Saturated fusion systems as idempotents in the double Burnside ring

K. Ragnarsson and R. Stancu

We give a new, unexpected 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, and reformulate fusion-theoretic phenomena in the language of idempotents. Interpreting our results in stable homotopy, we answer a long-standing question on stable splittings of classifying spaces of finite groups, and generalize the Adams--Wilkerson criterion for recognizing rings of invariants in the cohomology of an elementary abelian $p$-group. Applying our results to $p$-local finite groups, we show that $p$-local finite groups are equivalent to retractive transfer triples.