Geometric cycles and characteristic classes of manifold bundles

Abstract

We produce new cohomology for non-uniform arithmetic lattices $\Gamma<SO(p,q)$ using a technique of Millson--Raghunathan. From this, we obtain new characteristic classes of manifold bundles with fiber a closed $4k$-dimensional manifold $M$ with indefinite intersection form of signature $(p,q)$. These classes are defined on a finite cover of $BDiff(M)$ and are shown to be nontrivial for $M=\#_g(S^{2k}\times S^{2k})$. In this case, the classes produced live in degree $g$ and are independent from the algebra generated by the stable (i.e. MMM) classes. We also give an application to bundles with fiber a K3 surface.