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 \(\mathscr{A}_g\). (Jesse proposed it as an argument for why (compactifications of) \(\mathscr{A}_g\) and its level covers have trivial \(H^1\) for \(g>1\), but a slight enhancement actually shows much more). Moreover, the argument is anabelian, which always gets me excited. A similar argument should work for many Shimura varieties of rank \(>1\), but I'll just sketch it in the case of \(\mathscr{A}_g\) with \(g>1\).

Let \(G\) be the etale fundamental group of \(\mathscr{A}_g\), namely \(Sp_{2g}(\hat{\mathbb{Z}})\). This is a pretty pathological group (for example, its center is infinitely generated), but whatever. Given any continuous representation $$\rho: G\to GL_n(\mathbb{Z}_\ell),$$ there is a natural map $$\iota: H^*(G, \rho)\to H^*(\mathscr{A}_g, \mathscr{F}_\rho),$$ where \(\mathscr{F}_\rho\) is the lisse sheaf on \(\mathscr{A}_g\) associated to \(\rho\), and the left hand side is continuous cohomology. Now suppose that \(\rho\) actually comes from some representation of \(Sp_{2g}(\mathbb{Z})\) into \(GL_n(\mathbb{Z})\).

I claim that in this case \(\mathscr{F}_\rho\) shows up in the cohomology of some power of the universal Abelian variety, and is cut out by absolute Hodge cycles, so in particular \(\rho\) canonically extends to a representation of the arithmetic fundamental group of \(\mathscr{A}_g\) (possibly after some finite field extension). Moreover, at any \(K\)-point \(x\) of \(\mathscr{A}_g\), the action of the Galois group of \(K\) on \(\rho\) factors through its tautological action on the Tate module of the fiber of the universal Abelian variety at \(x\).

Let's choose a geometric point at some CM point \(x\in \mathscr{A}_g\), with associated CM field \(E\). Then the action of the Galois group of \(E\) on \(G\) factors through the Artin reciprocity map by the main theorem on CM; hence the action on \(H^*(G, \rho)\) is Abelian. But the action on the etale cohomology of \(\mathscr{A}_g\) is independent of basepoint, so running over all CM basepoints, we can see that anything in the image of \(\iota\) has to be of Tate type! In particular, none of the interesting cohomology of \(\mathscr{A}_g\) comes from its etale fundamental group. The same argument works for any level cover of \(\mathscr{A}_g\). Thus in some sense these things are as far from \(K(\pi, 1)\)'s as possible.

Of course, \(H^1(\mathscr{A}_g)=H^1(G)\) more or less by definition, so this shows that \(H^1(\mathscr{A}_g)\) (and any level cover thereof) is pure of Tate type --- in particular, any smooth compactification of (a level cover of) \(\mathscr{A}_g\) must have \(H^1=0\).

Exercise -- how did I use that \(g>1\)? (Obviously this last sentence is untrue when \(g=1\)). Hint: It's when I said that the action of Galois on \(G\) factors through the Artin reciprocity map; in this step, I used the solution to the congruence subgroup problem, which is of course false for \(g=1\).