Saturated fusion systems as idempotents in the double Burnside ring

Abstract

We give a new characterization of saturated fusion systems on a $p$–group $S$ in terms of idempotents in the $p$–local double Burnside ring of $S$ that satisfy a Frobenius reciprocity relation. Interpreting our results in stable homotopy, we characterize the stable summands of the classifying space of a finite $p$–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 $p$–group. This work is partly motivated by a conjecture of Haynes Miller that proposes $p$–tract groups as a purely homotopy-theoretical model for $p$–local finite groups. We show that a $p$–tract group gives rise to a $p$–local finite group when two technical assumptions are made, thus reducing the conjecture to proving those two assumptions.