#### Volume 21, issue 5 (2017)

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Stable homology of surface diffeomorphism groups made discrete

### Sam Nariman

Geometry & Topology 21 (2017) 3047–3092
##### 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.

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##### Keywords
Discrete diffeomorphisms, Haefliger classifying space, Homological stability, infinite loop space
##### Mathematical Subject Classification 2010
Primary: 58D05, 57R32, 55P35, 55R40, 57R19, 57R32, 57R50
Secondary: 57R20