Is there a smooth $4$manifold homeomorphic but not diffemorphic to $CP^2$? Are there known nonsmooth examples homeomorphic $CP^2$?
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This is a notorious open problem. For the moment the simplest compact fourmanifold 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 bihomolorphism). While for $S^2\times S^2$ this is still unknown (is there a surface of general type homeomorphic to $S^2\times S^2$?) 

