Is there a smooth $4$manifold homeomorphic but not diffemorphic to $CP^2$? Are there known nonsmooth examples homeomorphic $CP^2$?
closed as off topic by Ryan Budney, Ian Agol, Dan Petersen, Loop Space, S. Carnahan♦ Aug 17 '11 at 8:44Questions on MathOverflow are expected to relate to research level mathematics within the scope defined by the community. Consider editing the question or leaving comments for improvement if you believe the question can be reworded to fit within the scope. Read more about reopening questions here.If this question can be reworded to fit the rules in the help center, please edit the question. 


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 $S^2\times S^2$?) 

