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): $$ V(a):=\{z\in S_\epsilon^{2m+1} \, : \, z_0^{a_0} + \dots + z_m^{a_n} =0 \},$$ 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.

Misha's answer provides examples (for certain $n$), of pairs of open subsets of $\mathbb{R}^n$ (both with the standard smooth structure), that are homeomorphic but not diffeomorphic to each other, thus giving a partial answer to the more general question 2). One of the open sets is built from an exotic $(n−2)$-dimensional sphere which can be realized as Brieskorn variety. Smoothly embedding in $\mathbb{R}^n$, it can be viewed as a "small exotic $\mathbb{S}^{n-2}\times\mathbb{R}^2$"

Meanwhile, I got the following partial answer from an expert (private communication; as above, dimension $4$ is excluded). It goes in the other sense, providing examples of large exotic structures on open subsets of $\mathbb{R}^n$ (again, not for all $n>4$):

"Strictly speaking, $\mathbb{R}^n\setminus 0$ answers your question for most values of $n$, since it is diffeomorphic to $\mathbb{S}^{n-1}\times\mathbb{R}$, and the first factor typically has exotic structures that remain nondiffeomorphic to the original after product with $\mathbb{R}$. However, the resulting exotic structures do not embed in $\mathbb{R}^n$. I am guessing that you want pairs of open subsets of $\mathbb{R}^n$ that are homeomorphic to each other but not diffeomorphic. I don't know if such pairs exist. Clearly, they would have to have trivial tangent bundles, but that does not preclude exoticness: there are exotic spheres $S$ that bound parallelizable manifolds, so $S\times\mathbb{R}$ has a trivial tangent bundle and is not diffeomorphic to a standard sphere $\times\mathbb{R}$. However, such an example cannot embed as an open subset of $\mathbb{R}^n$. (Otherwise, $S$ would bound a ball and hence be standard.)"

Two short comments:

1) this construction provides examples of manifolds homeomorphic to simple open subsets of $\mathbb{R}^n$, but which do not smoothly embed in $\mathbb{R}^n$ (summing up, an exotic structure in the $(n-1)$-sphere gives rise to an exotic structure on the punctured $\mathbb{R}^n$, which is "large" in the sense that it does not embed smoothly in $\mathbb{R}^n$).

2) It would be also interesting to know analogue (topologically) simple examples in every dimension $n>4$ for which the $(n-1)$-sphere exhibits no exotic structure (in the case $n=5$, for which no exotic structure is known).