Published online by Cambridge University Press: 13 March 2017
We study tautological rings for high-dimensional manifolds, that is, for each smooth manifold
$M$
the ring
$R^{\ast }(M)$
of those 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.
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