Timeline for Is there a smooth $4$-manifold homeomorphic but not diffemorphic to $CP^2$?
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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 |