Homological stability and stable moduli of flat manifold bundles

We prove that the group homology of the diffeomorphism group of #gSn×Sn\int(D2n) 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 lift to a secondary R/Z-invariants for flat (#gSn×Sn)-bundles provided g≫0.