Geometry & Topology

Stable homology of surface diffeomorphism groups made discrete

Sam Nariman

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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.

Article information

Geom. Topol., Volume 21, Number 5 (2017), 3047-3092.

Received: 3 March 2016
Revised: 17 October 2016
Accepted: 6 December 2016
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 58D05: Groups of diffeomorphisms and homeomorphisms as manifolds [See also 22E65, 57S05] 57R32: Classifying spaces for foliations; Gelfand-Fuks cohomology [See also 58H10] 55P35: Loop spaces 55R40: Homology of classifying spaces, characteristic classes [See also 57Txx, 57R20] 57R19: Algebraic topology on manifolds 57R32: Classifying spaces for foliations; Gelfand-Fuks cohomology [See also 58H10] 57R50: Diffeomorphisms
Secondary: 57R20: Characteristic classes and numbers

Discrete diffeomorphisms Haefliger classifying space Homological stability infinite loop space


Nariman, Sam. Stable homology of surface diffeomorphism groups made discrete. Geom. Topol. 21 (2017), no. 5, 3047--3092. doi:10.2140/gt.2017.21.3047.

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