Timeline for exotic differentiable structures on manifolds in dimensions 5 and 6
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
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| Jul 22, 2010 at 11:57 | comment | added | Yalcin | Akhmedov-D. Park paper on exotic $\Bbb CP^{2} \# k\overline{\Bbb CP^2}$ for $k \geq 2$ long predate Fintushel-Stern's preprint. Their papers written in 2007 and published in Inven. Math. Also, Akhmedov-Park recently posted a preprint on exotic $S^2 \times S^2$. I am wondering if their method extends to $\Bbb CP^{2} \# \overline{\Bbb CP^{2}}$ or $\Bbb CP^2$? | |
| Apr 22, 2010 at 17:13 | comment | added | Tim Perutz | Jeff, exotic K3's long predate Chen's work - examples were identified using Donaldson theory (circa 1990) and Fintushel-Stern distinguished infinitely many using knot surgery in a 1998 paper (MR165030). | |
| Apr 22, 2010 at 16:03 | comment | added | Jeffrey Giansiracusa | Just to add to the list, Weimin Chen (e.g. arXiv:0709.1710) has constructed and studied infinitely many exotic smooth structures on a K3 surface. | |
| Apr 22, 2010 at 15:52 | comment | added | Tim Perutz | Andrey, 6 is correct; you could replace it by any $n\geq 2$ (cf. Fintushel-Stern's latest...). Moreover the number of different smooth structures is in each case countably infinite, while for simply connected manifolds of higher dimension it is always finite. | |
| Apr 22, 2010 at 15:34 | comment | added | Andrey Gogolev | My understanding is that it is rather large industry. There're exotic smooth structures on compact simply connected 4-manifolds. For example $\Bbb{CP}^2\#6\overline{\Bbb{CP}^2}$ (not sure about 6) admits exotic structure. People are constantly making progress making the example "smaller" in second homology. Some names here are Park, Stipsicz, Szabo, Akhmedov. | |
| Apr 22, 2010 at 15:13 | comment | added | symplectomorphic | While I'm at it, I might as well ask whether there's a simple compact or closed counterexample in dimension 4. As far as I know, this question for the 4-sphere is unresolved: I doubt it would be any simpler for any other closed 4-manifold, but perhaps I'm wrong? | |
| Apr 22, 2010 at 14:37 | comment | added | Andrey Gogolev | Yes, that's right, spheres in dimensions 5 and 6 are diffeomorphic to the standard $S^5$ ($S^6$). This is what I meant by calling them standard. I would like to point out that my and Igor's answer do not contradict each other. They complement each other very nicely. | |
| Apr 22, 2010 at 4:53 | vote | accept | symplectomorphic | ||
| Apr 22, 2010 at 4:37 | comment | added | symplectomorphic | Wikipedia tells me that the result actually does hold for spheres in dimensions 5 and 6, contrary to what you say: en.wikipedia.org/wiki/… That's the main reason I asked the question. Or perhaps by calling the spheres "standard" you didn't mean they were standard counterexamples, but only that they carry unique differential structures. Otherwise, thanks! | |
| Apr 22, 2010 at 4:10 | history | answered | Andrey Gogolev | CC BY-SA 2.5 |