Operations on stable moduli spaces
type: Item
instance of: scholarly article
title: Operations on stable moduli spaces
zbMATH Open document ID: 1452.55012
author: Søren Galatius
author: Oscar Randal-Williams
published in: Research in the Mathematical Sciences
publication date: 2020-05-26T00:00:00Z
full work available at URL: https://arxiv.org/abs/1912.10239
review text: Quillen was the first to note the phenomenon of homological stability for groups [unpublished, 1971], and the topological version, due to \textit{G. Segal} [Invent. Math. 21, 213--221 (1973; Zbl 0267.55020)], was applied by Harer to moduli spaces of surfaces in [\textit{J. L. Harer}, Ann. Math. (2) 121, 215--249 (1985; Zbl 0579.57005)]. The authors have studied moduli spaces of higher dimensional manifolds in a series of papers, starting with [\textit{S. Galatius} and \textit{O. Randal-Williams}, Geom. Topol. 14, No. 3, 1243--1302 (2010; Zbl 1205.55007)] and continuing with [\textit{S. Galatius} and \textit{O. Randal-Williams}, J. Am. Math. Soc. 31, No. 1, 215--264 (2018; Zbl 1395.57044); Ann. Math. (2) 186, No. 1, 127--204 (2017; Zbl 1412.57026)]. See also their survey [\textit{O. Randal-Williams} and \textit{S. Galatius}, ``Moduli spaces of manifolds: a user's guide'', in: Handbook of homotopy theory. Boca Raton, FL: CRC Press. 43 p. (2020; Zbl 1476.57058)]. The question addressed in this paper is to what extent the stabilization is topological -- that is, coming from a map between the moduli space constructed by a topological operation on the manifolds under consideration (as is the case with surfaces, where one takes the connected sum with a torus to increase genus). The authors consider the moduli space \(\mathcal{M}(W)\) of fiber bundles with fiber a given smooth closed \(d\)-manifold \(W\) (that is, \(B\mathrm{Diff}(W)\)), when \(d=2n\), under stable diffeomorphims (defined by taking connected sums with copies of \(S^{n}\times S^{n}\)). More generally, they consider moduli spaces \(\mathcal{M}^{\Lambda}(W,\lambda)\) of manifolds \(W\) with \(\Lambda\)-structures \(\lambda:\mathrm{Fr}(TW)\to\Lambda\) on their framed tangent bundles (for a suitable space \(\Lambda\)). Their main theorem states that for any abelian group \(A\), there is a canonical isomorphism \(H^{i}(\mathcal{M}^{\Lambda}(W,\lambda);A)\cong H^{i}(\mathcal{M}^{\Lambda}(W',\lambda');A)\) (induced…
reviewed by: David Blanc
Mathematics Subject Classification ID: 55P47
Mathematics Subject Classification ID: 55R40
Mathematics Subject Classification ID: 57S05
Mathematics Subject Classification ID: 57R15
Mathematics Subject Classification ID: 57R90
zbMATH DE Number: 7204097
zbMATH Keywords: moduli spaces
zbMATH Keywords: diffeomorphism groups
zbMATH Keywords: homological stability
zbMATH Keywords: characteristic classes
MaRDI profile type: Publication
OpenAlex ID: W3016733864
Wikidata QID: Q94554990
arXiv ID: 1912.10239
cites work: The transfer map and fiber bundles
cites work: Localization and homological stability of configuration spaces
cites work: On homological stability for configuration spaces on closed background manifolds
cites work: Homological stability of automorphism groups of quadratic modules and manifolds
cites work: Stable moduli spaces of high-dimensional manifolds
cites work: Homological stability for moduli spaces of high dimensional manifolds. II.
cites work: Homological stability for moduli spaces of high dimensional manifolds. I
cites work: Moduli spaces of manifolds: a user's guide
cites work: Stability of the homology of the mapping class groups of orientable surfaces
cites work: Homological stability of topological moduli spaces
cites work: The stable mapping class group and \(Q(\mathbb C P^ \infty_ +)\).
cites work: The stable moduli space of Riemann surfaces: Mumford's conjecture
cites work: On the homotopy of the stable mapping class group
DOI: 10.1007/S40687-020-00207-6
Recommended article: Moduli spaces of manifolds: a user's guide
Recommended article: Resolutions of moduli spaces and homological stability
Recommended article: Homological stability for moduli spaces of high dimensional manifolds. II.
Recommended article: Homological stability of automorphism groups of quadratic modules and manifolds
Recommended article: Stability for closed surfaces in a background space
Recommended article: Homological stability for spaces of embedded surfaces
Recommended article: Q3075581
Recommended article: Homological stability for moduli spaces of disconnected submanifolds. I
Recommended article: The stable topology of self-dual moduli spaces
Recommended article: Q5285020
description: scientific article