Extension between functors from groups

Motivated in part by the study of the stable homology of automorphism groups of free groups, we consider cohomological calculations in the category F(gr) of functors from finitely generated free groups to abelian groups. In particular, we compute the groups Ext F(gr) (T ◦a, T◦a) where a is the abelianization functor and T is the n-th tensor power functor for abelian groups. These groups are shown to be non-zero if and only if ∗ = m − n ≥ 0 and Ext F(gr) (T ◦ a, T ◦ a) = Z[Surj(m,n)] where Surj(m,n) is the set of surjections from a set having m elements to a set having n elements. We make explicit the action of symmetric groups on these groups and the Yoneda and external products. We deduce from these computations those of rational Ext-groups for functors of the form F ◦a where F is a symmetric or an exterior power functor. Combining these computations with a recent result of Djament we obtain explicit computations of stable homology of automorphism groups of free groups with coefficients given by particular contravariant functors. Stable homology with twisted coefficients of various families of groups can be computed thanks to functor homology in a suitable category (see [7] [17] [4] [5]). In particular, stable homology of automorphism groups of free groups with coefficients given by a reduced polynomial covariant functor is trivial (see [5]). Recently, Djament proved in [2] that stable homology (resp. cohomology) of automorphism groups of free groups with coefficients given by a reduced polynomial contravariant (resp. covariant) functor is governed by Tor groups (resp. Ext groups) in the category of functors from finitely generated free groups to abelian groups. The aim of this paper is to calculate Ext and Tor groups between concrete functors from groups to abelian groups in order to obtain explicit computations of stable homology of automorphism groups of free groups with coefficients given by a contravariant functor. Let gr be a small skeleton of the category of finitely generated free groups, F(gr) the category of functors from gr to abelian groups and a the abelianization functor in F(gr). The main result of this paper is: Theorem 1. Let n and m be natural integers, we have an isomorphism: ExtF(gr)(T n ◦ a, T ◦ a) ≃ { Z[Surj(m,n)] if ∗ = m− n 0 otherwise where Surj(m,n) is the set of surjections from a set having m elements to a set having n elements. The actions of the symmetric groups Sm and Sn on Ext ∗ F(gr)(T n ◦ a, T ◦ a) are induced by the composition of surjections via the previous isomorphism, up to a sign (see Proposition 2.5 for the precise signs). The Yoneda product is induced by the composition of surjections, up to a sign, (see Proposition 3.1 for the precise signs) via the previous isomorphism and the external product is induced by the disjoint union of sets. We remark that this theorem can be expressed elegantly in terms of symmetric sequences (see Proposition 3.2). Mathematics Subject Classification (2010): 18G15, 18A25, 20J06 Date: September 20, 2016. 1