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Aug 17, 2011 at 7:14 history edited Dmitri Panov CC BY-SA 3.0
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Aug 17, 2011 at 5:15 comment added Yang Yes, the original question was for compact and simply connected case. The examples mentioned above have the smallest possible Euler characteristics which admit exotic smooth structures. It open for 4-manifolds $S^4$, $CP^2$, and $CP^2#(-CP)^2$.
Aug 17, 2011 at 5:01 comment added David Roberts Perhaps you meant compact smooth manifold?
Aug 17, 2011 at 5:00 comment added David Roberts What about $\mathbb{R}^4$? This is a simpler 4-manifold than $S^2\times S^2$, and has uncountably many exotic smooth structures.
Aug 17, 2011 at 4:30 comment added Yang 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.
Aug 17, 2011 at 2:52 history edited Dmitri Panov CC BY-SA 3.0
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Aug 16, 2011 at 18:41 comment added Yang Yes, that was my question.
Aug 16, 2011 at 18:36 comment added Robert Bryant 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?"
Aug 16, 2011 at 17:42 comment added Yang Thanks Dmitri. What about the second part of my question? Are there non-smooth examples homeo to $CP^2$?
Aug 16, 2011 at 17:39 vote accept Yang
Aug 16, 2011 at 17:23 history answered Dmitri Panov CC BY-SA 3.0