explicit "exotic" charts can someone provide explicit charts for non-standard differentiable structures on, for instance $\mathbb{R}^4$ (or some other manifold)?
 A: Here is a Kirby Diagram for an exotic $\mathbb R^4,$ taken from Gompf and Stipsicz's book, "4-Manifolds and Kirby Calculus." It's not given in the form of an atlas, but it is a nice explicit description.

A: Not really helpful I guess, but I am interested as well in this subject so: A while ago I heard Matthew Baker (from Georgia Tech, on an entirely different subject, namely the Berkovich Projective Line of non-archimedean fields) describing a technique knows as the observers' topology (i.e. take a point in the space, look around and describe what you see). It would be interesting to know if one could see a difference with another differentiable structure on $\mathbb{R}^4$. It doesn't give you an explicit atlas though.
This ties in with Exotic Manifolds from the inside as well. I am afraid that this doesn't give you a straight answer as well, but a hint of where to look further.
By the way, did you know that only a small portion of the exotic $\mathbb{R}^4$'s can be represented by Kirby diagrams directly (by varying things in the diagram, you get a countably many non-diffeomorphic copies I guess [though I haven't seen a proof of this]). At hinsight, Bob Gompf has proved that there are uncountably many of exotic $\mathbb{R}^4$'s, so this doesn't help much.
