Cohomology with twisted coefficients of the classifying space of a fusion system

TL;DR

This paper extends the understanding of cohomology with twisted coefficients for classifying spaces of fusion systems, showing how to compute such cohomology using stable elements and generalizing previous results.

Contribution

It generalizes the notion of stable elements to twisted coefficients and demonstrates how to compute cohomology via these elements for classifying spaces of fusion systems.

Findings

Cohomology with twisted coefficients can be computed using ^c-stable elements.

The result extends previous work to locally constant coefficients.

Cohomology of classifying spaces of p-local finite groups is accessible through these stable elements.

Abstract

We study the cohomology with twisted coefficients of the geometric realization of a linking system associated to a saturated fusion system $\mathcal{F}$. More precisely, we extend a result due to Broto, Levi and Oliver to twisted coefficients. We generalize the notion of $\mathcal{F}$-stable elements to $\mathcal{F}^c$-stable elements in a setting of cohomology with twisted coefficients by an action of the fundamental group.% or, in other word, with locally constant coefficients. We then study the problem of inducing an idempotent from an $\mathcal{F}$-characteristic $(S,S)$-biset and we show that, if the coefficient module is nilpotent, then the cohomology of the geometric realization of a linking system can be computed by $\mathcal{F}^c$-stable elements. As a corollary, we show that for any coefficient module, the cohomology of the classifying space of a $p$-local finite group can be computed by these $\mathcal{F}^c$-stable elements.