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 ${\mathrm{BDiff}}^{+}(M)$ 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.

23 pages, major revision (included results for topological bundles, removed technical condition in main results, simplified sections 1 and 3), to appear in Journal of the London Mathematical Society

References in corpus (1)

• The non-multiplicativity of the signature modulo 8 of a fibre bundle is an Arf-Kervaire invariant

Cited by in corpus (1)

• Mapping class groups of highly connected $(4k+2)$-manifolds