Homological infiniteness of Torelli groups

The rational homology of the Torelli group of genus g relative to n distinguished points and r fixed embedded disks is proved to be infinite dimensional if g is sufficiently large relative to n+r. In particular, the rational homology of the (classical) Torelli group of genus g is infinite dimensional when g⩾7. In addition, the rational homology of the subgroup of the Torelli group of genus g generated by all the Dehn twists along separating simple closed curves is proved to be infinite dimensional when g⩾2.