TITLE: Cohomology of the quotient of a Lie group by a finite subgroup
QUESTION [11 upvotes]: Let $G$ denote the $\operatorname{Spin}(n)$ group with $n>4$ and let $\Gamma$  be a cyclic subgroup $G$  of a prime order $p >2$. When does the projection $G \to G/\Gamma$ induce a surjection
between  cohomology groups $H^3$ with integral coefficients?

REPLY [7 votes]: Here's another approach using the Leray-Serre spectral sequence.  Using the fact that $\Gamma$ acts freely on $G$, the map $G \to G/\Gamma$ is a covering space, and so the cohomology of $G/\Gamma$ may be computed by the spectral sequence
$$E_2^{s, t} = H^s(\Gamma, H^t(G)) \implies H^{s+t}(G/\Gamma).$$
Here $H^s(\Gamma, M)$ is the s'th group cohomology of $\Gamma$ with coefficients in the $\Gamma$-representation $M$.  We record three facts to start:

If $\Gamma \cong \mathbb{Z} / p$, then $H^s(\Gamma, \mathbb{Z})$ is 0 for $s$ odd, and $\mathbb{Z} / p$ for $s$ even.
For $G=Spin(n)$, $H^t(G) = \mathbb{Z}, 0, 0, \mathbb{Z}$, for $t=0,1,2,3$.
For $p$ odd, $\mathbb{Z} / p$ cannot act nontrivially on $\mathbb{Z}$, since $Aut(\mathbb{Z}) = \mathbb{Z} / 2$.

Let's compute part of the $E_2$-term of the spectral sequence.  Fact 2 implies that the $t=1$ and $2$ rows vanish entirely.  Fact 3 implies that $H^0(G) = \mathbb{Z}$ and $H^3(G) = \mathbb{Z}$ are trivial $\Gamma$-modules, so the $t=0$ and $3$ rows are the group cohomology described in Fact 1.
Thus the only possible term that can contribute to $H^3(G/\Gamma)$ is $E_2^{0, 3} = H^3(G) = \mathbb{Z}$.  There is, however, a possiblity of a single differential in this region of the spectral sequence, namely
$$d_4: E_2^{0, 3} = \mathbb{Z} \longrightarrow E_2^{4, 0} = H^4(\Gamma, \mathbb{Z}) = \mathbb{Z} / p.$$
Therefore, $H^3(G/\Gamma)$ surjects onto $H^3(G)$ precisely when this differential $d_4= 0$.  We note that if it's not 0, it is surjective; thus at worst $H^3(G/\Gamma)$ may be identified with an index $p$ subgroup of $H^3(G)$.
So, how do we compute $d_4$?  I claim that it's given as:
$$H^3(G) \cong H^4(BG) \to H^4(B\Gamma) = H^4(\Gamma)$$
where the map is the restriction in cohomology, induced by the (inclusion) homomorphism  $\Gamma \subseteq G$.  This can be seen, for instance, by comparing this with the spectral sequence for the (rather dumb) fibration $G \to G/G=pt$.
So a long winded answer to your question is: The map is surjective if and only if none of the $H^4(BG) = \mathbb{Z}$ is supported on $H^4(\Gamma) = \mathbb{Z} /p$.  I would imagine that determining when that is the case is highly dependent upon the subgroup in question.