jats:titleAbstract</jats:title> jats:pWe describe the ring structure of the rational cohomology of the Torelli groups of the manifolds $\#gSn\times Sn$ in a stable range, for $2n\ge 6$. Some of our results are also valid for $2n=2$, where they are closely related to unpublished results of Kawazumi and Morita.</jats:p>