  We study tautological rings for high dimensional manifolds, that is, for each
smooth manifold $M$ the ring $R^*(M)$ of those of characteristic classes of
smooth fibre bundles with fibre $M$ which is generated by generalised
Miller--Morita--Mumford classes. We completely describe these rings modulo
nilpotent elements, when $M$ is a connected sum of copies of $S^n \times S^n$
for $n$ odd.
