I would like to know what are the group cohomology classes $H^d[Z_n, Z_2]$, $H^d[U(1), Z_2]$, $H^d[SO(n), Z_2]$, $H^d[SU(n), Z_2]$, etc. Thanks! (Here the group cohomology $H^d[G, M]$ for a group $G$ is the topological cohomology of the classifying space $BG$, $H_{top}^d[BG, M]=H^d[G, M]$.)
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For the latter three, here is the integer-coefficients (apply Kunneth formula to get your mod-2 coefficients:
Group cohomology of compact Lie group with integer coeffient
As for the first: $H^i(\mathbb{Z}_n)$ is $\mathbb{Z}_n$ for $i$ even, and zero otherwise. Again, apply Kunneth formula for $\mathbb{Z}_2$-coefficients.
In general, for a finite group $G$, $H^*(G,M)$ is a $\mathbb{Z}_{|G|}$-module for $n>0$. And If $M$ has exponent $p$ (prime) then $H^*(G,M)$ is a $\mathbb{Z}_p$-vector space. So for $gcd(n,2)=1$ we must have $H^*(\mathbb{Z}_n,\mathbb{Z}_2)=0$ (in positive dimensions).
answered Oct 16, 2011 at 0:20
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$\begingroup$ Dear Chris: Kunneth formula relates $H^*[G_1\times G_2, M]$ to $H^*[G_1, M]$ and $H^*[G_2, M]$. I do not know how Kunneth formula relates $H^*[G, Z_2]$ to $H^*[G, Z]$. $\endgroup$ Commented Oct 16, 2011 at 0:34
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2$\begingroup$ .... No, it relates $H^*(G_1\times G_2,M\otimes M')$ to $H^*(G_1,M)$ and $H^*(G_2,M')$. Take $G_2=0$ and $M=\mathbb{Z}_2$ and $M'=\mathbb{Z}$. $\endgroup$ Commented Oct 16, 2011 at 1:24
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1$\begingroup$ (flip the M and M' entries around actually; but I guess the typo is evident) $\endgroup$ Commented Oct 17, 2011 at 8:05