Invariance properties of Miller-Morita-Mumford characteristic numbers of fibre bundles (Q2851199)

Let \(\pi: E\rightarrow B\) be a fibre bundle in the category of closed oriented manifolds, with smooth closed oriented \(d\)-dimensional \((d\geq 2)\) fibres. For any generalized Miller-Morita-Mumford (briefly MMM) characteristic class, in degree equal to \(\dim(B)\), of this fibre bundle, one has an associated (rational) MMM characteristic number. \textit{T. Church} et al. [J. Topol. 5, No. 3, 575--592 (2012; Zbl 1260.57047)] proved, in the case of \(d=2\), that certain MMM characteristic numbers depend on the oriented topological cobordism class of the total space \(E\), but not on the fibre, the base space \(B\) or the projection \(\pi\). The paper under review generalizes this result. More precisely, its main theorem (Theorem C) completely classifies, for \(d\geq 2\), the MMM characteristic numbers which depend only on the oriented cobordism class of the total space \(E\). In the proof, using the fact that the generalized MMM classes of smooth bundles with closed oriented manifold fibres can be replaced with classes in the cohomology of the infinite loop space of the Madsen-Tillmann-Weiss spectrum, the authors apply the corresponding results of \textit{J. Ebert} [Algebr. Geom. Topol. 11, No. 1, 69--105 (2011; Zbl 1210.55012)].NEWLINENEWLINEIn the stably almost complex category, the authors derive an analogous result (Theorem E).