TITLE: Topology of SU(3)
QUESTION [24 upvotes]: $U(1)$ is diffeomorphic to $S^1$ and $SU(2)$ is to $S^3$, but apparently it is not true that $SU(3)$ is diffeomorphic to $S^8$ (more bellow). Since $SU(3)$ appears in the standard model I would like to understand its topology.
By one of the tables here $SU(3)$ is a compact, connected and simply connected 8-dimensional manifold. This MO post says that its $\pi_5$ is $\mathbb{Z}$ thus it can not be homeomorphic to $S^8$(e.g.: see this wiki article). Even if it was a homotopy sphere Poincaré conjecture would not be helpful (at least in the smooth category: there exists exotic 8-spheres, right?).   
I guess that this is what the author of this question was trying to know... 
Anyway, is it known any manifold diffeomorphic to $SU(3)$?

REPLY [5 votes]: As explained in the survey paper The Geometry of Compact Lie Groups by L.J. Boya, and by several other people here, $SU(3)$ is an $SU(2)\cong S^{3}$ fiber bundle over $S^{5}$, but in this case in particular, you can say a little more: The $SU(2)$-bundles over $S^{5}$ are classified by $\pi_{4}(S^{3})=\mathbb{Z}/2\mathbb{Z}$. We know that $SU(3)$ is not the trivial bundle, hence, it should be the unique square root of the trivial principal bundle.