Timeline for Free group actions on complex projective spaces
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 4, 2011 at 7:41 | vote | accept | COhrt | ||
| Dec 3, 2011 at 21:01 | answer | added | Vitali Kapovitch | timeline score: 10 | |
| Dec 3, 2011 at 16:29 | comment | added | Vitali Kapovitch | When $n=2$ I believe the following is a free $\mathbb Z_2$ action. Look at the action on $\mathbb C^\infty$ with the generator acting by $(z_1,z_2,z_3,z_4,\ldots)\mapsto (-\bar z_2,\bar z_1,-\bar z_4,\bar z_3,\ldots)$. The induced action on $\mathbb{CP}^\infty$ is free. Of course, nothing similar can possibly work for $n>2$ because if the action leaves any $\mathbb{CP}^N$ with $N<\infty$ invariant then the square of the generator acts trivially on cohomology and hence has a nonzero Lefschetz number. | |
| Dec 3, 2011 at 7:03 | comment | added | Will Sawin | As an EML space, a $\mathbb Z/n$ action on $\mathbb CP^{\infty}$ is a $\mathbb $Z/n$ action on every cohomology group $H^2(X,\mathbb Z)$. Also, it permutes the complex line bundles on every space. But neither of these facts seems to help. | |
| Dec 3, 2011 at 6:40 | answer | added | Alain Valette | timeline score: 2 | |
| Dec 3, 2011 at 3:35 | comment | added | Dylan Wilson | One thing we know is that the action of the generator is homotopic to a map that induces $\pm 1$ on $H^2$. This is stronger than what we can usually say since we happen to know that $[CP^\infty, CP^\infty] = H^2CP^\infty$. I don't see how this can help, though, since Lefschetz fails badly for non-compact spaces... | |
| Dec 2, 2011 at 19:31 | comment | added | COhrt | continuous would be enough. Thanks! | |
| Dec 2, 2011 at 19:29 | comment | added | Will Sawin | Linear or continuous? (or some other criterion?) Linear, there are none for any projective space, so I doubt there are any here. | |
| Dec 2, 2011 at 18:35 | history | asked | COhrt | CC BY-SA 3.0 |