# Is there any kind of relationship concerning sensible homology of X and also X/G

When $G$ is limited, the sensible cohomology of $X/G$ is the set factor set $H^*(X;\mathbb{Q})^G$. This is confirmed in Grothendieck is Tohoku paper (Theorem 5.3.1 and also the Corollary to Proposition 5.3.2).

So if the sensible cohomology of $X$ is unimportant, the very same holds true for $X/G$. And also reasonably the cohomology and also homology are isomorphic.

For paracompact Hausdorff rooms, these cohomology teams can be required the Cech cohomology teams. Keep in mind that if $X$ is homotopy equal to a CW facility, after that Cech cohomology concurs with single cohomology. You could additionally intend to consider Oscar Randall - Williams remarks below : https://mathoverflow.net/questions/18898/grothendiecks-tohoku-paper-and-combinatorial-topology/30015#30015.

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