exotic smooth structures I have difficulty imagining how two manifolds can be homeomorphic but not diffeomorphic, it seems for a homeomorphism $f:M\to N$ and charts $(U,\phi)$ on $M$ and $(V,\psi)$ on $N$, one can always approximate $\phi^{-1} \circ f\circ \psi:U\to V$ by a smooth function and do this in such a way that although one purturbs $\phi$ and $\psi$,but eventually it is possible to reconcile the changes on overlap of any two charts, so that this gives a diffeomorphism as well. Of course this is not correct, but I am not sure how I should think about exotic smooth structures to make it apparent how two manifolds potentially can have different smooth structures.
Here there is a huge difference between low-dimensional manifolds ($1$, $2$, $3$) and high-dimensional manifolds ($\geq 4$). This difference is illustrated by the following theorem.


  Theorem: Let $M$ and $N$ be smooth manifolds of like-dimension $1$, $2$, or $3$. Then $M$ and $N$ are homeomorphic if and only if they are diffeomorphic as smooth manifolds.


The important detail here is that $M$ and $N$ are smooth manifolds, in higher dimensions this result does not hold.



For a simple, visual example of two homeomorphic but non-diffeomorphic manifolds, consider $\Bbb S^1$, the standard unit circle and $C^1$, the square. Clearly these are homeomorphic, but does it seem sensible to claim that they are diffeomorphic? After all, the square has those pointy vertices...

Why doesn't the theorem apply to $\Bbb S^1$ and $C^1$? Because $C^1$ is not a smooth manifold, it is simply a topological manifold.

I'm afraid I will be little help in visualizing exotic structures, since they only exist in dimension $4$ and up!
Here there is a huge difference between low-dimensional manifolds ($1$, $2$, $3$) and high-dimensional manifolds ($\geq 4$). This difference is illustrated by the following theorem.


  Theorem: Let $M$ and $N$ be smooth manifolds of like-dimension $1$, $2$, or $3$. Then $M$ and $N$ are homeomorphic if and only if they are diffeomorphic as smooth manifolds.


The important detail here is that $M$ and $N$ are smooth manifolds, in higher dimensions this result does not hold.



For a simple, visual example of two homeomorphic but non-diffeomorphic manifolds, consider $\Bbb S^1$, the standard unit circle and $C^1$, the square. Clearly these are homeomorphic, but does it seem sensible to claim that they are diffeomorphic? After all, the square has those pointy vertices...

Why doesn't the theorem apply to $\Bbb S^1$ and $C^1$? Because $C^1$ is not a smooth manifold, it is simply a topological manifold.

I'm afraid I will be little help in visualizing exotic structures, since they only exist in dimension $4$ and up!