Up to dimension 3, homeomorphic manifolds are diffeomorphic (in particular, homeomorphic open subsets of $\mathbb{R}^n$ ($n\leq 3$) are diffeomorphic). It is known that there are uncountably many mutually non-diffeomorphic open subsets of $\mathbb{R}^4$ that are homeomorphic to $\mathbb{R}^4$ (Demichelis & Freedman, 1992). For $n\neq 4$, $\mathbb{R}^n$ admits a unique smooth structure (up to diffeomorphism), thus the previously mentioned phenomenon is unique to dimension 4. 


**Questions**: In dimension $n\geq 5$, 


1. is it possible for a manifold to be homeomorphic to an open subset of $\mathbb{R}^n$ but not diffeomorphic to it?
2. is it possible that two homeomorphic open subsets of $\mathbb{R}^n$ may be non-diffeomorphic?


Are there examples of such open sets, in each dimension $n\geq 5$, having a simple description up to homeomorphism? 


(the smooth structure on each open subset of $\mathbb{R}^n$ being the standard one).


**Edit:** If I understand it well, Misha Kapovich points out in his comment below that Robion Kirby told him that the answer to question (2) above is positive, in every dimension $n\geq 5$ (and thus in every dimension $n\geq 4$, with the difference that for $n\geq 5$ no such sets can be homeomorphic to $\mathbb{R}^n$). As (2) implies (1) this would settle the question.


It would be interesting to know if other mathematicians heard about these examples of Kirby. From the topological point of view (i.e. up to homeomorphism), what are these sets?



Open subsets of Euclidean space in dimension 5 and higher admitting exotic smooth structures
I finally talked to Rob and did some literature search. Here are some examples of open subsets of Euclidean spaces which are homeomorphic but not diffeomorphic. 


Let $\Sigma$ be an exotic $(n-1)$-dimensional sphere which can be realized as a Brieskorn variety (see [here](http://www.map.mpim-bonn.mpg.de/Exotic_spheres#Brieskorn_varieties)):
$$
V(a):=\{z\in S\_\epsilon^{2m+1} \, : \, z\_0^{a\_0} + \dots + z\_m^{a\_n} =0 \},$$
where $2m=n$ and $S\_\epsilon^{2m+1}=\{z: ||z||=\epsilon\}$ for some sufficiently small $\epsilon>0$. 


For instance, any 7-dimensional exotic sphere will do the job. 


Next, if $M^{n-1}$ is an $(n-1)$-dimensional (smooth) Brieskorn variety $V(a)$, it has a natural embedding in $S^{n+1}$ and the image has trivial normal bundle (since it is the regular level set of a smooth map to ${\mathbb C}$; this smooth map comes from the defining equation of $V(a)$). Hence, our $\Sigma$ embeds in $R^{n+1}$ such that the tubular neighborhood $U\_\Sigma$ of the image is diffeomorphic to $\Sigma\times R^2$. Let $S$ be the standard $n-1$-dimensional sphere; it also embeds in $R^{n+1}$ with trivial normal bundle, of course, and we get a diffeomorphism $U\_S\to S\times R^2$ for the tubular neighborhood $U\_S$ of $S$ in $R^{n+1}$. It is then clear that $U\_\Sigma$ is homeomorphic to $U\_S$. 


**Claim.** The open subsets $U\_\Sigma$ and $U\_S$ of $R^{n+1}$ are not diffeomorphic; equivalently, the manifolds $\Sigma\times R^2$ and $S\times R^2$ are not diffeomorphic. 


This is where the story gets a bit complicated. Rob just says that he knows this, but does not remember a reference. The claim about nonexistence of a diffeomorphism appears on page 150 (Theorem 1) in 


S. Kwasik, R. Schultz, Multiplicative stabilization and transformation groups. Current trends in transformation groups, (2002) 147–165.


But instead of a proof they provide six references none of which contains this claim (at least in a recognizable form). I have no reason to doubt that the proof can be extracted from some combination of their references. Instead, I looked in their other paper:


S. Kwasik, R. Schultz, Toral and exponential stabilization for homotopy spherical spaceforms. Math. Proc. Cambridge Philos. Soc. 137 (2004), no. 3, 571–593. 


In that paper they pretty much provide a proof of this claim in section 4. (Their argument is used for space-forms.) 


Here is a sketch of their argument. Suppose that there is a diffeomorphism
$$
f: \Sigma\times R^2\to S\times R^2. 
$$
Let $S\_a^1\subset R^2, S\_b^1\subset R^2$ be concentric circles of radii $a$ and $b$, where $a$ is much larger than $b$. Then the submanifolds $S\times S^1\_b$ and $f(\Sigma\times S\_a^1)$ cofound a smooth compact submanifold $W\subset S\times R^2$. Next, one verifies that $W$ is an h-cobordism; since $\pi\_1(S\times S^1)\cong {\mathbb Z}$, this h-cobordism is actually an s-cobordism and, hence, by the s-cobordism theorem, is diffeomorphic to the product. In particular, $\Sigma\times S^1$ is diffeomorphic to $S\times S^1$. Now, one can either refer to Theorem 1.9 in 


R. Schultz, Smooth Structures on $S^p\times S^q$, Annals of Mathematics, Second Series, Vol. 90 (1969), p. 187-198


or recycle the above argument, to conclude that $\Sigma$ is diffeomorphic to $S$. This proves the claim. 


