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 $\widehat{A}$‐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 $M$ and discuss the resulting obstructions to smoothing them.

中文翻译：

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具有高度连接的纤维的表面上的歧管束的特征数

我们研究具有高度连接的几乎平行的光纤的表面上的平滑束 $\mathrm{中号}$具有均匀的尺寸，为歧管提供了必要的条件以使其与这种管束的总空间相匹配，并表明在大多数情况下，这些条件也是足够的。使用此方法，我们确定由这种类型的束的总空间实现的特征数，并推导对其签名的可除性约束和$\widehat{\mathrm{一种}}$genera，并计算 ${BDiff}^{+}（\mathrm{中号}）$根据广义的Miller–Morita–Mumford类，直至扭转。我们还证明了纤维表面上的拓扑束的相似结果$\mathrm{中号}$ 并讨论如何消除障碍。