If we know the rational homology of $X$ is $0,$ can we get some information about the rational homology of $X/G,$ where $G$ is a finite group? Thank you very much for the answers!
Is there any relation about rational homology of X and X/G
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algebraic-topology
homology-cohomology
homological-algebra
1 Answers
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When $G$ is finite, the rational cohomology of $X/G$ is the fixed point set $H^*(X;\mathbb{Q})^G$. This is proven in Grothendieck's Tohoku paper (Theorem 5.3.1 and the Corollary to Proposition 5.3.2).
So if the rational cohomology of $X$ is trivial, the same is true for $X/G$. And rationally the cohomology and homology are isomorphic.
For paracompact Hausdorff spaces, these cohomology groups can be taken to be the Cech cohomology groups. Note that if $X$ is homotopy equivalent to a CW complex, then Cech cohomology agrees with singular cohomology. You might also want to look at Oscar Randall-Williams comments here: https://mathoverflow.net/questions/18898/grothendiecks-tohoku-paper-and-combinatorial-topology/30015#30015.
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1I edited here and on MO to include the more precise reference. Just to be clear, I am referring to the same part of the paper as Mariano (Chapter 5 is sort of the 6th section, if you count the introduction). – 2010-09-17