A friend of mine told me that the cohomology of $\pi_1(M)$ was isomorphic to the cohomology of the manifold $M$. Is that true (maybe there are some hypothesis) ? Does someone know a reference for this result, and maybe an explanation why this should be intuitively true ?
Thanks in advance.
It is true if you assume that $M$ is an Eilenberg-Maclane space, i.e. that $M$ is a $K(\pi_1 M,1)$ space. You can find a proof in the book "Cohomology of groups" by Brown, in a general context where $M$ is a CW-complex.
Without that hypothesis it is false, take $M=S^2$ for example.