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Today one of my professors told me that $\mathbb R^4$ admits uncountably many non-diffeomorphic differential structures. When I asked him whether there's an intuitive reason to expect a result like that he said it's just hard $4$-dimensional topology. I thought perhaps I'd look for intuition here, so

Why, intuitively, should one expect many non-diffeomorphic differential structures on $\mathbb R^4$? Why should the other Euclidean spaces admit just one smooth structure?

aes
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    The best answers that I have heard seem to come down to low dimensional accidents. http://mathoverflow.net/questions/47569/what-makes-four-dimensions-special – Baby Dragon Dec 30 '14 at 15:54
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    These are difficult results whose combined proof takes several hundred pages. I do not think you will ever find an intuitive explanation. Start by reading Freed and Uhlenbeck's book, this will help you to get at least some idea of the proofs. – Moishe Kohan Dec 31 '14 at 22:06

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The intuition is basically that $4$ dimensions is large enough that $4$-manifolds exhibit great variety (one example: there are $4$ manifolds with arbitrary finitely-generated fundamental groups), but small enough that the high-dimensional-manifold surgery theory apparatus doesn't apply to smooth manifolds, only topological manifolds.

The key idea for the latter is the Whitney trick. This trick allows one to isotop apart embedded submanifolds of complementary dimension which algebraically have intersection number zero under certain conditions. The rough idea is to cancel intersection points of opposite signs in pairs by finding a Whitney disk, a disk with half its boundary on one submanifold and the other half on the other, meeting at the two points. Once you have one, if the Whitney disk and each submanifold has codimension greater than two, the whitney disk generically is embedded and doesn't meet either of them on its interior. Then you simply push one of the submanifolds across the disk. (There are other cases which work, but this is the simplest to explain.)

In dimension four this trick doesn't work, because the Whitney disk has codimension two, so it may have self intersections which can't be wiggled away, and may also intersect each other submanifold similarly, if they're each dimension two.

Bizarrely, one can make this trick work topologically (but not smoothly), roughly by iterating the construction infinitely many times. See: Casson handle and Freedman's theorem on these.

(Then Donaldson came along with an amazing way to tell smooth $4$-manifolds apart, but I'm not sure this belongs to the intuition regarding this.)

aes
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  • This answer will become more valuable to me as I learn more. I would love to hear a little about this "way to tell smooth 4-manifolds apart". –  Jan 09 '15 at 10:23
  • @Exterior The keywords for Donaldson's theory are Yang-Mills theory, gauge theory, and, well, Donaldson theory. It also inspired the later (and I think more successful?) Seiberg-Witten theory. –  Jan 09 '15 at 15:18
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    @Exterior Mike Miller above gave the keywords. The very rough idea: You can add extra structure to your manifold (e.g. a metric) and then extract numbers (or more complicated algebraic invariants) by counting solutions to special partial differential equations. If you do this just right, the numbers are both interesting and don't depend much on the extra structure. Even better, you can tell how the numbers will change if you make certain changes to your manifold. (This is vaguely analogous to how e.g. betti numbers from standard homology behave, but more complicated and coming from analysis.) – aes Jan 09 '15 at 18:02