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A major questions in the theory of p-local finite groups was whether any saturated fusion system over a finite p-group admits an associated centric linking system, and when it does, whether it is unique. Both questions were answered in the affirmative by A. Chermak, using the theory of partial groups and localities he developed. Using Chermak’s ideas combined with the techniques of obstruction theory, Bob Oliver gave a different proof of Chermak’s theorem. In this paper we generalise Oliver’s proof to the context of fusion systems over discrete p-toral groups, thus positively resolving the analogous questions in p-local compact group theory. A p-local compact group is an algebraic object designed to encode in an algebraic setup the p-local homotopy theory of classifying spaces of compact Lie groups and p-compact groups, as well as some other families of a similar nature [BLO3]. The theory of p-local compact groups includes, and in many aspects generalises, the earlier theory of p-local finite groups [BLO2]. A p-local compact group is thus a triple (S,F ,L), where S is a discrete p-toral group (Definition 1.1(c)), F is a saturated fusion system over S (Definition 1.4), and L is a centric linking system associated to F [BLO3, Definition 4.1]. In [Ch] A. Chermak showed that for any saturated fusion system F over a finite p-group S, there exists an associated centric linking system, which is unique up to isomorphism. To do so he used the theory of partial groups and localities, which he developed in order to provide an alternative, more group theoretic approach, to p-local group theory. Armed with Chermak’s ideas and techniques of obstruction theory, B. Oliver [O, Theorem 3.4] proved that the obstructions to the existence and uniqueness of a centric linking system associated to a saturated fusion system all vanish. In particular, this implies Chermak’s theorem. For a fusion system F over a discrete p-toral group S, let O(F) denote the associated orbit category of all F-centric subgroup P ≤ S, and let Z : O(F) → Ab denote the functor which associates with a subgroup its centre, [BLO3, Section 7]. Throughout this paper we will write H∗(C;F ) for lim ←− ∗ C F where F : C → Ab is a functor from a small category C. The main result of this paper is the following generalisation of [O, Theorem 3.4] to saturated fusion systems over discrete p-toral groups. Theorem A. Let F be a saturated fusion system over a discrete p-toral group S. Then H(O(F),Z) = 0 for all i > 0 if p is odd, and for all i > 1 if p = 2. The following result (cf. [Ch], and [O, Theorem A]) now follows from Proposition 1.7 below. Theorem B. Let F be a saturated fusion system over a discrete p-toral group. Then there exists a centric linking system associated to F which is unique up to isomorphism. 2000 Mathematics Subject Classification 55R35 (primary), 20J05, 20N99, 20D20 (secondary).. Page 2 of 24 RAN LEVI AND ASSAF LIBMAN The proof of Theorem A follows very closely Oliver’s argument in [O], adapting his methods to the infinite case. The main new input in this paper is the re-definition of best offenders in the context of discrete p-toral groups (Definition 2.2). Chermak, in his original solution of the existence-uniqueness problem, relies on a paper by Meierfrankenfeld and Stellmacher [MS], which in turn depends on the classification theorem of finite simple groups. Oliver’s interpretation of Chermak’s work, and as a consequence our result, remain dependent on the classification theorem. The paper is organised as follows. In Section 1 we collect the definitions, notation and background material needed throughout the paper. Section 2 introduces the Thompson subgroups and offenders in the context of discrete p-toral groups, and analyses the properties of these objects along the lines of [O]. Finally in Section 3 we prove Theorem A, which will be restated there as Theorem 3.6. In Section 4 we give an outline of Oliver’s proof and highlight the changes necessary to adapt it to the infinite case we deal with. Readers who are familiar with [O] may find it useful to read this section first. The crucial observations that led to Definition 2.2, without which this paper could not have been written, were made by Andy Chermak, and we are deeply indebted to him for his interest in these results. |
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