Geometry & Topology

Homological stability for spaces of embedded surfaces

Federico Cantero and Oscar Randal-Williams

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We study the space of oriented genus-g subsurfaces of a fixed manifold M and, in particular, its homological properties. We construct a “scanning map” which compares this space to the space of sections of a certain fibre bundle over M associated to its tangent bundle, and show that this map induces an isomorphism on homology in a range of degrees.

Our results are analogous to McDuff’s theorem on configuration spaces, extended from 0–dimensional submanifolds to 2–dimensional submanifolds.

Article information

Geom. Topol., Volume 21, Number 3 (2017), 1387-1467.

Received: 14 July 2014
Revised: 14 March 2016
Accepted: 29 May 2016
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55R40: Homology of classifying spaces, characteristic classes [See also 57Txx, 57R20] 57R20: Characteristic classes and numbers 57R40: Embeddings 57R50: Diffeomorphisms 57S05: Topological properties of groups of homeomorphisms or diffeomorphisms

submanifolds characteristic classes homology stability embedding spaces mapping class groups scanning


Cantero, Federico; Randal-Williams, Oscar. Homological stability for spaces of embedded surfaces. Geom. Topol. 21 (2017), no. 3, 1387--1467. doi:10.2140/gt.2017.21.1387.

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