I am reading an essay by Armand Borel, titled "On the Place of Mathematics in Culture", and in it at one point he mentions something that caught my eye:
There are even findings about our objects which surprise us enormously and make an impression not unlike that made on physicists by the discovery of an elementary particle. A few years ago, it was shown that the euclidean space in four dimensions carries several differentiable structures. I shall not try to define those terms. Let me just say that euclidean space is one of the most basic structures in mathematics. There is one in each dimension and it was known to have a unique differentiable structure except possibly in dimension 4. It was quite a shock when it was shown that this last case was indeed exceptional. The feeling of many mathematicians may have been akin to that of the physicist I. Rabi who, when apprised of the discovery of a new particle, the muon, somewhat unwelcome since it was unexpected and did not fit into any existing theory, exclaimed: "Who ordered that?".
Despite not quite knowing what is meant by "differentiable structure" (I assume it's about precisely how derivatives are defined in $\Bbb R^n$ for higher $n$'s), I was immediately curious as to how 4-dimensional Euclidean space is "special" to have several of those, while the others have only one. Could someone illumine me regarding what Borel is talking of and exactly why $\Bbb R^4$ is the exceptional one, i.e. the one whose structure differs from the rest?