Question

This is my first question.

Take a simple, connected, compact, simply connected Lie group $ G$ (dim $G\geq 3$).
The cohomology of $G$ with integer coefficients is $H^{1,2}(G,\mathbb{Z})\cong 0$, $H^{3}(G,\mathbb{Z})\cong\mathbb{Z}$.
I'm looking for a generator of $H^{3}(G,\mathbb{Z})$.
It should be something of the form $\alpha K(\theta,[\theta,\theta])$, where $\theta$ denotes the left Maurer-Cartan-Form, $K$ is the Killing-Form and $\alpha$ is a scaling factor.

Answer 1

The third cohomology group of the Lie algebra (with coefficients in the trivial module) is generated by the cocyle $\phi(x,y,z)=K(x,[y,z])$. The usual proof that the de Rham cohomology of the group is the same as the Lie algebra cohomology of the corresponding Lie algebra gives you a more or less explicit $3$-form generating $H^3_{\mathrm{dR}}(G,\mathbb R)$. It gives you a generator of integral cohomology up to scaling. I don't know what the factor should be, though...