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.