On the geometric nature of characteristic classes of surface bundles

Abstract

Each Morita–Mumford–Miller (MMM) class en assigns to each genus g⩾2 surface bundle Σg→E2n+2→M2n an integer en# (E→M):=〈 en, [M]〉∈ℤ. We prove that when n is odd the number en# (E→M) depends only on the diffeomorphism type of E, not on g, M or the map E→M. More generally, we prove that en# (E→M) depends only on the cobordism class of E. Recent work of Hatcher implies that this stronger statement is false when n is even. If E→M is a holomorphic fibering of complex manifolds, we show, that for every n, the number en# (E→M) only depends on the complex cobordism type of E. We give a general procedure to construct manifolds fibering as surface bundles in multiple ways, providing infinitely many examples to which our theorems apply. As an application of our results, we give a new proof of the rational case of a recent theorem of Giansiracusa–Tillmann [7, Theorem A] that the odd MMM classes e2i−1 vanish for any surface bundle that bounds a handlebody bundle. We show how the MMM classes can be seen as obstructions to low-genus fiberings. Finally, we discuss a number of open questions that arise from this work.