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Oct 26, 2017 at 18:19 comment added mme This should also follow from the Lefschetz fixed point theorem, since $\Bbb{RP}^{2n}$ has the rational homology of a point. (Compact simplicial complexes whose homology is all torsion fall to the same argument.)
Oct 26, 2017 at 17:20 history answered Anubhav Mukherjee CC BY-SA 3.0