  We prove a well known conjecture of Nikolai Ivanov which states that if $X$
is a surface of genus $\geq 3$ (with any number of punctures and boundary
components), $\rm{Mod}(X)$ is the mapping class group of $X$, and $K <
\rm{Mod}(X)$ is a finite-index subgroup, then $K$ does not virtually surject to
$\mathbb{Z}$. As a corollary of this we get that $H_1(Z; \mathbb{Q}) = 0$
whenever $Z$ is a finite cover of $\mathcal{M}_{g,n}$, the moduli space of
complex algebraic curves of genus $g\geq 3$ with $n$ marked points.
