dg.differential geometry - A reference for smooth structures on R^n

You can handle the case of $n leq 3$ one at a time, and so the question really is about $n geq 5$. Two important names in this regard are Kirby and Siebenmann. The Wikipedia article on the Hauptvermutung is a good place to start.

If M is an $n$-dimensional topological manifold (and $n geq 5$), then $M$ admits a PL structure if and only if a special cohomology class, the Kirby-Siebenmann class, in $H^4(M; mathbb{Z}_2)$ vanishes. If this class vanishes, then the different PL structures are parametrized up to concordance by $H^3(M; mathbb{Z})$. (Note: The Wikipedia article on the Hauptvermutung assumes that $M$ is compact, but I don't believe that this is a necessary assumption.)

So what does this say about $M = mathbb{R}^n$? Well, we already know that $mathbb{R}^n$ has a PL structure, and since $H^3(mathbb{R}^n; mathbb{Z}_2)=0$, it follows that this structure is unique up to concordance. Since concordance implies diffeomorphism, and since every smooth structure gives us a PL structure, it follows that there can be only one smooth structure on $mathbb{R}^n$ up to diffeomorphism.

Here are the main references (you can find them both here):

1. Kirby and Siebenmann, On the triangulation of manifolds and the Hauptvermutung. Bull. Amer. Math. Soc. 75 1969 742--749.




2. Kirby and Siebenmann, Foundational essays on topological manifolds, smoothings, and triangulations. Annals of Mathematics Studies 88 (1977). (I did some MathSciNet investigating, and the relevant essays are IV and V.)

This expository article by Rudyak, which I found through Wikipedia, also seems interesting.

Finally, I learned all of this from Scorpan's wonderful book, "The Wild World of 4-Manifolds".