Existence and uniqueness of classifying spaces for fusion systems over discrete p-toral groups

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.