The Randal-Williams–Wahl Machine
Date:
I gave this talk at the Talbot 2025 workshop on homological stability, mentored by Alexander Kupers and Nathalie Wahl. My talk notes can be found here, and are mainly based off the original article by Randal-Williams and Wahl and Wahl’s ICM paper.
A sequence of groups \(G_1 \to G_2 \to \cdots\) exhibits homological stability if the induced maps on homology \(H_i(G_n) \to H_i(G_{n+1})\) are isomorphisms for \(n\) large relative to \(i\). In the talk, we explored an axiomatization due to Oscar Randal-Williams and Nathalie Wahl of conditions on the groups \(G_n\) that imply homological stability, and showed how this applies to classical groups of interest. We then explained Quillen’s spectral sequence argument and how it gives a proof of homological stability in the Randal-Williams–Wahl setup.