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can someone provide explicit charts for non-standard differentiable structures on, for instance $\mathbb{R}^4$ (or some other manifold)?

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I'm a little confused by the nature of your question. Are you interested in simple examples of smooth manifolds with "non-standard" differentiable structures, or are you interested in clutching constructions with charts? I don't see how the Brieskorn example satisfies your original question -- neither Brieskorn nor Kervaire-Milnor provided explicit chart constructions for such manifolds. What are you looking for, precisely? – Ryan Budney Mar 29 '11 at 22:19
i wanted explicit charts (if any are known). i added the other part to maybe drum up interest in the question or to expand it to include constructions (since no one had written anything). – yoyo Mar 30 '11 at 0:25

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.

Exotic $\mathbb R^4$

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Great! You've given me motivation to finally go find out what those diagrams actually mean :) – Mariano Suárez-Alvarez Apr 20 '11 at 2:50
@Mariano: You're constructing the 4-manifold by taking $\mathbb R^3 \times (-\infty,0]$ then attaching handles on $\mathbb R^3 \times \{0\}$ according to the link diagram. The unknotted dotted circles encode a way of attaching a 1-handle. The possibly knotted circles with number decorations indicate a 2-handle attachment. These conventions are given in detail in Gompf and Stipsicz book, sections 5.1 through 5.4. – Ryan Budney Jul 24 '11 at 17:05

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.

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what do you mean when you say only a small portion of exotic $\mathbb R^4$'s can be represented diagrammatically? Do you mean: a) there are some with no Kirby digram, or b) You can't draw the whole diagram because it's infinite? – Grumpy Parsnip Jul 24 '11 at 19:54
There are some with no Kirby diagram (the thing on this page is also infinite). I am at work now with no reference books, but I think there was a line on that in either Gompf/Stipsicz (4-manifolds and Kirby Calculus) or/and Scorpan (The wild would of ...) – Willem Noorduin Jul 25 '11 at 10:48
I am at home and on page 366 of "Gompf and Stpsicz" it says that "The known large exotic $\mathbb{R}^4$'s require infinitely many 3-handles in any handly decompositions, but – Willem Noorduin Jul 25 '11 at 15:31
and there is presently no clue as to how one might draw explicit handle diagrams of them". They wrote that in 1999, so there is chance that something has changed. If I have time, I will try and get some answers via this site, by asking this as a question, and not as a comment. – Willem Noorduin Jul 25 '11 at 15:42
Thanks. To me that means there are Kirby diagrams for these exotic $\mathbb R^4$s, but nobody knows how to actually construct them. – Grumpy Parsnip Jul 25 '11 at 15:54

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