I am interested in computing the Cech cohomology (with integer coefficients) of the group $SU(3)$. I particularly care about $H^k(SU(3),\mathbb{Z})$ with $k=7$, although ideally I would like to be able to calculate all the other groups as well.
My attempt so far: First principles seems like a terrible idea, so I should come up with a better way. I know that I can think of $SU(3)$ is a principal $SU(2)$ bundle over $S^5$, so fiber bundle theory seems like a good idea, but I'm not really familiar with computing cohomology groups using fiber bundles.
I remember reading somewhere that $SU(3)$ has the same cohomology groups as $S^3 \times S^5$, but I am not sure if this holds for integer Cech cohomology (if it does, then I can simply use the Kunneth theorem to get $H^7 (SU(3),\mathbb{Z}) = 0$).
I suspect that the answer is either $0$ or $\mathbb{Z}_2$ (from physics), but I am unsure how to go about computing this.
It is correct that $SU(3)$ has the same cohomology as $S^3\times S^5$, so $H^7(SU(3),\mathbb{Z})=0$. You can prove this using the Serre spectral sequence for the fibration you describe, since $SU(2)\cong S^3$. For degree reasons, there can be no nontrivial differentials in the spectral sequence, and so it degenerates and gives an isomorphism $$H^*(SU(3),\mathbb{Z})\cong H^*(S^5,H^*(S^3,\mathbb{Z}))\cong H^*(S^5,\mathbb{Z})\otimes H^*(S^3,\mathbb{Z})$$ (the last isomorphism being by the universal coefficient theorem since the homology of $S^3$ and $S^5$ is finitely generated and free).
(Note that it does not matter that you are interested in Cech cohomology here; since $SU(3)$ is a smooth manifold and in particular a CW-complex, all ordinary cohomology theories such as Cech or singular cohomology are naturally isomorphic for it.)