Characteristic numbers of manifold bundles over surfaces with highly connected fibers


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 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 MM and discuss the resulting obstructions to smoothing them

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oaioai:www.repository.cam.ac.uk:1810/303604Last time updated on 3/26/2020View original full text link

This paper was published in Apollo.

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