Group cohomology of compact Lie group with integer coeffient  It is known that group cohomology class $H^d[U(1),Z]$ is Z for even d and 0 for odd d.
Do we know $H^d[G,Z]$ for $G=SO(3)$, $SU(2)$ and other compact Lie group?
Also is the Borel-group-cohomology class $H^d[G,R]$ alway trivial for compact Lie group $G$?
 A: A collection of relevant references (general, standard as well as specific ones) is here:
http://ncatlab.org/nlab/show/group+cohomology#OnTopologicalGroups
A: For the group $SU(2)=S^3$ we just have $H^*(BSU(2);\mathbb{Z})=\mathbb{Z}[c_2]$ (where $c_2\in H^4$).  More generally, for all $n$ we have
\begin{align*}
 H^*(BU(n);\mathbb{Z}) &= \mathbb{Z}[c_1,\dotsc,c_n] \\\\
 H^*(BSU(n);\mathbb{Z}) &= \mathbb{Z}[c_2,\dotsc,c_n] \\\\
 H^*(BSp(n);\mathbb{Z}) &= \mathbb{Z}[p_1,\dotsc,p_n]
\end{align*}
with $c_i\in H^{2i}$ and $p_i\in H^{4i}$.  
Now let $V$ be the tautological $3$-plane bundle over the space $X=BSO(3)$.  This has Stiefel-Whitney classes $w_2\in H^2(X;\mathbb{Z}/2)$ and $w_3\in H^3(X;\mathbb{Z}/2)$.  There is also a Bockstein element $v=\beta(w_2)\in H^3(X;\mathbb{Z})$ (which satisfies $2v=0$) and a Chern class $c=c_2(\mathbb{C}\otimes V)\in H^4(X;\mathbb{Z})$.  The mod two reduction map $\rho$ satisfies $\rho(v)=Sq^1(w_2)=w_3$ and $\rho(c)=w_2^2$.  If I've got everything straight, one can check using the Bockstein spectral sequence that
$$ H^*(BSO(3);\mathbb{Z}) = \mathbb{Z}[v,c]/(2v). $$
It is not possible to be similarly explicit about $H^*(BSO(n);\mathbb{Z})$ for general $n$ (although $H^*(BSO(n);\mathbb{Z}/2)$ and $H^*(BSO(n);\mathbb{Q})$ are fairly straightforward).
A: Check out 
http://www.math.cornell.edu/~hatcher/SO/comments.pdf
