# Geometric topology – Wild character of the codimension 1 subvarieties of the Euclidean space

This question arose from this pile trading post. I am writing a thesis on the $$s$$-Cobordism and Siebenmann's work on the end obstructions. Combined, they give a quick proof of the uniqueness of the smooth structure for $$mathbb {R} ^ n$$, $$n geq 6$$. Siebenmann's theorem says pretty much that for $$n geq 6$$ a contractible $$n$$-collecteur $$M$$ which is simply connected to infinity integrates (smoothly) inside a compact collector. Since this compact variety is contractible, by the $$s$$-cobordism, it is diffeomorphic to the standard $$n$$-disk $$D ^ n$$ (see Minor Conferences on the $$h$$-cobordisme 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 the Euclidean space but I'm looking for a way to deal with the $$n = 5$$ case via Siebenmann's end theorem and the good $$s$$theorem of -cobordism (see link to the question mse). This brings me to the next question, which is interesting in itself

Since codimension 1 is well integrated and smooth $$S subset mathbb {R} ^ {n + 1}$$, is there a diffeomorphism auto $$mathbb {R} ^ {n + 1} rightarrow mathbb {R} ^ {n + 1}$$ who wears $$S$$ in a region limited to one dimension $$mathbb {R} ^ n times (-1, 1)$$ ?

Now if $$M$$ is a multiple that is homeomorphic to $$mathbb {R} ^ 5$$, the product $$M times mathbb {R}$$ is homeomorphic to $$mathbb {R} ^ 6$$, and therefore also diffeomorphic. Subject to the existence of diffeomorphism in my question, we could find a diffeomorphism $$f: M times mathbb {R} rightarrow mathbb {R} ^ 6$$ that cards $$M times 0$$ in $$mathbb {R} ^ 5 times (-1, 1)$$. This would produce a good $$h$$-cobordism between $$M$$ and $$mathbb {R} ^ 5$$ taking the area between $$f (M times 0)$$ and $$mathbb {R} ^ 5 times 1$$ in $$mathbb {R} ^ 5 times mathbb {R}$$. Since $$M$$ is simply connected, the good $$s$$-cobordism theorem applies and shows that $$M$$ and $$mathbb {R} ^ 5$$ are really diffeomorphic.