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 of surfaces as family of discrete groups exhibit homological stability. We show that the stable homology of $C^{\infty }$–diffeomorphisms 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 and codimension-$2$ foliations.