# Useful topology on space of smooth structures on $\mathbb R^4$?

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
– 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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