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

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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 '11 at 19:29
continuous would be enough. Thanks! –  COhrt Dec 2 '11 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 '11 at 3:35