Homology of the curve complex and the Steinberg module of the mapping class group

Harer has shown that the mapping class group is a virtual duality groupmirroring the work of Borel-Serre on arithmetic groups in semisimple Q-groups. Just as the homology of the rational Tits building provides the dualizing module for any torsion-free arithmetic group, the homology of the curve complex is the dualizingmodule for any torsion-free, finite index subgroup of the mapping class group. The homology of the curve complex was previously known to be an infinitely generated free abelian group, but to date, its structure as a mapping class group module has gone unexplored. In this paper we give a resolution for the homology of the curve complex as a mapping class group module. From the presentation coming from the last two terms of this resolution we show that this module is cyclic and give an explicit single generator. As a corollary, this generator is a homologically nontrivial sphere in the curve complex.