$U(1)$ 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](http://en.wikipedia.org/wiki/Table_of_Lie_groups) $SU(3)$ is a compact, connected and simply connected 8-dimensional manifold. [This](https://mathoverflow.net/questions/35125/original-references-for-the-homotopy-groups-pi-5-of-su3-and-pi-4-of-su2) MO post says that its $\pi\_5$ is $\mathbb{Z}$ thus it can not be homeomorphic to $S^8$(e.g.: see [this](http://en.wikipedia.org/wiki/Homotopy_groups_of_spheres) 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](https://mathoverflow.net/questions/57034/manifolds-isomorphic-to-su3-closed) question was trying to know... 


Anyway, is it known any manifold diffeomorphic to $SU(3)$? 



Topology of SU(3)
Apart from jokes, an answer which may satisfy you is the following: $SU(3)$ is a $S^3$-bundle over $S^5$. To see this just consider the defining representation of $SU(3)$ on $\mathbb{C}^3$; this induces a transitive action of $SU(3)$ on the unit sphere of $\mathbb{C}^3$, which is $S^5$. Since the stabilizer of a point for this action is $SU(2)$ this exhibits $SU(3)$ as an $SU(2)$-bundle over $S^5$, and as you wrote $SU(2)$ is diffeomophic to $S^3$. Now, the next question is: which $SU(2)$-bundle over $S^5$ is $SU(3)$? to answer this, recall that isomorphic classes of principal $SU(2)$-bundles over (a not too wild) topological space $X$ are in bijection with the set $[X,BSU(2)]$ of homotopy classes of maps from $X$ to the classifying space of $SU(2)$. So in the case at hand you are interested in $[S^5,BSU(2)]= \pi\_5(BSU(2))= \pi\_4(SU(2))= \pi\_4(S^3)= \mathbb{Z}/2\mathbb{Z}$. So there are only two $S^3$-bundles over $S^5$, the trivial one and the nontrivial one: $SU(3)$ is the nontrivial one (otherwise one would have $\pi\_4(SU(3))=\mathbb{Z}/2\mathbb{Z}$, which is not the case: it is $\pi\_4(SU(3))=\{0\}$).


