Geometry & Topology

Homological stability for spaces of embedded surfaces

Federico Cantero and Oscar Randal-Williams

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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.gt/1510859205

Digital Object Identifier
doi:10.2140/gt.2017.21.1387

Mathematical Reviews number (MathSciNet)
MR3650077

Zentralblatt MATH identifier
1383.55011

Subjects
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

Keywords
submanifolds characteristic classes homology stability embedding spaces mapping class groups scanning

Citation

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. https://projecteuclid.org/euclid.gt/1510859205


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