In general: if one knows the cohomology group of some manifold ${\cal M}$, i.e. $H^n ({\cal M})$, are there known results for the same cohomology group $H^n (X)$ of a submanifold $X \subset {\cal M}$? If so I would really appreciate any pointers to the literature.

I am in particular interested in the following case of an empty second cohomology group of a Lie group $G$. Given that $H^2 (G) = \emptyset$ and $F$ is a simply connected subgroup of $G$, is it obvious what is $H^2(F)$? Is it empty as well? Not sure if the following helps to narrow it down but in the particular example I am studying, $Lie(F)$ is also isotropic with respect to a symmetric bilinear form on $Lie(G)$.

In general: if one knows the cohomology group of some manifold ${\cal M}$, i.e. $H^n ({\cal M})$, are there known results for the same cohomology group $H^n (X)$ of a submanifold $X \subset {\cal M}$? If so I would really appreciate any pointers to the literature.

I am in particular interested in the following case of an empty second cohomology group of a Lie group $G$. Given that $H^2 (G) = 0$ and $F$ is a simply connected subgroup of $G$, is it obvious what is $H^2(F)$? Is it vanishing as well? Not sure if the following helps to narrow it down but in the particular example I am studying, $Lie(F)$ is also isotropic with respect to a symmetric bilinear form on $Lie(G)$.