Sphericity of $\kappa$-classes 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 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 these bounds are attained. The proof is 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 spin, has a non-trivial rational Pontryagin class and admits such a metric. This is done by constructing $M$-bundles over spheres with non-vanishing $\hat{A}$-genus. Furthermore, we give a vanishing theorem for generalised Morita--Miller--Mumford classes for fiber homotopy trivial bundles over spheres. In the appendix co-authored by Jens Reinhold it is (partially) determined which classes of the rational oriented cobordism ring contain an element that fibers over a sphere of a given dimension.