  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.
