Algebra/Topology Seminar

Speaker: Christine Vespa (Université d'Aix-Marseille)

Title: Integral extensions between functors on free groups

Abstract: Functors on the category of free groups appear in several areas of algebra and topology. Understanding Ext-groups between such functors is a natural problem, motivated in part by the study of the stable homology of automorphism groups of free groups.

For the tensor powers of the abelianization functor, integral Ext-groups can be computed using an explicit resolution of the abelianization functor together with a Künneth formula. From these computations one obtains, in particular, that the rational Ext-groups between the abelianization functor and its symmetric powers vanish.

Recently, Arone used homotopical methods to compute integral Ext-groups between functors of free groups. He constructed a complex whose homology gives the integral Ext-groups between the abelianization functor and its symmetric powers. Arones’s results show that these Ext-groups are far more intricate than their rational counterparts.

In this talk, I will revisit Arone’s complex from a functorial perspective and show how it can be used to derive explicit computations. As an application, I will present joint work with Minkyu Kim determining the first and second Ext groups between the abelianization functor and its symmetric powers. I will also describe ongoing joint work with Gregory Arone and Minkyu Kim establishing a surprising connection between integral Ext groups between the abelianization functor and its symmetric powers and the homology of braid groups.