Homological stability and stable moduli of flat manifold bundles

Abstract

We prove that group homology of the diffeomorphism group of $\#^g S^n \times S^n$ as a discrete group is independent of $g$ in a range, provided that $n>2$. This answers the high dimensional version of a question posed by Morita about surface diffeomorphism groups made discrete. The stable homology is isomorphic to the homology of a certain infinite loop space related to the Haefliger's classifying space of foliations. One geometric consequence of this description of the stable homology is a splitting theorem that implies certain classes called generalized Mumford-Morita-Miller classes can be detected on flat $(\#^g S^n \times S^n)$-bundles for $g$ large enough.