Geometric cycles and characteristic classes of manifold bundles

TL;DR

This paper introduces new cohomology classes for arithmetic lattices and manifold bundles, revealing nontrivial characteristic classes for certain high-dimensional manifolds and applications to K3 surface bundles.

Contribution

It develops new cohomology for non-uniform arithmetic lattices and constructs novel characteristic classes for manifold bundles with indefinite intersection forms.

Findings

New cohomology classes for $ ext{SO}(p,q)$ lattices.

Nontrivial characteristic classes for bundles with specific high-dimensional fibers.

Distinct from stable (MMM) classes, especially for connected sums of spheres.

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.