SEGAL’S CONJECTURE AND THE BURNSIDE RINGS OF FUSION SYSTEMS ANTONIO D ´ IAZ AND ASSAF LIBMAN Abstract. For a given saturated fusion system F we define the ring A(S) F of the F -invariants of the Burnside ring functor. We show how this ring is related to the Burnside ring of the fusion system F and how it appears naturally in the analogue of Segal’s conjecture for the classifying spectrum BF . We give an explicit description of A(S) F and we prove it is a local ring. 1. Introduction A saturated fusion system F on a finite p-group S is a small category whose ob- jects are the subgroups of S. Its morphism sets F (P,Q) consist of group monomor- phisms P → Q, where P,Q ≤ S, which are subject to a certain set of axioms listed in §2. Isomorphic objects in F are called F -conjugate. Puig was the first to de- fine these objects but in this note we will use the formulation of Broto-Levi-Oliver in [2]. The model is the category F S (G) associated to a Sylow p-subgroup S of a finite group G. In this case the objects are the subgroups of S and the mor- phisms in F S (G) are those monomorphisms P → Q which are restrictions of inner automorphisms of G. For any contravariant functor H : F→C we can consider the inverse limit lim ←−F H. We call this limit the F -invariants of H(S) because, as it is easy to see, it consists of the elements x ∈ H(S) such that ϕ(x)= ψ(x) for every ϕ, ψ ∈F (P,S) and every subgroup P of S. We will use the suggestive notation H(S) F to denote this inverse limit. For example, by [2, Theorem 5.8], the cohomology of the classifying space of a p-local finite group is isomorphic to H ∗ (S; F p ) F for the obvious functor P → H ∗ (P ; F p ) which assigns to P ≤ S its mod-p cohomology. In this paper we study the F -invariants A(S) F of the functor A : F→ Rings which maps P ≤ S to its Burnside ring A(P ). We call this ring “the ring of F -invariant virtual S-sets”. Clearly A(S) F is a subring of A(S) which contains the identity and therefore the standard augmentation map ǫ : A(S) → Z restricts to an augmentation epimorphism ǫ : A(S) F → Z whose kernel is denoted I (S) F . Proposition 3.6 and Lemma 4.2 give an alternative description of A(S) F and I (S) F . Recall that the Burnside ring A(G) of a finite group G is the Grothendieck group of the monoid B(G) of the isomorphism classes of finite G-sets. In symbols, A(G)= Gr(B(G)). The multiplication in this ring is induced by cartesian product of G- sets. As an abelian group A(G) is free with one basis element for each conjugacy class of subgroups of G. In [3] we construct the Burnside ring A(F ) of saturated Date : June 19, 2009. 2000 Mathematics Subject Classification. 55Q55, 19A22, 20C20 . Key words and phrases. Fusion systems, Burnside ring. The authors were supported by an EPSRC grant EP/D506484/1. 1