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Mathoverflow is intimidating, so I thought I'd ask here first (second). If I don't get any useful answers here in a few days, I'll ask there.

$Q_0$: Is there any use for a topology on the (continuum of) smooth structures on $\mathbb{R}^4$?

$Q_1$: If so, is a useful topology known?

The only reference I have found is a paper by K. Kuga, "A note on Lipschitz distance for smooth structures on noncompact manifolds", MR1117158 (92f:57025). He apparently shows that several obvious topologies are discrete. One might be able to metrize the maximal exotic $\mathbb{R}^4$ and use some variant of the Hausdorff-Gromov pseudometric, but that's such an obvious idea that I'd be surprised if it works, given that I haven't seen it.

Edit 2: The previous edit was edited into the comments below. Edit 3: I may as well clarify "is there any use?" in the OP as the zeroth question.

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Useful for what purposes -- what do you want to use this topology for? For example, does Kuga answer your question, and if not, why not? –  Ryan Budney Sep 18 '11 at 18:17
    
@Ryan: Useful for anything, really. I do not know enough to know if there is a potential use for this, aside from having a vague feeling that since all 4-manifolds are modeled on $\mathbb{R}^4$, knowing more about the smooth structures on $\mathbb{R}^4$ should give information about 4-manifolds in general. (Perhaps I should rephrase my question, "$\exists$ use for a topology on the space of smooth structures on $\mathbb{R}^4$?") –  Neal Sep 18 '11 at 20:33
    
All I know about Kuga's paper is the review on Mathscinet; the paper doesn't appear to be on JSTOR, according to Mathscinet no paper has referenced it, and the top Google hit is this very thread. –  Neal Sep 18 '11 at 20:33
    
@Surly: I have converted your answer to a comment. To post a comment, look on the bottom left of the comment thread; there is a grey button "add comment". You didn't see it earlier because you didn't log in using the same account as you used to post the question; I have now merged your accounts (to avoid future issues, it should help if you register your account). Note that because you do not have 50 reputation points yet, you can only comment on your own questions and answers. –  Zev Chonoles Sep 18 '11 at 21:11
    
@Zev: Thank you! I'm registering now. We'll see if this works! –  Neal Sep 18 '11 at 22:04
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