Stable homology of surface diffeomorphism groups made discrete

TL;DR

This paper proves homological stability for surface diffeomorphism groups when made discrete and identifies their stable homology with that of a related infinite loop space, advancing understanding of flat surface bundle characteristic classes.

Contribution

It establishes homological stability for discrete surface diffeomorphism groups and links their stable homology to an infinite loop space associated with foliations.

Findings

Homological stability holds for discrete surface diffeomorphism groups.

Stable homology matches that of a specific infinite loop space.

Results inform the (non)triviality of characteristic classes of flat surface bundles.

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.