Rational characteristic classes of bundles with fibre a product of spheres

Abstract

We prove the existence of many non-trivial characteristic classes of smooth oriented bundles with fibre a product $ S^{n}\times S^{n} $ of odd-dimensional spheres. We do so by proving injectivity of the map from the ring of rational characteristic classes of oriented fibrations with fibre $ S^{n}\times S^{n} $; the latter is proven by Berglund--Zeman to be isomorphic to the group cohomology of the symmetric powers of the standard representation of a certain finite-index subgroup $ \Gamma $ of $ \mathrm{SL}_{2}(\mathbb{Z}) $. These characteristic classes of smooth bundles are not generalised Miller--Morita--Mumford classes, and they exist in arbitrarily large cohomological degrees. Inspired by an example given by Morita, we provide a collection of smooth oriented $ S^{n}\times S^{n} $-bundles, indexed by cyclic subgroups of $ \Gamma $, which detect any given non-zero characteristic class of such fibrations.