On the geometric nature of characteristic classes of surface bundles

Abstract

Each Morita--Mumford--Miller (MMM) class e_n assigns to each genus g >= 2 surface bundle S_g -> E^{2n+2} -> M^{2n} an integer e_n^#(E -> M) := <e_n,[M]> in Z. We prove that when n is odd the number e_n^#(E -> M) depends only on the diffeomorphism type of E, not on g, M, or the map E -> M. More generally, we prove that e_n^#(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 e_n^#(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 that the odd MMM classes e_{2i-1} vanish for any surface bundle which 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.