Cohomology on the centric orbit category of a fusion system

TL;DR

The paper addresses the problem of higher derived limits for mod $p$ cohomology on the centric orbit category of a saturated fusion system. It develops a Mackey-functor framework, leveraging simple Mackey functors $S_{T,V}$ and a pruning/filtration strategy to reduce the problem to the acyclicity of contravariant parts of these factors for $j \le p-2$. Using nilpotent-action bounds and trace-vanishing arguments, it proves ${\lim}^i H^j(-, F_p)|_{O(F^c)} = 0$ for all $i \ge 1$ and $j \le p-2$, which in particular gives acyclicity in the classical $j=1$ case for odd $p$. This sharpens understanding of the subgroup decomposition and stable-element formulations in fusion-system cohomology and has implications for the cohomology of linking systems and related homotopy decompositions.

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.

Theorems & Definitions (7)

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• proof : Proof of \ref{['STVacyc']}
• proof : Proof of \ref{['T:main']}