Timeline for Is the mapping class group of $\Bbb{CP}^n$ known?
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 7, 2020 at 15:58 | comment | added | archipelago | Regarding your last comment: Wall (MR0156354, MR0177421) and Kreck (MR0561244) have computed various mapping class groups of highly connected manifolds up to extension problems. | |
| Jan 5, 2020 at 10:52 | comment | added | skupers | Cool, I wasn't aware of that paper! | |
| Jan 5, 2020 at 5:33 | comment | added | mme | @skupers Kreck points out to me that the oriented smooth mapping class group $\pi_0 \text{Diff}^+(\Bbb{CP}^3)$ was calculated by Brumfiel to be $\Bbb Z/4$, all of whose elements are represnted by diffeomorphisms supported in a ball. I did not check but presumably $\pi_0 \text{Diff}$ is the dihedral group on 8 elements. | |
| Dec 29, 2019 at 17:17 | comment | added | skupers | Kreck and Su have announced a paper containing the case n=3, see Remark 1.4 of arxiv.org/abs/1907.05693. You could try asking one of them. | |
| Dec 29, 2019 at 17:09 | comment | added | mme | Only small homotopy groups seem relevant in Hatcher's computation, though, so I hope that doesn't cause too much trouble. Thanks for the reference! | |
| Dec 29, 2019 at 17:02 | comment | added | Igor Belegradek | There are spectral sequences developed by Schultz, and Becker-Schultz that converge to homotopy groups of the identity component of the space of self-maps of $CP^n$. See section 6 in arxiv.org/abs/0912.4874 and references therein. The computational difficulties are substantial, e.g., in the linked paper we only manage to get partial information on $\pi_7$. | |
| Dec 29, 2019 at 15:40 | history | asked | mme | CC BY-SA 4.0 |