The rational characteristic class ring for oriented S^n x S^n-fibrations injects into smooth bundle cohomology, producing non-trivial classes in all degrees detected by bundles from cyclic subgroups of a finite-index subgroup of SL_2(Z).
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Rational characteristic classes of bundles with fibre a product of spheres
The rational characteristic class ring for oriented S^n x S^n-fibrations injects into smooth bundle cohomology, producing non-trivial classes in all degrees detected by bundles from cyclic subgroups of a finite-index subgroup of SL_2(Z).