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Is there a smooth $4$-manifold homeomorphic but not diffemorphic to $CP^2$? Are there known non-smooth examples homeomorphic $CP^2$?

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 Your question is answered on the "smooth structure" Wikipedia page. en.wikipedia.org/wiki/Differential_structure As it's an open problem, moreover of the type we likely won't resolve quickly, I'm voting to close. Please click on the "faq" and "how to ask" links above for further context. – Ryan Budney Aug 16 2011 at 17:35
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 Dear Yang, I'm afraid famous open problems are off-topic on MathOverflow. This site seems to work best for questions where you think an expert somewhere might know the answer. – S. Carnahan♦ Aug 17 2011 at 8:48
5 
 Yang, I really don't think the second part of your question is meaningful. If one constructs a manifold homeomorphic to $CP^2$, but the construction is only of a topological manifold, then one can smooth it by declaring the homeomorphism to be a diffeomorphism. I don't think there's anything more to be said. – Tim Perutz Aug 17 2011 at 14:51
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 Can you always smooth topological 4-manifold? I know that not all the topological 4 manifolds are smoothable. – Yang Aug 17 2011 at 15:07
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 There are plenty of non-smoothable 4-manifolds. However, one that is homeomorphic to $CP^2$ has a smooth structure: that of $CP^2$. – Tim Perutz Aug 17 2011 at 15:47
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closed as off topic by Ryan Budney, Agol, Dan Petersen, Andrew Stacey, S. Carnahan♦ Aug 17 2011 at 8:44

1 Answer

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This is a notorious open problem. For the moment the simplest compact four-manifold that is announced to admit (infinite number of) exotic smooth structures is $S^2\times S^2$. This result is contained here : http://arxiv.org/abs/1005.3346

I have to say that I am not at all an expert in the area (also it seems that the above paper is not yet published). On the other hand there are several published papers showing that $CP^2\sharp 3\overline{CP^2}$ admit exotic smooth structures.

Also, it might be worth to recall that by a theorem of Yau a complex surface homeomrophic to $CP^2$ always has the standard smooth structure (in other words $CP^2$ admits a unique holomorphic structure up to bi-homolorphism). While for $S^2\times S^2$ this is still unknown (is there a surface of general type homeomorphic to $S^2\times S^2$?)

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 It's not clear what the second part of your question means. Do you mean, "Are there constructions of bare topological 4-manifolds that are known to be homeomorphic to $\mathbb{CP}^2$ that don't have any obvious or natural candidate for a smooth structure?" – Robert Bryant Aug 16 2011 at 18:36
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 Yes, that was my question. – Yang Aug 16 2011 at 18:41
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 I did google about this, and find out a lot known about exotic smooth and symplectic structures on $CP^2#\n(-CP^2)$ for $n>1$. Best records ($n = 2,3,4$) are due to A. Akhmedov and B. D. Park and proved in the following published papers: springerlink.com/content/b27550567381396t springerlink.com/content/901221167510182u The cases $n = 0$ and $n = 1$ are open as Dmitri pointed out. – Yang Aug 17 2011 at 4:30
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 What about $\mathbb{R}^4$? This is a simpler 4-manifold than $S^2\times S^2$, and has uncountably many exotic smooth structures. – David Roberts Aug 17 2011 at 5:00
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 Perhaps you meant compact smooth manifold? – David Roberts Aug 17 2011 at 5:01
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