Homology of the moduli spaces and mapping class groups of framed, \(r\)-spin and pin surfaces
type: Item
instance of: scholarly article
zbMATH Open document ID: 1322.55006
author: Oscar Randal-Williams
published in: Journal of Topology
publication date: 2014-09-05T00:00:00Z
full work available at URL: https://arxiv.org/abs/1001.5366
Mathematics Subject Classification ID: 55R40
Mathematics Subject Classification ID: 57R15
Mathematics Subject Classification ID: 57R50
Mathematics Subject Classification ID: 57M07
Mathematics Subject Classification ID: 57S05
Mathematics Subject Classification ID: 57R20
Mathematics Subject Classification ID: 55P47
zbMATH DE Number: 6340745
zbMATH Keywords: moduli space
zbMATH Keywords: surface bundle
zbMATH Keywords: mapping class group
zbMATH Keywords: tangential structure
zbMATH Keywords: \(r\)-Spin
zbMATH Keywords: Pin
zbMATH Keywords: framing
zbMATH Keywords: Arf invariant
zbMATH Keywords: homological stability
zbMATH Keywords: stable homology
MaRDI profile type: Publication
OpenAlex ID: W3099369851
arXiv ID: 1001.5366
DOI: 10.1112/JTOPOL/JTT029
title: Homology of the moduli spaces and mapping class groups of framed, \(r\)-spin and pin surfaces
review text: Moduli spaces and mapping class groups of oriented surfaces have been studied intensively in recent times, as well as moduli spaces and mapping class groups of surfaces equipped with other tangential structures, including unoriented surfaces and Spin surfaces. Here, a \textit{tangential structure} means a Serre fibration \(\theta: B\to BO(2)\), which in the above cases is induced by a group homomorphism to the topological group \(O(2)\). As the title suggests, this paper studies the moduli spaces and mapping class groups for certain other tangential structures.NEWLINENEWLINEFor a closed surface \(F\), the moduli space \(\mathcal{M}^\theta(F)\) of \(\theta\) structures on \(F\) is the homotopy quotient by \(\mathrm{Diff}(F)\) of the space of bundle maps from \(TF\) to \(\theta^*\gamma_2\), the pullback of the universal bundle \(\gamma_2\) over \(BO(2)\). For a compact surface with boundary the definition is the same, except the bundle maps are prescribed on \(\partial F\), and the moduli space is written \(\mathcal{M}^\theta(F;\delta)\) for a boundary condition \(\delta\). The \(\theta\) considered in this paper are as follows. For \(r\geqslant 0\) the map \((-)^r : U(1) \to U(1) = SO(2) \hookrightarrow O(2)\) induces a Serre fibration over \(BO(2)\) called \(\theta^r\). When \(r=0\) this is the structure of a framing, when \(r=1\) it is an orientation and when \(r=2\) it is a (usual) Spin structure. When \(r=2\) the double covering is usually denoted \(\mathrm{Spin}(2) \to SO(2)\), and can be extended in two different ways to \(O(2)\), written \(\mathrm{Pin}^+(2)\to O(2)\) and \(\mathrm{Pin}^-(2)\to O(2)\). These induce Serre fibrations over \(BO(2)\) called \(\theta=\mathrm{Pin}^\pm\).NEWLINENEWLINEThe main result is that the moduli spaces of \(\theta^r\) and \(\mathrm{Pin}^\pm\) structures are homologically stable, with respect to the genus of a surface, in explicitly given ranges. The \(\theta^r\) structures only apply to orientable surfaces, and the results for …
reviewed by: Martin Palmer
description: scientific article; zbMATH DE number 6340745