Timeline for Is there a smooth $4$-manifold homeomorphic but not diffemorphic to $CP^2$? [closed]
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14 events
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| Sep 19, 2017 at 10:02 | comment | added | user74900 | note that Yau has shown in 1977 that if a complex surface is homeomorphic to $\mathbb{C}P^2$ then it is biholomorphic to it en.wikipedia.org/wiki/Fake_projective_plane | |
| Aug 17, 2011 at 16:13 | comment | added | Yang | My question was is that is there a construction which doesn't have a natural smooth structure. My guess is that there is no such 4-manifold. The question would be more interesting if I consider the same intersection form of $CP^2$, rather than the homeomorphism type of $CP^2$. | |
| Aug 17, 2011 at 15:47 | comment | added | Tim Perutz | There are plenty of non-smoothable 4-manifolds. However, one that is homeomorphic to $CP^2$ has a smooth structure: that of $CP^2$. | |
| Aug 17, 2011 at 15:31 | comment | added | Yang | I think extra argument is needed to prove that. The claim all compact topological 4 manifolds are smoothable is NOT correct. More examples follows from Donaldson's Theorem. | |
| Aug 17, 2011 at 15:13 | comment | added | Yang | For example, the Freedman's famous $E_8$ manifold is not smoothable. en.wikipedia.org/wiki/E8_manifold | |
| Aug 17, 2011 at 15:07 | comment | added | Yang | Can you always smooth topological 4-manifold? I know that not all the topological 4 manifolds are smoothable. | |
| Aug 17, 2011 at 14:51 | comment | added | Tim Perutz | 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. | |
| Aug 17, 2011 at 8:48 | comment | added | S. Carnahan♦ | 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. | |
| Aug 17, 2011 at 8:44 | history | closed |
Ryan Budney Ian Agol Dan Petersen Andrew Stacey S. Carnahan♦ |
off topic | |
| Aug 16, 2011 at 17:39 | vote | accept | Yang | ||
| Aug 16, 2011 at 17:35 | comment | added | Ryan Budney | 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. | |
| Aug 16, 2011 at 17:31 | history | edited | Ryan Budney |
added open problem tag, removed a few less informative tags
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| Aug 16, 2011 at 17:23 | answer | added | Dmitri Panov | timeline score: 22 | |
| Aug 16, 2011 at 17:12 | history | asked | Yang | CC BY-SA 3.0 |