Computing higher limits over the fusion orbit category via amalgams

TL;DR

This paper introduces a new technique using amalgams to study higher limits over fusion orbit categories, proving the Diaz-Park sharpness conjecture for certain fusion systems and relating it to signalizer functors.

Contribution

It presents a novel method for analyzing higher limits in fusion systems, successfully proving the Diaz-Park sharpness conjecture for specific cases and connecting it with existing theories.

Findings

Proved the Diaz-Park sharpness conjecture for all the Clelland-Parker and Parker-Stroth fusion systems.

Established a relationship between higher limits over fusion orbit categories and signalizer functors.

Developed a technique that can serve as a basis for future induction-based proofs.

Abstract

We study higher limits over the centric orbit category of a fusion system realized by an amalgamated product. In so doing we provide a novel technique for studying the Diaz-Park sharpness conjecture and prove it (in the case of the cohomology Mackey functors) for all the Clelland-Parker and Parker-Stroth fusion systems. This complements previous work from Henke, Libmand and Lynd. We further use the developed technique to study the Benson-Solomon fusion systems thus relating higher limits over the centric fusion orbit category of these systems with the signalizer functors described by Aschbacher and Chermak. We believe that the proposed technique can, in future work, be used as a first step in an induction argument that can bring us closer to providing an answer to this conjecture.