As is well known, a differential manifold has trivial degree-$0$ de Rham cohomology $H^0(M) = 0$ if and only if it is connected.
It seems that the degree-$1$ de Rham cohomology group $H^1(M)$ being non-trivial implies that $M$ is not simply connected. I guess the converse is not true, but I would like to know of a proof.
Do there exist similar relationships between higher de Rham cohomology groups and the homotopy groups of differential manifold?