Higher limits over the fusion orbit category

Abstract

The fusion orbit category $\overline{\mathcal F _{\mathcal C}} (G)$ of a discrete group $G$ over a collection $\mathcal C$ is the category whose objects are the subgroups $H$ in $\mathcal C$, and whose morphisms $H \to K$ are given by the $G$-maps $G/H \to G/K$ modulo the action of the centralizer group $C_G(H)$. We show that the higher limits over $\overline{\mathcal F _{\mathcal C}} (G)$ can be computed using the hypercohomology spectral sequences coming from the Dwyer $G$-spaces for centralizer and normalizer decompositions for $G$. If $G$ is the discrete group realizing a saturated fusion system $\mathcal F$, then these hypercohomology spectral sequences give two spectral sequences that converge to the cohomology of the centric orbit category ${\mathcal O}^c (\mathcal F)$. This allows us to apply our results to the sharpness problem for the subgroup decomposition of a $p$-local finite group. We prove that the subgroup decomposition for every $p$-local finite group is sharp (over $\mathcal F$-centric subgroups) if it is sharp for every $p$-local finite group with nontrivial center. We also show that for every $p$-local finite group $(S, \mathcal F, \mathcal L)$, the subgroup decomposition is sharp if and only if the normalizer decomposition is sharp.