Geometry & Topology

Homological stability of topological moduli spaces

Manuel Krannich

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Abstract

Given a graded E1–module over an E2–algebra in spaces, we construct an augmented semi-simplicial space up to higher coherent homotopy over it, called its canonical resolution, whose graded connectivity yields homological stability for the graded pieces of the module with respect to constant and abelian coefficients. We furthermore introduce a notion of coefficient systems of finite degree in this context and show that, without further assumptions, the corresponding twisted homology groups stabilise as well. This generalises a framework of Randal-Williams and Wahl for families of discrete groups.

In many examples, the canonical resolution recovers geometric resolutions with known connectivity bounds. As a consequence, we derive new twisted homological stability results for various examples including moduli spaces of high-dimensional manifolds, unordered configuration spaces of manifolds with labels in a fibration, and moduli spaces of manifolds equipped with unordered embedded discs. This in turn implies representation stability for the ordered variants of the latter examples.

Article information

Source
Geom. Topol., Volume 23, Number 5 (2019), 2397-2474.

Dates
Received: 29 January 2018
Revised: 18 September 2018
Accepted: 26 December 2018
First available in Project Euclid: 22 October 2019

Permanent link to this document
https://projecteuclid.org/euclid.gt/1571709628

Digital Object Identifier
doi:10.2140/gt.2019.23.2397

Mathematical Reviews number (MathSciNet)
MR4019896

Zentralblatt MATH identifier
07121754

Subjects
Primary: 55P48: Loop space machines, operads [See also 18D50] 55R40: Homology of classifying spaces, characteristic classes [See also 57Txx, 57R20] 55R80: Discriminantal varieties, configuration spaces 57R19: Algebraic topology on manifolds 57R50: Diffeomorphisms

Keywords
homological stability $E_n$–algebras operads configuration spaces moduli spaces of manifolds automorphism groups representation stability

Citation

Krannich, Manuel. Homological stability of topological moduli spaces. Geom. Topol. 23 (2019), no. 5, 2397--2474. doi:10.2140/gt.2019.23.2397. https://projecteuclid.org/euclid.gt/1571709628


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