Characteristic numbers of manifold bundles over surfaces with highly connected fibers

Abstract: We study smooth bundles over surfaces with highly connected almost parallelizable fiber $M$ of even dimension, providing necessary conditions for a manifold to be bordant to the total space of such a bundle and showing that, in most cases, these conditions are also sufficient. Using this, we determine the characteristic numbers realized by total spaces of bundles of this type, deduce divisibility constraints on their signatures and $\hat{A}$-genera, and compute the second integral cohomology of ${\rm BDiff}^+(M)$ up to torsion in terms of generalized Miller--Morita--Mumford classes. We also prove analogous results for topological bundles over surfaces with fiber $M$ and discuss the resulting obstructions to smoothing them.