Course 1 : Homological stability, by Nathalie Wahl (6h)
Abstract :
A family of groups G_n satisfies homological stability if the homology of G_n in a given degree is independent of n when n is large enough. Examples of families satisfying stability are symmetric groups, braid groups, certain mapping class groups, or general linear groups. The part of the homology that becomes independent of n is called the stable homology, and the power of homological stability is that, often, the stable homology is easier to compute. This has lead to numerous group cohomology computations in recent years, as for example the proof of the Mumford conjecture by Madsen and Weiss for mapping class groups of surfaces.
The goal of the course is to address the following questions:
- what sort of families of groups satisfy homological stability?
- how does one prove a stability result?
- how does one compute the stable homology?
We will discuss possible twisted coefficients for stability results, and see a direct connection to the parallel lectures by Christine Vespa.
While the course will focus on homological stability for groups, we will discuss a little bit stability in other contexts.
Course 2 : Twisted stable homology via functor homology, by Christine Vespa (6h)
Abstract:
For G_n a family of groups and M_n G_n-modules, we consider the homology of G_n with coefficients in M_n: H_*(G_n,M_n). For a nice family of groups G_n and for modules M_n coming from a functor satisfying a polynomial property, this satisfies homological stability (see the parallel lecture by Nathalie Wahl). The part of the homology that becomes independent of n corresponds to the twisted stable homology. It turns out that the twisted stable homology of a nice family of groups G_n can be computed from two ingredients: the stable homology of G_n and functor homology. Functor homology is a short way to talk about homological algebra in functor categories.
The goal of the course is to address the following questions:
- What is twisted stable homology ?
- What is functor homology and how to compute it?
- How to relate twisted stable homology and functor homology?
As a common thread throughout this course, we will develop the example of the family of the automorphism groups of free groups.
