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Stable homology of surface diffeomorphism groups made discrete

Sam Nariman

Geometry & Topology 21 (2017) 3047–3092

We answer affirmatively a question posed by Morita on homological stability of surface diffeomorphisms made discrete. In particular, we prove that C–diffeomorphisms of surfaces as family of discrete groups exhibit homological stability. We show that the stable homology of C–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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Discrete diffeomorphisms, Haefliger classifying space, Homological stability, infinite loop space
Mathematical Subject Classification 2010
Primary: 58D05, 57R32, 55P35, 55R40, 57R19, 57R32, 57R50
Secondary: 57R20
Received: 3 March 2016
Revised: 17 October 2016
Accepted: 6 December 2016
Published: 15 August 2017
Proposed: Shigeyuki Morita
Seconded: Stefan Schwede, Dmitri Burago
Sam Nariman
Department of Mathematics
Northwestern University
Evanston, IL
United States