This is a website with additional reading for the electronic Computational Homotopy Theory minicourse on homological stability. This minicourse of four lectures is an introduction to a useful tool in an algebraic topologist’s arsenal: homological stability. Our goal is to explain what homological stability is, give you the tools to prove your own homological stability results, and provide an overview of the state-of-the-art. We will start by working out in detail homological stability for symmetric groups, originally due to Nakaoka. This example displays all the features of the “usual” homological stability argument, as formalised by Randal-Williams and Wahl. After we explain their argument and its input, we end by surveying more refined stability phenomena. These lectures are intended to be accessible to graduate students with a background in algebraic topology.
Lecture notes
The combined lecture notes can be found here.
- Lecture 1: The homology of symmetric groups.
- Lecture 2: Homological stability for symmetric groups.
- Lecture 3: Homological stability for automorphism groups.
- Lecture 4: More subtle stability phenomena.
Recommended reading
- An older but still relevant overview of stability phenomena in topology can be found in this survey by R. Cohen.
- We will use some tools from algebraic topology:
- Spectral sequences and Serre classes: Hatcher.
- Classifying spaces of categories: Chapter 1 of Quillen.
- Semi-simplicial spaces and the geometric realisation spectral sequence: Segal and Ebert & Randal-Williams.
- In the first few lectures, our approach to homological stability will parallel that of Randal-Williams & Wahl.