The rational homology of quotients of the curve complex

Abstract

Let ${\mathrm{Mod}}_{g,n}$ be the mapping class group of a genus-$g$ surface with $n$ punctures and let ${\mathcal{C}}_{g,n}$ be its curve complex. For a finite-index subgroup $G<{\mathrm{Mod}}_{g,n}$, Boggi proved that ${\tilde{H}}_k({\mathcal{C}}_{g,n}/G;\mathbb{Q})=0$ for $g+n\gg k$. This plays an important role in the work of Putman-Wieland on the virtual first Betti number of ${\mathrm{Mod}}_{g,n}$. The proof of Boggi's theorem uses mixed Hodge theory and is embedded in his flawed paper purporting to show that the mapping class group has the congruence subgroup property. We give a detailed exposition of Boggi's proof aimed at geometric topologists.

arXiv comment41 pages, 7 figures