Cohomology of linking systems with twisted coefficients by a $p$-solvable action

TL;DR

This paper investigates the cohomology of linking systems with twisted coefficients in p-local finite groups, focusing on the effects of p-solvable actions and constrained fusion systems.

Contribution

It introduces a comparison between the cohomology of linking systems with twisted coefficients and stable elements in the cohomology of the p-group, especially for p-solvable actions.

Findings

Cohomology of linking systems can be compared to stable elements in group cohomology.

Results are specialized for constrained fusion systems.

Analysis includes the case of p-solvable actions on coefficients.

Abstract

In this paper we study the cohomology of the geometric realization of linking systems with twisted coefficients. More precisely, given a prime $p$ and a $p$-local finite group $(S,\mathcal{F},\mathcal{L})$, we compare the cohomology of $\mathcal{L}$ with twisted coefficients with the submodule of $\mathcal{F}^c$-stable elements in the cohomology of $S$. We start with the particular case of constrained fusion systems. Then, we study the case of $p$-solvable actions on the coefficients.