## Geometry & Topology

### Homological stability for spaces of embedded surfaces

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

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

#### 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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