Higher limits over the fusion orbit category via centralizers of amalgams

TL;DR

This paper investigates the Diaz-Park sharpness conjecture for fusion systems, establishing a 4-term exact sequence for higher limits of Mackey functors, and applies it to Benson-Solomon fusion systems to advance understanding of their properties.

Contribution

It introduces a new exact sequence relating higher limits over fusion systems and applies it to Benson-Solomon systems, offering a novel approach to the sharpness conjecture.

Findings

Established a 4-term exact sequence for higher limits in fusion systems

Applied the sequence to Benson-Solomon fusion systems

Provided new insights into the sharpness conjecture for these systems

Abstract

We study the D\'iaz-Park sharpness conjecture for fusion systems and prove that, under certain circumstances, there exists a 4 terms exact sequence relating the first two higher limits of the contravariant part of a Mackey functor over certain fusion systems. We show how this result can be applied to the family of Benson-Solomon fusion systems thus providing another approach to studying the sharpness for this family of fusion systems.