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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 $Lie(G)$.

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    $\begingroup$ The second cohomology of any LIe group is trivial. $\endgroup$
    – Ben McKay
    Commented Jun 3, 2021 at 10:27
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    $\begingroup$ (Slight correction to Ben McKay's comment: this is true of simply connected Lie groups.) Also, a group is not empty. You mean to say $H^2(G) = 0$. $\endgroup$
    – mme
    Commented Jun 3, 2021 at 11:01
  • $\begingroup$ Thanks! That makes the second part of my question quite trivial :) $\endgroup$
    – cherzieandkressy
    Commented Jun 3, 2021 at 12:23
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    $\begingroup$ There are of course relations, given by Mayer-Vietoris etc. But these answers will need a lot more imput of what $X$ is: Whitney's theorem says that any manifold is a submanifold of $\mathbb{R}^n$. The characteristic classes of the tangent/normal bundle of $X$ also have strong relations with characteristic classes of $M$. $\endgroup$
    – Thomas Rot
    Commented Jun 3, 2021 at 12:37
  • $\begingroup$ One possibility would be to relate the cohomology of $G$ with the cohomology of $F$ by means of the Leray spectral sequence associated with the quotient map $\pi : G \mapsto G/F$. $\endgroup$
    – Max Reinhold Jahnke
    Commented Jul 13, 2021 at 14:40

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