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My question is whether all manifolds that can be embedded in $\mathbb R^n$ are homeomorphic to a smooth manifold?

I know that every smooth manifold can be triangulated which I think is a result of Whitehead and I think every manifold in $\mathbb R^n$ can be triangulated so this lends plausibility I think. (If the dimension 4 E8 manifold can be embedded in $\mathbb R^n$ then we have a counterexample but I'm not sure if it can be. All manifolds of dimension up to 3 can be triangulated).

I appreciate if this is a basic question but it doesn't seem to be spelt out explicitly anywhere. Most textbooks define a manifold and then a smooth manifold but don't say in which cases these concepts may be equivalent.

Ivan Meir
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The answer is negative. As you observe yourself, if we allow abstract manifolds, then the E8 manifold provides a counterexample. However, it turns out that any abstract manifold can be embedded into $\mathbb R^n$ for a suitable $n$ (for instance, twice the dimension plus one, see e.g. here), so in particular the E8 manifold is a submanifold of $\mathbb R^n$ (for $n=9$) which is not homeomorphic to a smooth manifold.

Wojowu
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    Thank you. It's perhaps worth noting that the result is trivially true for manifolds of dimension 1,2 and 3 simply because all manifolds in these lower dimensions have a unique differential and hence smooth structure. – Ivan Meir Aug 30 '19 at 09:52
  • This is Nash embedding theorem – debabrata chakraborty Aug 30 '19 at 10:21
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    @debabratachakraborty Nash embedding theorem concerns Riemannian manifolds, which ought to be $C^1$ for the Riemannian structure to even make sense. – Wojowu Aug 30 '19 at 10:35
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    @IvanMeir I don't think one can call the uniqueness of smooth structures on three manifolds "trivial". I think it is the main result of an influential Annals paper. – ssx Aug 30 '19 at 10:52
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    @Spiritofperpetualnegation I meant that my result in dims 1,2 and 3 was a trivial deduction from the proven fact that all manifolds in these dimensions have smooth structure. – Ivan Meir Aug 30 '19 at 11:13