Classical homological stability from the point of view of cells

Randal-Williams, Oscar

doi:10.2140/agt.2024.24.1691

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ISSN (electronic): 1472-2739

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Classical homological stability from the point of view of cells

Oscar Randal-Williams

Algebraic & Geometric Topology 24 (2024) 1691–1712

DOI: 10.2140/agt.2024.24.1691

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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Volume 24, issue 3 (2024)