TITLE: A reference for smooth structures on R^n
QUESTION [17 upvotes]: There is a theorem stating that there is essentially one smooth structure on $R^n$ for every n other than 4. Does anybody know where i could find the proof of this? Not so much of what happens in dimension four, where there are infinitely many, but of the uniqueness in other dimensions? Thanks!

REPLY [13 votes]: For $n \geq 5$ , this was first proven in Stallings' The piecewise-linear structure of Euclidean space. It actually proves the PL case and applies smoothing theory. Anyway, Theorem 5.1 of it says

Let $M^n$ be a contractible differentiable manifold which is 1-connected at infinity. If $n \geq 5$, then $M$ is diffeomorphic to Euclidean space $\mathbb{R}^n$.

A related result appears in Lashof's ICM address on smoothing theory. His Corollary says

Every contractible open topological manifold is smoothable.

and he follows by remarking that this smooth structure is unique if $n \geq 5$.
For $n = 2,3$, there is Moise, but his articles may be hard to read. There are now easier proofs using Kirby-Siebenmann techniques. For $n = 2$, one can use Hatcher's The Kirby torus trick for surfaces. For $n = 3$, one can use Hamilton's The triangulation of 3-manifolds in the PL case and apply smoothing theory.