On the geometric nature of characteristic classes of surface bundles

Each Morita-Mumford-Miller (MMM) class en assigns to each genus g >= 2 surface bundle Sigma(g) -> E2n+2 -> M-2n an integer e(n)(#) (E -> M) := < e(n), [M]> is an element of 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 [7, Theorem A] that the odd MMM classes e(2i-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.