Classical homological stability from the point of view of cells
type: Item
instance of: scholarly article
title: Classical homological stability from the point of view of cells
author: Oscar Randal-Williams
published in: Algebraic \& Geometric Topology
publication date: 2024-08-26T00:00:00Z
review text: In this paper, the author shows how the approach to homological stability using cellular \(E_k\)-algebras [\textit{S. Galatius} et al., ``Cellular \(E_k\)-algebras'', Preprint, \url{arXiv:1805.07184}] relates to and can be applied to more classical approaches.\N\NWorking in \(\mathbb{N}\)-graded topological spaces, the author introduces an associative algebra \(\mathbf{S}\), model for the free unital \(E_2\)-algebra on a generator \(1_*(*)\) of degree one. This provides a framework for homological stability; namely, working with a right \(\mathbf{S}\)-module \(\mathbf{M}\), one has the stabilization map \(\sigma : \mathbf{M} \otimes 1_*(*) \rightarrow \mathbf{M}\), and stability is related to the (high) homological connectivity of the cofibre \(\mathbf{M}/ \sigma\) as an \(\mathbb{N}\)-graded space.\N\NThe author strictifies \textit{M. Krannich}'s canonical resolution [Geom. Topol. 23, No. 5, 2397--2474 (2019; Zbl 1432.55022)] to give an augmented object \(R_\bullet \mathbf{M} \rightarrow \mathbf{M}\) and hence \(\varepsilon_\mathbf{M} : |R_\bullet \mathbf{M} |\rightarrow \mathbf{M}\). Expanding upon Krannich's work, he shows that the map \(\varepsilon_{\mathbf{S}} : |R_\bullet \mathbf{S} |\rightarrow \mathbf{S}\) is an equivalence onto \(\mathbf{S}_{>0}\). This is proved by using a contractibility result due to \textit{C. Damiolini} for certain arc complexes [``The braid group and the arc complex'', Master's Thesis, Universiteit Leiden (2013); available at \url{https://hdl.handle.net/1887/3597297}].\N\NUsing this, he shows that the homotopy cofibre of \(\varepsilon_\mathbf{M}\) is the derived \(\mathbf{S}\)-module indecomposables of \(\mathbf{M}\). He applies this to classical homological stability, relating the homological connectivity of the \(\mathbb{N}\)-graded space \(\mathbf{M}/ \sigma\) with coefficients in a commutative ring \(\mathbb{K}\) to that of the relative homology for \((\mathbf{M}, |R_\bullet (\mathbf{M})|)\). The proof uses a \(\mathbf{S}_\mathbb{…
reviewed by: Geoffrey Powell
Mathematics Subject Classification ID: 55P48
Mathematics Subject Classification ID: 20J05
zbMATH DE Number: 7901551
zbMATH Keywords: homological stability
zbMATH Keywords: \(E_k\)-algebras
zbMATH Keywords: central stability homology
MaRDI profile type: Publication
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DOI: 10.2140/AGT.2024.24.1691
description: scientific article; zbMATH DE number 7901551