Let $X$ a manifold and $G$ a group that acts on X freely, properly, and smoothly. The quotient $X/G$ has a manifold structure such the projection $\pi:X\rightarrow X/G$ is a submersion. My question is: how are related the De Rham´s cohomology groups of $X$ with the De Rham´s cohomology groups of the quotient?

Thank you for your time.

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    $\begingroup$ There is a spectral sequence going from the cohomology $H^*(G,H^*(X))$ to that of $X/G$. Similarly, the projection map $X\to X/G$ is a fibration, and there is the spectral sequence of that fibration. $\endgroup$ – Mariano Suárez-Álvarez Oct 19 '17 at 13:26
  • $\begingroup$ Where can I read about that? Do you know any reference? $\endgroup$ – user320224 Oct 19 '17 at 13:32
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    $\begingroup$ Bott's book on algbraic topology. McCleary's book on spectral sequences. $\endgroup$ – Mariano Suárez-Álvarez Oct 19 '17 at 13:35
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    $\begingroup$ What's your goal? What do you want to do with this relationship? $\endgroup$ – user98602 Oct 19 '17 at 14:50

If the group $G$ is finite, $\pi ^* = H_k^*(X /G, {\bf R} )\to H_k^*(X, {\bf R})$ is injective. You can prove it by an averaging argument or say that (for the pull back Riemannian metric) the pull back of an harmonic form is harmonic.

In fact $ H_k^*(X/G,{\bf R})\subset H_k^*(X, {\bf R})$ is the set of $G$-invariant vectors of $H_k^*(X,{\bf R})$ viewed as a finite dimensional representation of $G$.

Another basic result is about the Euler-Poincaré characteristic : $\chi (X/G) \times \vert G \vert= \chi (X)$.

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