TITLE: Wildness of codimension 1 submanifolds of euclidean space
QUESTION [8 upvotes]: This question arose out of this stack exchange post. I am wirting a thesis about the $s$-cobordism theorem and Siebenmann's work about end obstructions. Combined they give a quick proof of the uniqueness of the smooth structure for $\mathbb{R}^n$, $n \geq 6$. Roughly Siebenmann's theorem says that for $n \geq 6$ a contractible $n$-manifold $M$ that is simply connected at infinity embeds (smoothly) as the interior of a compact manifold. Since this compact manifold is contractible, by the $s$-cobordism theorem, it is diffeomorphic to the standard $n$-disk $D^n$ (see Minor's Lectures on the $h$-cobordism theorem for example). It follows that $M = \text{int } D^n$ is diffeomorphic to $\mathbb{R}^n$.
The problem is that the case $n = 5$ is not covered. I am aware of Stallings beautifully written On the Piecewise-Linear Structure of Euclidean Space but I am searching for a way to deal with the $n = 5$ case via Siebenmann's end theorem and the proper $s$-cobordism theorem (see link to the mse question). This leads me to the following question, which is interesting in itself

Given a smooth properly embedded codimension 1 submanifold $S \subset \mathbb{R}^{n+1}$, is there a self diffeomorphism $\mathbb{R}^{n+1} \rightarrow \mathbb{R}^{n+1}$ which carries $S$ into a one dimensional bounded region $\mathbb{R}^n \times (-1, 1)$ ?

Now if $M$ is a manifold that is homeomorphic to $\mathbb{R}^5$, the product $M \times \mathbb{R}$ is homeomorphic to $\mathbb{R}^6$, and hence also diffeomorphic. Granted the existence of the diffeomorphism in my question, we could find a diffeomorphism $f : M \times \mathbb{R} \rightarrow \mathbb{R}^6$ that maps $M \times 0$ into $\mathbb{R}^5 \times (-1, 1)$. This would produce a proper $h$-cobordism between $M$ and $\mathbb{R}^5$ by taking the region between $f(M \times 0)$ and $\mathbb{R}^5 \times 1$ in $\mathbb{R}^5 \times \mathbb{R}$. Since $M$ is simply connected, the proper $s$-cobordism theorem applies and shows that $M$ and $\mathbb{R}^5$ are actually diffeomorphic.

REPLY [3 votes]: As you probably know, Siebenmann's thesis can be extended to dimension 5 (subject to a fundamenal group condition) by using the 4-dimensional work of Freedman. Unfortunately, one loses smoothness in the final conclusion, so your hope for a diffeomorphism would fall short.
An alternative to using Siebenmann's thesis might be to apply his Open Collar Theorem from the later paper "On detecting open collars". The beauty of that approach is that it is valid in the smooth category down to and including dimension 5.