Stable homology of surface diffeomorphism groups made discrete

Nariman, Sam

doi:10.2140/gt.2017.21.3047

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Stable homology of surface diffeomorphism groups made discrete

Sam Nariman

Geometry & Topology 21 (2017) 3047–3092

DOI: 10.2140/gt.2017.21.3047

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.

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

References

Publication

Received: 3 March 2016

Revised: 17 October 2016

Accepted: 6 December 2016

Published: 15 August 2017

Proposed: Shigeyuki Morita

Seconded: Stefan Schwede, Dmitri Burago

Authors

Sam Nariman
Department of Mathematics Northwestern University Evanston, IL United States

Volume 21, issue 5 (2017)