Timeline for Free group actions on complex projective spaces
Current License: CC BY-SA 3.0
7 events
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| Dec 3, 2011 at 20:38 | comment | added | Alain Valette | @Vitali: Thanks a lot for your comment, which taught me things I was completely unaware of. But coming back to non-existence of linear free actions on $\mathbb{CP}^\infty$ (now with your model), I think that the same argument as for Hilbert space works, since the projections onto isotypic components can be defined purely algebraically, as elements of the complex group ring of $\mathbb{Z}/n$, and therefore make sense in any $\mathbb{Z}/n$-module... | |
| Dec 3, 2011 at 19:41 | comment | added | Vitali Kapovitch | @Alain I'm aware of the homotopy equivalence of the two models but I always considered the canonical model of $\mathbb{CP}^\infty$ to be the direct limit of $\mathbb{CP}^n$. And in this case a particular model makes a big difference because if we are only concerned with the homotopy type then there is always a free action of $\mathbb Z_n$ via Borel construction. In fact with your definition there is such an action on the original space because then $\mathbb{CP}^\infty$ and $\mathbb{CP}^\infty\times \mathbb S^\infty$ are homeomorphic since they are homotopy equivalent $l_2$-manifolds. | |
| Dec 3, 2011 at 19:18 | comment | added | Alain Valette | @Vitali: Your comment made me smile, because this is a recurring discussion between analysts and topologists (my definition of $\mathbb{CP}^\infty$ being the analysts'one - we are aware that we do not get a CW-complex!). The homotopy equivalence between the analyst's model and the topologist's model is proved e.g. by Guido Mislin in the appendix of my book "Introduction to the Baum-Connes conjecture", ETHZ Lecture Notes, Birkhauser 2000. | |
| Dec 3, 2011 at 18:54 | comment | added | Dylan Wilson | Also, you may as well prove the stronger statement that there is no such continuous action, via the Lefschetz fixed point theorem... the cohomology is concentrated in even degrees, so the Lefschetz number has to be nonzero. But I agree with Vitali, I don't think either fo these arguments can be extended to the $CP^\infty$ case | |
| Dec 3, 2011 at 16:20 | comment | added | Vitali Kapovitch | I don't think this works without some extra assumptions for $N=\infty$ because $\mathbb{CP}^\infty$ (at least the canonically constructed one) is smaller than the set of complex lines in a Hilbert space. | |
| Dec 3, 2011 at 7:39 | history | edited | Dan Ramras | CC BY-SA 3.0 |
added ticks to make brackets show up
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| Dec 3, 2011 at 6:40 | history | answered | Alain Valette | CC BY-SA 3.0 |