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.

Final version, to appear in Advances in Mathematics

References in corpus (3)

• Homological stability for moduli spaces of high dimensional manifolds. I
• Stable homology of surface diffeomorphism groups made discrete
• Tautological rings for high dimensional manifolds

Cited by in corpus (4)

• Stable homology of surface diffeomorphism groups made discrete
• Braid groups and discrete diffeomorphisms of the punctured disk
• Abelian quotients of mapping class groups of highly connected manifolds
• A local to global argument on low dimensional manifolds