Characteristic numbers of manifold bundles over surfaces with highly connected fibers

TL;DR

This paper investigates the characteristic numbers of smooth manifold bundles over surfaces with highly connected, almost parallelizable fibers, establishing conditions for bordism, and analyzing their topological and smooth structures.

Contribution

It provides necessary and sufficient conditions for manifolds to be bordant to total spaces of such bundles, and computes their characteristic numbers and cohomology.

Findings

Determines characteristic numbers realized by total spaces of these bundles.

Establishes divisibility constraints on signatures and $\,\hat{A}$-genera.

Computes the second integral cohomology of ${\rm BDiff}^+(M)$ in terms of generalized Miller--Morita--Mumford classes.

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.