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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Homological stability for spaces of embedded surfaces

Federico Cantero Morán and Oscar Randal-Williams

Geometry & Topology 21 (2017) 1387–1467
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.

Keywords
submanifolds, characteristic classes, homology stability, embedding spaces, mapping class groups, scanning
Mathematical Subject Classification 2010
Primary: 55R40, 57R20, 57R40, 57R50, 57S05
References
Publication
Received: 14 July 2014
Revised: 14 March 2016
Accepted: 29 May 2016
Published: 10 May 2017
Proposed: Shigeyuki Morita
Seconded: Peter Teichner, Benson Farb
Authors
Federico Cantero Morán
Institut de Recherche en Mathématique et Physique
Université Catholique de Louvain
Chemin du Cyclotron, 2
1348 Louvain-la-Neuve
Belgium
Oscar Randal-Williams
Centre for Mathematical Sciences
University of Cambridge
Wilberforce Road
Cambridge CB3 0WB
United Kingdom