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

Rémi Molinier

Published 2015-04-13Version 1

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 twisted coefficient cohomology 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.