite number of choices for differentiability. Coincidence?

Spacetime can be modelled using a four-dimensional topological manifold. Say we denote the manifold using $$(M, \mathcal{O})$$ where $$\dim M =d$$. The structure $$(M,\mathcal{O})$$ is not sufficient for talking about a notion of differentiability of a curve on the manifold. We need to make more choices or more information before we can talk about the differentiability.

Now Schuller says (within the first 5 minutes) that for one, two and three-dimensional manifolds we have the only one choice for a notion of differentiability (or rather all the possible choices are equivalent). For five, six, seven and higher dimensional topological manifolds there's a finite number of choices (in each case) for talking about a notion of differentiability. According to chapter 4 of this set of lecture notes, the former is a corollary of Morse-Radon theorems, whereas the latter is a result from surgery theory.

Number of $$C^{\infty}$$-manifolds one can make out of given $$C^0$$-manifolds (if any) - up to diffeomorphisms:

$$\begin{array}{l | c | r } \dim M & \# & \\ \hline 1 & 1 & \text{Morse-Radon theorems} \\ 2 & 1 & \text{Morse-Radon theorems} \\ 3 & 1 & \text{Morse-Radon theorems} \\ 4 & \text{uncountably infinite} & \\ 5 & \text{finite} & \text{surgery theory} \\ 6 & \text{finite} & \text{surgery theory} \\ \vdots & \text{finite} & \text{surgery theory} \end{array}$$

This chart probably has some exceptions. For instance, @QiaochuYuan points out that:

Donaldson showed that Dolgachev surfaces have countably many smooth structures.

My questions basically are:

1. Is it simply coincidence that only four-dimensional topological manifolds have infinite choices for talking about a notion of differentiability? Or does this have a physical significance?

2. Which fundamental postulates of physics would change if say our spacetime were (say) a three-dimensional or a five-dimensional topological manifold with only a finite number of choices for a notion of differentiability?

• More on exotic differential structures: physics.stackexchange.com/q/264033/2451 , physics.stackexchange.com/q/111144/2451 and links therein. – Qmechanic Jan 12 '19 at 10:40
• 1. I do not see how this is supposed to be a physics question. To the best of my knowledge, physics does not generally use non-standard differentiable structures - there are a few speculative appearances of exotic $\mathbb{R}^4$'s in theoretical physics and that's it. 2. The physics of universes with a different number of dimensions are discussed already at e.g. physics.stackexchange.com/q/110876/50583, physics.stackexchange.com/q/10651/50583, mathoverflow.net/q/47569 and their linked questions. – ACuriousMind Jan 12 '19 at 10:51
• Why are Dolgachev surfaces an exception? The link you gave says that they can be used to construct 4-manifolds with many different diff. structures. – MBN Jan 12 '19 at 10:52
• @MBN Because for Dolgachev surfaces it's "countably infinite" whereas for other 4-dimensional manifolds it's "uncountably infinite". – S.D. Jan 12 '19 at 12:21

The following is just my quick and dirty attempt to say the most simple, sensible things I can say based on a quick browse through the references from this question. I'm not a specialist, and this may be wrong.

Is it simply coincidence that only four-dimensional topological manifolds have infinite choices for talking about a notion of differentiability? Or does this have a physical significance?

Manifolds turn up in lots of places in physics, not just as the manifold representing spacetime. Any time you write down a set of continuously varying parameters to describe a system, that's a manifold. There are well-established applications of exotic smoothness in proving certain things about QFT, but if I'm understanding correctly, this doesn't have anything to do with the structure of the underlying spacetime, which is still the ordinary one of special relativity.

Carl Brans has done some work on what happens in classical GR if you replace the standard smooth structures of spacetime itself with exotic ones, but this seems to be extremely speculative, and it probably doesn't help that these smooth structures have never been explicitly constructed, only proved to exist.

So if my read on the state of the art is correct, then we don't know if these exotic smooth structures have any significance at all as models of spacetime. Therefore it's clearly not possible to say anything about whether the special properties of 4 dimensions are or are not a coincidence.

Which fundamental postulates of physics would change if say our spacetime were (say) a three-dimensional or a five-dimensional topological manifold with only a finite number of choices for a notion of differentiability?

No postulates would change. GR works on any manifold and doesn't postulate anything about one manifold being more legal than another.

Also note that there would be no effect on local physics, since by definition every 4-manifold is locally the same as every other. There are no consequuences that could be tested in accelerator experiments, for example. Only the global structure of spacetime would be different.