Spectra of subrings of cohomology generated by characteristic classes for fusion systems
Ian J. Leary, Jason Semeraro
TL;DR
The paper extends Quillen–type spectrum descriptions from finite groups to saturated fusion systems by introducing the Chern subring $Ch(\mathcal{F})$ generated by Chern classes of $\mathcal{F}$-stable representations and establishing a Green–Leary style stratification over categories of elementary abelian subgroups. It also develops parallel results for subrings generated by characteristic classes of $\mathcal{F}$-stable $S$-sets (permutation representations). The core method shows these subrings are large and natural, enabling Colimit-Descent: the associated varieties $V_R(k)$ are homeomorphic to colimits of elementary-abelian subgroups' varieties $X_E(k)$ over the appropriate categories (e.g., $\mathcal{E}'(\mathcal{F})$, $\mathcal{E}_{\mathbb{R}}'(\mathcal{F})$, $\mathcal{E}_P'(\mathcal{F})$, or $\mathcal{A}(\mathcal{F})$). The results unify and extend Quillen’s description to fusion systems, including real and permutation character theories, and connect cohomology of fusion systems with classifying spaces of linking systems via precise stratifications and isogeny relations. The approach leverages Reeh’s bases of stable permutation characters and Linckelmann’s orbit-category framework to realize explicit topological descriptions of spectra.
Abstract
If $\mathcal{F}$ is a saturated fusion system on a finite $p$-group $S$, we define the Chern subring $Ch(\mathcal{F})$ of $\mathcal{F}$ to be the subring of the mod-$p$ cohomology $H^*(S)$ of $S$ generated by the Chern classes of $\mathcal{F}$-stable representations of $S$. We show that $Ch(\mathcal{F})$ is contained in $H^*(\mathcal{F})$ and apply a result of Green and the first author to describe its spectrum in terms of a certain category of elementary abelian subgroups of $S$. We obtain similar results for various related subrings, including those generated by characteristic classes of $\mathcal{F}$-stable $S$-sets.