# De Rham cohomology of a quotient manifold

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?

• 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. – Mariano Suárez-Álvarez Oct 19 '17 at 13:26
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 : $\chi (X/G) \times \vert G \vert= \chi (X)$.