Let $\mathscr{A}_g$ be the moduli space of principally polarized Abelian varieties of dimension $g$.  The complex-analytic space (stack) associated to $\mathscr{A}_g$ is a $K(\pi,1)$; that is, its only non-vanishing homotopy group is $\pi_1$, which is $Sp_{2g}(\mathbb{Z})$.  In particular, the cohomology of $\mathscr{A}_g$ is the same as the cohomlogy of $Sp_{2g}(\mathbb{Z})$.

Jesse Silliman, a graduate student at Stanford, has told me an argument that shows that this is in some sense maximally untrue when one considers the "etale homotopy type" of $\mathscr{A}_g$.