Stable homology of surface diffeomorphism groups made discrete

Abstract

We answer affirmatively a question posed by Morita on homological stability of surface diffeomorphisms made discrete. In particular, we prove that $C^{\infty }$-diffeomorphisms and volume preserving diffeomorphisms of surfaces as family of discrete groups exhibit homological stability. We show that the stable homology of $C^{\infty }$-diffeomorphims of surfaces as discrete groups is the same as homology of certain infinite loop space related to Haefliger's classifying space of foliations of codimension 2. We use this infinite loop space to obtain new results about (non)triviality of characteristic classes of flat surface bundles.

arXiv comment34 pages. Final submitted version, to appear in Geometry and Topology

References in corpus (6)

• Homological stability for moduli spaces of high dimensional manifolds. I
• Monoids of moduli spaces of manifolds
• Homological stability for mapping class groups of surfaces
• Signatures of foliated surface bundles and the symplectomorphism groups of surfaces
• Homological stability and stable moduli of flat manifold bundles
• Braid groups and discrete diffeomorphisms of the punctured disk

Cited by in corpus (4)

• Homological stability and stable moduli of flat manifold bundles
• Representation stability, secondary stability, and polynomial functors
• Quasi-Complementary Foliations and the Mather-Thurston Theorem
• Braid groups and discrete diffeomorphisms of the punctured disk