Cohomology with twisted coefficients of the classifying space of a fusion system

Abstract

Cohomology with twisted coefficients of the classifying space of a fusion system RÉMI MOLINIERWe study the cohomology with twisted coefficients of the geometric realization of a linking system associated to a saturated fusion system F . More precisely, we extend a result due to Broto, Levi and Oliver to twisted coefficients. We generalize the notion of F -stable elements to F c -stable elements in a setting of cohomology with twisted coefficients by an action of the fundamental group.We then study the problem of inducing an idempotent from an F -characteristic (S, S)-biset and we show that, if the coefficient module is nilpotent, then the cohomology of the geometric realization of a linking system can be computed by F c -stable elements. As a corollary, we show that for any coefficient module, the cohomology of the classifying space of a p-local finite group can be computed by these F c -stable elements. 55R40, 55N25, 55R35, 20J06, 20D20, 20J15The notion of saturated fusion systems was introduced by Puig in the 90s in a context of modular representation theory. In topology, saturated fusion systems are used in the study of p-completed classifying spaces of groups. A p-local finite group is a triple (S, F, L) where S is a p-group, F a saturated fusion system over S and L an associated centric linking system. For a p-local finite group (S, F, L), the p-completed nerve of L, |L| ∧ p , is called its classifying space. The theory of p-local finite group has been studied in detail by Broto, Levi, Oliver and others (see [BLO2], [OV1],[5a1] and [5a2]). The linking system and its geometric realization, even without p-completion, play here a fundamental and central role. In fact, for a given saturated fusion system, the existence and uniqueness of a linking system associated to a saturated fusion system were shown more recently by Chermak [Ch] (using the theory of partial groups). The proof of this important conjecture highlights the deep link between group theory and homotopy theory and the importance of linking systems to study p-local structures. We refer to Aschbacher, Kessar and Oliver [AKO] for more details about fusion systems in general.A well-known result of Cartan and Eilenberg (see [CE] Chap XII, Theorem 10.1) expresses the cohomology with mod p coefficients of a finite group as the submodule of stable elements in the cohomology of a Sylow p-subgroup. This submodule of stable elements corresponds to the inverse limit over the fusion system of the group cohomology functor. One important result in the theory of p-local finite groups is an extension of this theorem to any p-local finite group which tells us that the cohomology of the geometric realization of a linking system with trivial coefficients can be computed by F -stable elements.

Cohomology of linking systems with twisted coefficients by a $p$-solvable action

“…Here we recall the definition of F c -stable elements in a context of twisted coefficients. We refer the reader to [Mo1] for more details. As in [Mo1], we will denote by ω : L → π L = π 1 (|L|, S) the functor which maps each object to the unique object in the target and sends each morphism ϕ ∈ Mor L (P, Q) to the class of the loop ι Q .ϕ.ι P where ι P = δ(1) ∈ Mor L (P, S), ι Q = δ(1) ∈ Mor L (Q, S) and ι P is the edge ι P followed in the opposite direction.…”

“…We already know, from a previous article ([Mo1, Theorem 4.3]) that, if M is a finite Z (p) [π L ]-module and the action of π L on M factor through a p-group, then δ S induces an isomorphism [Mo1,Theorem 4.3] because, any action of a p-group on an abelian p-group is nilpotent). Hence, with the same arguments, we get another corollary of Theorem 4.3.…”

“…Levi and Ragnarsson [LR] already give some tools along these lines. In an other paper [Mo1], the author extends Theorem 1.1 to the case of nilpotent actions on the coefficients. The main ingredient is to construct, as in the trivial coefficient case, an idempotent of H * (S, M ) with image H * (F c , M ).…”

“…In this paper, we also want to extend Theorem 1.1 to twisted coefficients but when the action factors through a p-solvable group. The methods used here are completely different from the ones used in [Mo1] and also more direct. We first have a look at constrained fusion systems.…”