An inductive approach to the Diaz-Park sharpness conjecture

TL;DR

This paper introduces new tools using fusion system techniques to analyze the Diaz-Park sharpness conjecture, proving cohomological sharpness for many classes of fusion systems and revealing a novel link between sharpness and cohomological properties.

Contribution

The authors develop a fusion system-based approach to study the Diaz-Park sharpness conjecture, proving cohomological sharpness in numerous cases and uncovering a surprising connection between sharpness and Mackey functors.

Findings

Proved cohomological sharpness for all saturated fusion systems over certain p-groups.

Established vanishing of higher limits for specific fusion systems, approximating sharpness.

Identified a new link between cohomological sharpness and fusion system building techniques.

Abstract

We develop tools which use common fusion systems building techniques in order to compute higher limits over the centric orbit category. We apply these tools in order to study both the Diaz-Park sharpness conjecture as well as the weaker cohomological sharpness conjecture which predicts vanishing of higher limits only for the cohomology Mackey functors . Our approach leads to proving cohomological sharpness (but not sharpness) for all saturated fusion systems over p-groups of either maximal nihlpotency or of rank 2 and all polynomial, Henke-Shpectorov and van Beek fusion systems. This list includes all but 2 of the cases for which cohomological sharpness was previously known as well as most currently known families of exotic fusion systems. For the polynomial, Henke-Shpectorov and 6 of the van Beek fusion systems, sharpness is also approximated by proving vanishing of all but the first higher limits of any Mackey functor. The distinction our approach makes between sharpness and cohomological sharpness is somewhat surprising and interesting by itself. Our approach draws a new connection between cohomological sharpness and fusion system building techniques. We believe that this connection will lead to a better understanding of both fusion systems and Mackey functors over them.