Cohomology on the centric orbit category of a fusion system

TL;DR

This paper investigates the higher derived limits of mod p cohomology on the centric orbit category of saturated fusion systems, establishing vanishing results for degrees up to p-2, advancing understanding of cohomological properties in fusion systems.

Contribution

It proves that higher limits of cohomology vanish for degrees up to p-2, extending known cases and providing new insights into the structure of fusion systems.

Findings

Higher limits of H^j vanish for j ≤ p-2.

The same vanishing holds for the contravariant part of simple Mackey composition factors.

Advances understanding of cohomological behavior in fusion systems.

Abstract

We study here the higher derived limits of mod $p$ cohomology on the centric orbit category of a saturated fusion system on a finite $p$-group. It is an open problem whether all such higher limits vanish. This is known in many cases, including for fusion systems realized by a finite group and for many classes of fusion systems which are not so realized. We prove that the higher limits of $H^j$ vanish provided $j \leq p-2$, by showing that the same is true for the contravariant part of a simple Mackey composition factor of $H^j$ under the same conditions.