Reference for the cohomology of SU

Question

Let SU be the infinite special group. Where can I find the following fact (state in part III of the Adam's blue book):

$H^{6}(SU,Z)=0$.

Thank you.

Answer 1

This follows, for instance, from Corollary 4D.3(b) of Hatcher's Algebraic Topology which says the cohomology of $SU(n)$ is an exterior algebra on generators in degree $3,5,\dots,2n-1$. It follows that the cohomology of $SU$ is an exterior algebra on generators in degrees $3,5,\dots$. There is no combination of these degrees that adds up to $6$ so $H^6$ is trivial.