Master class : Homology of groups and functors

Research talks

Andrea Bianchi (University of Copenhaguen)

Title: Polynomial stability of the homology of Hurwitz spaces

 

Abstract: This is joint work with Jeremy Miller. For a conjugation-invariant subset Q of a finite group G we consider the Hurwitz spaces Hur_n(Q), parametrising G-branched covers of the plane with exactly n branch points and local monodromies in Q. We are interested in stability phenomena of the homology groups H_i(Hur_n(Q)) for fixed i and increasing n. There are several stabilisation maps Hur_n(Q)--->Hur_{n+1}(Q), one for each element of Q; our main result is that for n large enough (compared to i), each class in H_i(Hur_{n+1}(Q)) is a combination of images of homology classes in H_i(Hur_n(Q)) along (possibly different) stabilisation maps. Taking coefficients in a field F, we show that the dimension of H_i(Hur_n(Q);F) agrees, for n large enough, with a quasi-polynomial function of n of controlled period and degree. Our work extends previous results of Ellenberg-Venkatesh-Westerland and rely on techniques introduced by them and by Hatcher-Wahl.

Benjamin Brück (ETH Zürich)

Title: Computing high-dimensional group cohomology via duality

 

Abstract: In recent years, duality approaches have been used to investigate "high-dimensional" stability phenomena for the cohomology of groups such as special linear groups or mapping class groups of surfaces.

I will give an example of this by explaining how Borel-Serre duality can be used to show that the rational cohomology of SL_n(Z) vanishes near its virtual cohomological dimension. This is based on joint work with Miller-Patzt-Sroka-Wilson and builds on results by Church-Farb-Putman. To put this into context, I will give an overview of analogous results and conjectures for mapping class groups of surfaces, automorphism groups of free groups and further arithmetic groups.

Aurélien Djament (Université Sorbonne Paris Nord)

Title : Functor categories over an additive category: some computation and comparison results

 

Abstract :  We will remind the relation between functor homology over an additive category with polynomial coefficients and twisted stable homology of general linear groups or stable multiplicative monoids of matrices. Then we will give some important comparison results involving several functor categories, following Franjou-Friedlander-Scorichenko-Suslin (Annals 1999) and recent joint work with Touzé. We will explain how to deduce some complete non-trivial computations. If time permits, we will also give some related homological finiteness properties for functors over an additive category.
 

 

Richard Hepworth (University of Aberdeen)

Title: Homology of Algebras

 

Abstract: Recent work has shown that topological techniques from group homology can be successfully adapted to study the homology of algebras. Sometimes the techniques generalise directly, but in other cases we encounter new phenomena and need new ideas. I will survey our work on Iwahori-Hecke algebras, Temperley-Lieb algebras (joint with Boyd), and Brauer and partition algebras (joint with Boyd and Patzt), and recent developments by other authors.  In case this sounds too algebraic, I'll include introductions to all these algebras and why you should care about them.

Martin Palmer (Mathematical Institute of the Romanian Academy)

Title: Homology of big mapping class groups
 

Abstract: Mapping class groups of infinite-type surfaces ("big mapping class groups") have recently become the subject of intensive study, inspired by their connections with dynamical systems and geometric group theory. However, their homology above degree one has so far been very little understood. I will talk about recent joint work with Xiaolei Wu that computes the homology, in all degrees, of the mapping class groups of surfaces that are "self-similar" in a precise sense, such as the plane minus a Cantor set. A key ingredient of the proof is a homological stability result for big mapping class groups, which takes inspiration from the work of Szymik and Wahl on the Higman-Thompson groups.