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Given a simply connected manifold $M$, we completely determine which rationalmonomial Pontryagin numbers are attained by fiber homotopy trivial $M$-bundlesover the $k$-sphere, provided that $k$ is small compared to the dimension of$M$. Furthermore we study the vector space of rational cobordism classesrepresented by such bundles. We give upper and lower bounds on its dimensionand we construct manifolds for which these bounds are attained. The proof isbased on the classical approach to studying diffeomorphism groups via blockbundles and surgery theory and we make use of ideas developed byKrannich--Kupers--Randal-Williams.  As an application, we show the existence of elements of infinite order in thehomotopy groups of the spaces of positive Ricci and positive sectionalcurvature, provided that $M$ is spin, has a non-trivial rational Pontryaginclass and admits such a metric. This is done by constructing $M$-bundles overspheres with non-vanishing $\hat{A}$-genus. Furthermore, we give a vanishingtheorem for generalised Morita--Miller--Mumford classes for fiber homotopytrivial bundles over spheres.  In the appendix co-authored by Jens Reinhold it is (partially) determinedwhich classes of the rational oriented cobordism ring contain an element thatfibers over a sphere of a given dimension.

Sphericity of $\kappa$-classes and positive curvature via block bundles