While solving some problems on Differential Topology I asked myself this question: Suppose a smooth manifold having atleast two exotic smooth structures can be embedded in some Euclidean space under one smooth structure; will the same thing happen for the other smooth structure?
In other words,will the smallest dimensional Euclidean spaces in which the manifold embeds under different smooth structures be the same?
Ps. I am asking this question just out of my curiosity.
On
Not necessarily. Here's one such (non-compact) example.
There are uncountably many smooth structures on $\mathbb{R}^4$. An exotic $\mathbb{R}^4$ is said to be small if it can be smoothly embedded in the standard $\mathbb{R}^4$, and large otherwise; both small and large $\mathbb{R}^4$'s exist. A small $\mathbb{R}^4$ (or standard $\mathbb{R}^4$) embeds in $\mathbb{R}^4$, large $\mathbb{R}^4$'s do not. So the smallest $n$ for which a large $\mathbb{R}^4$ embeds into $\mathbb{R}^n$ satisfies $n > 4$ (and $n \leq 8$ by the Whitney Embedding Theorem). In fact, every large $\mathbb{R}^4$ embeds in $\mathbb{R}^5$, see this question.
Every exotic sphere (of dimension $\ge 7$) is an example. Indeed, suppose $\Sigma$ is an exotic $n$-dimensional sphere and that it embeds (smoothly) in $R^{n+1}$. Then it bounds a contractible compact submanifold $W$ of $R^{n+1}$. Now, remove a small ball $B$ from $W$. The result is an h-cobordism between $\Sigma$ and the boundary of $B$, which is the usual sphere $S^n$. Hence, by the smooth h-coboridism theorem, $\Sigma$ is diffeomorphic to $S^n$.