We give a new characterization of saturated fusion systems on a –group in terms of idempotents in the –local double Burnside ring of that satisfy a Frobenius reciprocity relation. Interpreting our results in stable homotopy, we characterize the stable summands of the classifying space of a finite –group that have the homotopy type of the classifying spectrum of a saturated fusion system, and prove an invariant theorem for double Burnside modules analogous to the Adams–Wilkerson criterion for rings of invariants in the cohomology of an elementary abelian –group. This work is partly motivated by a conjecture of Haynes Miller that proposes –tract groups as a purely homotopy-theoretical model for –local finite groups. We show that a –tract group gives rise to a –local finite group when two technical assumptions are made, thus reducing the conjecture to proving those two assumptions.
"Saturated fusion systems as idempotents in the double Burnside ring." Geom. Topol. 17 (2) 839 - 904, 2013. https://doi.org/10.2140/gt.2013.17.839