How to prove that $\mathbb{R}^n$ for every n > 4 has a unique smooth structure up to diffeomorphism? I have read Gauge Theory on Asymptotically Periodic 4-Manifolds by Clifford Henry Taubes where an uncountable family of diffeomorphism classes of oriented 4-manifolds which are homeomoprphic to $\mathbb{R}^4$ is constructed.
Furthermore, I know that for 1-, 2-, and 3-manifolds homeomorphic manifolds are already diffeomorphic. Thus, all $\mathbb{R}^n$ with n < 4 have a unique smooth structure up to diffeomorphism. The same holdes for n > 4 as many textbooks and wikipedia claim without proof.
Unfortunately, I have not any idea how to prove that $\mathbb{R}^n$ for every n > 4 has a unique smooth structure up to diffeomorphism. Does anybody know the proof or a paper where it has been proved?
 A: (To kick it from the unanswered queue)
This question has multiple answers from MO (I would just present the highest voted one). I strictly follow the approach mentioned in What to do with questions that are exact duplicates from MathOverflow?.


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):

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


*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".


The answer is by written by Faisal (I cannot ping him as he doesn't have an MSE account)
See Original MO link for the other two answer: A reference for smooth structures on R^n
