On the geometric nature of characteristic classes of surface bundles

TL;DR

This paper investigates the geometric and cobordism invariance properties of Morita--Mumford--Miller classes in surface bundles, revealing new invariance results for odd classes and applications to complex and handlebody bundles.

Contribution

It proves that odd MMM classes depend only on the cobordism type of the total space, introduces a method to construct multiple surface bundle examples, and offers new proofs and insights into their obstructions.

Findings

Odd MMM classes depend only on the cobordism class of the total space.

Constructed multiple fiberings of manifolds as surface bundles.

Provided a new proof that odd MMM classes vanish for certain handlebody-boundary 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.