SOCIÉTÉ MATHÉMATIQUE DE FRANCE quatrième série-tome 43 fascicule 3 mai-juin 2010 Aurélien DJAMENT & Christine VESPA Sur l'homologie des groupes orthogonaux et symplectiques à coefficients tordus

Important motivations of this paper come from the homology of congruence groups and of subgroups IA of the automorphisms of free groups, whose functorial study is part in a natural way of this formalism. These two examples are studied in section 5. This paper is organized as follows. In Section 1 we give the definitions of strong polynomial functors and weak polynomial functors and give them properties. In Section 2 we show that the notion of strong polynomial functors can be described in terms of cross effects and prove that it extends the usual notion of polynomial functors introduced by Eilenberg and Mac Lane. Section 3 concerns the construction of a universal symmetric monoidal category where the unit is a null object from a symmetric monoidal category where the unit is an initial object. In Section 4 we study the polynomial functors on the category of finite sets with injections. In Section 5 we give examples of polynomial functors from the category of finitely generated free groups where the morphisms are monomorphisms with a given splitting. In the last section we prove our main result concerning polynomial functors on categories of hermitians spaces.

RésuméNous montrons que l'homologie stable des groupes d'automorphismes des groupes libres à coefficients tordus par un foncteur covariant polynomial est trivial. Pour le foncteur d'abélianiation, qui est polynomial de degré 1, nous retrouvons par des méthodes algébriques un résultat précédemment obtenu par Hatcher-Wahl, par des méthodes topologiques et géométriques. Pour les coefficients donnés par un foncteur polynomial contravariant se factorisant par l'abélianisation, nous calculons la valeur stable du premier groupe d'homologie des groupes d'automorphismes des groupes libres, qui est généralement non nul. AbstractWe prove that the stable homology of automorphism groups of free groups with twisted coefficients given by a polynomial covariant functor is trivial. For the abelianization functor, which is polynomial of degree 1, we recover by algebraic methods a result previously obtained by HatcherWahl by topological and geometrical methods. For coefficients given by a contravariant polynomial functor factorizing through the abelianization, we compute the stable value of the first homology group of automorphism groups of free groups, which is generally nonzero.

Dedicated to the memory of Jean-Louis Loday for his generosity and benevolence.Abstract. In this paper, we give a description of polynomial functors from (finitely generated free) groups to abelian groups in terms of non-linear Mackey functors generalizing those given in a paper of Baues-Dreckmann-Franjou-Pirashvili published in 2001. This description is a consequence of our two main results: a description of functors from (finitely generated free) P-algebras (for P a set-operad) to abelian groups in terms of non-linear Mackey functors and the isomorphism between polynomial functors on (finitely generated free) monoids and those on (finitely generated free) groups. Polynomial functors from (finitely generated free) P-algebras to abelian groups and from (finitely generated free) groups to abelian groups are described explicitely by their cross-effects and maps relating them which satisfy a list of relations.Mathematics Subject Classification: 18D; 18A25; 55UKeywords: polynomial functors; non-linear Mackey functors; set-operads Polynomial functors play a prominent role in the representation theory of algebraic groups, algebraic K-theory as well as in the theory of modules over the Steenrod algebra (see, for instance, [11]). In particular it is a main computational tool for computing the stable cohomology of classical groups (considered as discrete groups) with twisted coefficients (see [4], [12] and [7]).The study of polynomial functors, in their own right, has a long history starting with the work of Schur in 1901 in [27], even before the notion of a functor was defined by Eilenberg and MacLane. In fact, without using the language of functors, Schur proved that over a field of characteristic zero any polynomial functor is a direct sum of homogeneous functors and the category of homogeneous functors of degree d is equivalent to the category of representations of the symmetric group on d letters. There were many attempts to generalize Schur's theorem for general rings. For polynomial functors from (finitely generated free) abelian groups to abelian groups a satisfactory answer is given in [2] where the authors obtained a description of those polynomial functors in terms of non linear Mackey functors.The principal aim of this paper is to generalize this result. The two main results of this paper are a description of functors from (finitely generated free) P-algebras (for P a set-operad) to abelian groups and a description of polynomial functors from (finitely generated free) groups to abelian groups, obtained from the first result and an isomorphism between polynomial functors on (finitely generated free) monoids and those on (finitely generated free) groups.More precisely, for P a set-operad, F ree(P) the category of (finitely generated free) Palgebras and Ab the category of abelian groups, we consider the category F unc(F ree(P), Ab) of functors F : F ree(P) → Ab. The first principal result of this paper is the following: where P Mack(Ω(P), Ab) is the category of pseudo-Mackey functors defined in Definition 1.28.T...

We study polynomial functors of degree 2, called quadratic, with values in the category of abelian groups Ab, and whose source category is an arbitrary category C with null object such that all objects are colimits of copies of a generating object E which is small and regular projective; this includes all pointed algebraic varieties. More specifically, we are interested in such quadratic functors F from C to Ab which preserve filtered colimits and suitable coequalizers; one may take reflexive ones if C is Mal'cev and Barr exact.A functorial equivalence is established between such functors F : C → Ab and certain minimal algebraic data which we call quadratic C-modules: these involve the values on E of the cross-effects of F and certain structure maps generalizing the second Hopf invariant and the Whitehead product.Applying this general result to the case where E is a cogroup these data take a particularly simple form. This application extends results of Baues and Pirashvili obtained for C being the category of groups or of modules over some ring; here quadratic C-modules are equivalent with abelian square groups or quadratic R-