Signalizer functors, existence, and applications to the fundamental group

Abstract

We solve the seventh problem of Oliver's list [M. Aschbacher, R. Kessar, B. Oliver, Fusion systems in algebra and topology, LMS Lecture Note Series: 31, Cambridge University Press, 2011] via an explicit signalizer functor construction in the sense of Aschbacher-Chermak for various group models. Moreover we prove the existence of centric linking systems via group models in certain cases which is the first problem and give applications to the fundamental group which is the eighth problem of the list respectively. We illustrate with many examples.

Assigning a classifying space to a fusion system up to F-isomorphism

“…A signalizer functor if it exists gives rise to a centric linking system [47]. All group models for fusion systems known so far [44], [64], [47], [69] have signalizer functors [47], [69]. Aschbacher and Chermak [4] introduced the notion of representation of a p-local finite group if there exists a group model for F and a signalizer functor on this group which induces L. All so far known group models [44], [64], [47], [69] have such a representation as shown by ourselves in two independent collaborations one together with Libman [47] and in [69].…”

“…The question whether for every p-local finite group (S, F , L) there exists a group model is open. We review the existing group model constructions [44], [64], [47], [69] for fusion and linking systems.…”

“…Group models for fusion systems involving only amalgams (for example Robinson [64], Libman and ourselves [47] ) are a special case of the following construction due to ourselves and Leip [69].…”

“…The elements x 1 and y 1 have the property x 1 y 1 = 0 reflecting the fact that D 8 has two non-conjugate elementary abelian 2−subgroups of rank 2. This was generalized by ourselves with Libman [47] as well as in [69]. Proposition 3.36 Let F be a saturated fusion system over a finite p-group S and G a group model for F such that we have a short exact sequence: 0 → K → H * (BG) → H * (F ) → 0 of unstable modules, K ∈ N il k for k a fixed integer, k ≥ 1.…”

“…Proof: The proof is based on results of Libman and ourselves [47] generalizing the case of fusion systems over finite p-groups [69]. Consider the category C consisting of objects l 0 , .…”