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Classical homological stability from the point of view of cells

Oscar Randal-Williams

Algebraic & Geometric Topology 24 (2024) 1691–1712
Abstract

We explain how to interpret the complexes arising in the “classical” homology stability argument (eg in the framework of Randal-Williams and Wahl) in terms of higher algebra, which leads to a new proof of homological stability in this setting. The key ingredient is a theorem of Damiolini on the contractibility of certain arc complexes. We also explain how to directly compare the connectivities of these complexes with that of the “splitting complexes” of Galatius, Kupers and Randal-Williams.

Keywords
homological stability, $E_k$–algebras
Mathematical Subject Classification
Primary: 20J05, 55P48
References
Publication
Received: 13 July 2022
Revised: 10 November 2022
Accepted: 4 December 2022
Published: 28 June 2024
Authors
Oscar Randal-Williams
Centre for Mathematical Sciences
University of Cambridge
Cambridge
United Kingdom

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