Characteristic numbers of manifold bundles over spheres and positive curvature via block bundles

Abstract Given a simply connected manifold M , we completely determine which rational monomial Pontryagin numbers are attained by fiber homotopy trivial M -bundles over the k -sphere, provided that k is small compared to the dimension and the connectivity of M . Furthermore, we study the vector space of rational cobordism classes represented by such bundles. We give upper and lower bounds on its dimension, and we construct manifolds for which the lower bound is attained. Our proofs are based on the classical approach to studying diffeomorphism groups via block bundles and surgery theory, and we make use of ideas developed by Krannich–Kupers–Randal-Williams. As an application, we show the existence of elements of infinite order in the homotopy groups of the spaces of positive Ricci and positive sectional curvature, provided that M is $\operatorname {Spin}$ , has a nontrivial rational Pontryagin class and admits such a metric. This is done by constructing M -bundles over spheres with nonvanishing ${\hat {\mathcal {A}}}$ -genus. Furthermore, we give a vanishing theorem for generalized Morita–Miller–Mumford classes for fiber homotopy trivial bundles over spheres. In the appendix coauthored by Jens Reinhold, we investigate which classes of the rational oriented cobordism ring contain an element that fibers over a sphere of a given dimension.