# Any “connection” between uncountably infinitely many differentiable manifolds of dimension 4 and the spacetime having dimension four?

Self-studying general relativity, I came across a rather mind-blowing statement (for a beginner like me). Maybe this question is a naive one because of my lack of knowledge in differential geometry.

The number of differential manifolds with up-to diffeomorpshism of dimension 1,2,3 is 1 (using Moise-Radon theorems) and that in any dimension more than four is finite (using surgery) but, that in four dimension is uncountably infinite and the dimension coincides with that of the dimension of spacetime. Is it just a coincidence and what are its consequences?

## 2 Answers

What the lecturer said seems rather obviously false. For instance, in dimension $2$ there are already up to homeomorphism infinitely many orientable, connected, compact manifolds. The Euler characteristic (or equivalently the genus) is a topological invariant that distinguishes them. If two manifolds are not homeomorphic they are definitely not diffeomorphic. If you allow the manifold to be disconnected then even in dimension $1$ disjoint unions of circles yield an infinite family of compact, orientable manifolds.

What the lecturer could have been referring to is that $\Bbb R^n$ has, up to diffeomorphism, exactly one smooth structure (the "standard" one) in any dimension $n\neq 4$. In dimension $4$, it follows from work by Freedman (on topological properties of $4$-manifolds) and Donaldson (on properties of $4$-manifolds that can detect the smooth structure) that there are uncountably many nondiffeomorphic smooth structures on $\Bbb R^4$: there are many "fake" or "exotic" $\Bbb R^4$'s.

The work by Donaldson has close connections to physics (one proves his main results using Yang-Mills theory or Seiberg-Witten theory, which provide invariants that can distinguish nondiffeomorphic smooth structures). However, there is currently no mainstream physical interpretation for the fact that there are fake $\Bbb R^4$'s. It is not clear that there is any physical relevance to this fact.

As a fun aside, exotic spheres are also a research topic in mathematics. The paper I just linked shows that in dimensions $5-61$, most spheres admit exotic structures. Thus, the phenomenon of "exotic structures" on manifolds is far from unique to dimension four. Note that the problem is open in dimension $4$. As pointed out in the comments by ACuriousMind, there is a paper by Witten that gives an attempt at physically interpreting exotic spheres, though I don't know anything about his arguments.

The usually cited reason why dimension $4$ is special has to do with certain differential-geometric theorems of extraordinary power that require "wiggle room", which causes them to work only in dimensions $\geq 5$. As far as I know, the h cobordism theorem by Smale is one of the main examples of this phenomenon. Thus, $n=4$ is the largest dimension in which "high-dimensional methods" do not work. Dimensions $1$ and $2$ are a bit easier to work with because the phenomena occurring there are comparatively "tame". From a geometric and topological point of view, things start to get really interesting in dimension $3$, while $4$ is "wild". I don't know much more about it than that, but there are several excellent books on $3$- and $4$-manifolds (for the former, I know especially Thurston's book is great, while the latter has books by Donaldson, Freedman, Freed & Uhlenbeck, Gompf & Stipsicz, etc.).

You misquoted the lecturer.

The statement was how many $C^\infty$ manifolds you can make from a given $C^0$ manifold ( up to a diffeomorphism ).