Relations among characteristic classes of manifold bundles

Abstract

We study relations among characteristic classes of smooth manifold bundles with highly connected fibers. For bundles with fiber the connected sum of $g$ copies of a product of spheres $S^d\times S^d$, where $d$ is odd, we find numerous algebraic relations among so-called “generalized Miller–Morita–Mumford classes”. For all $g>1$, we show that these infinitely many classes are algebraically generated by a finite subset.

Our results contrast with the fact that there are no algebraic relations among these classes in a range of cohomological degrees that grows linearly with $g$, according to recent homological stability results. In the case of surface bundles ($d=1$), our approach recovers some previously known results about the structure of the classical “tautological ring”, as introduced by Mumford, using only the tools of algebraic topology.