Classical homological stability from the point of view of cells (Q6593009)

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 \(\mathbf{S}_\mathbb{K}\)-module cellular approximation to \(\mathbf{M}_\mathbb{K}\).\N\NThe author extends this to \(\mathsf{G}\)-graded topological spaces, for \(\mathsf{G}\) a braided monoidal groupoid (in place of \(\mathbb{N}\)). He generalizes the identification of the homotopy cofibre of the canonical resolution \(\varepsilon\) and applies this to show how the (high) connectivity of the spaces of destabilization [\textit{O. Randal-Williams} and \textit{N. Wahl}, Adv. Math. 318, 534--626 (2017; Zbl 1393.18006)] (corresponding to fibres of the augmentation \(\varepsilon\)) imply homological stability in this setting.\N\NThis allows for the treatment of homological stability with twisted coefficients. For instance, the author uses this framework to give an interpretation of central stability homology [\textit{A. Putman} and \textit{S. V. Sam}, Duke Math. J. 166, No. 13, 2521--2598 (2017; Zbl 1408.18003)] for discrete coefficient systems and then, using this, to derive a version of a result of [\textit{P. Patzt}, Math. Z. 295, No. 3--4, 877--916 (2020; Zbl 1442.18004)].\N\NFinally, under appropriate hypotheses, he explains the relationship between the homological connectivity of certain spaces of destabilizations and that of the \(E_1\)- and \(E_2\)-splittings occurring in the cellular approach.