Classical homological stability from the point of view of cells

TL;DR

This paper offers a new proof of classical homological stability by interpreting complexes through higher algebra and comparing their connectivities, leveraging recent theorems on arc complex contractibility.

Contribution

It introduces a novel approach to homological stability proofs using higher algebra and connectivity comparisons of complexes.

Findings

New proof of homological stability using higher algebra.

Comparison of connectivities between different complexes.

Application of Damiolini's theorem on arc complex contractibility.

Abstract

We explain how to interpret the complexes arising in the "classical" homology stability argument (e.g. in the framework of Randal-Williams--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--Randal-Williams.