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In particular, I am wondering if $H^1_{DR}(G)\neq 0$ implies that the group can written as a semidirect product of $\mathbb{S^1}$ and something else, with the $\mathbb{S^1}$ factor being responsible for the first cohomology group. I have no idea if this is true. In case it is not clear by $H^1_{DR}(G)$, I mean the De Rham cohomology of the underlying manifold associated to a group $G$. (I understand this is isomorphic to some kind of Lie algebra cohomology, but as of now know nothing about Lie algebra cohomology).

Note that I originally asked this on mathstackexchange and received no answers so I hope this was because perhaps it was more appropriate for here... I have deleted the question on stackexchange.

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  • $\begingroup$ I guess if you state your question in terms of fundamental group instead of the De-Rham cohomology, more people will take a look at your question. As a matter of fact, there is no reason to choose De-Rham cohomology over other cohomologies (Cech, Singular, etc.), so you might want to remove $_{DR}$ from the notation anyway. $\endgroup$ – user43326 Apr 18 '18 at 10:28
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    $\begingroup$ @user43326 As long as you consider real (or rational) coefficients. For example $H^1(SO(3); \mathbb{F}_2) \neq 0$. $\endgroup$ – Najib Idrissi Apr 18 '18 at 10:32
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    $\begingroup$ There can be a good reason to choose de Rham cohomology, namely that the OP's familiar with it more than with other theories. I don't think it requires to restate the question. Comments giving useful restatements of the question are very welcome. Side note: such a notation as $H^*(G,\mathbf{R})$ would give a confusion with the group cohomology. $\endgroup$ – YCor Apr 19 '18 at 8:21
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The de Rham cohomology is non-trivial if and only if the connected component of $G$ contains a non-trivial central torus.

We can assume that $G$ is connected. Then $H^1_{dR}(G)$ is isomorphic with $H_1(G,{\mathbb R})^*$ by Poincare duality. By Hurwitz's Theorem, $H_1(G)$ is the maximal abelian quotient of $\Gamma$, so $H^1_{dR}(G)\cong Hom(\Gamma,{\mathbb R})$, where $\Gamma$ is the fundamental group. Now by, say, Theorem 7.1 of the book by Broecker and tom Dieck, the fundamental group is a quotient of two lattices, so is of the form $F\times {\mathbb Z}^r$, where $F$ is a finite abelian group and $r$ is the dimension of a maximal central torus in $G$. It follows that $H_{dR}^1(G)\cong{\mathbb R}^r$.

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