MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

## Free group actions on complex projective spaces

Hi,

is there a free group action of the cyclic group $\mathbb{Z}/n\mathbb{Z}$ on the infinite dimensional projective space $\mathbb{CP}^\infty$ for every $n\in \mathbb{N}$? And if there is one, how does it work?

Thanks

-
Linear or continuous? (or some other criterion?) Linear, there are none for any projective space, so I doubt there are any here. – Will Sawin Dec 2 2011 at 19:29
continuous would be enough. Thanks! – COhrt Dec 2 2011 at 19:31
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... – Dylan Wilson Dec 3 2011 at 3:35