Saturated fusion systems as idempotents in the double Burnside ring

Details Saturated fusion systems as idempotents in the double Burnside ring Saturated fusion systems as idempotents in the double Burnside ring

2009-10-31

arxiv.org/abs/0911.0085v3

Mathematics

Algebraic Topology

Added citation to Puig; clarified discussion of Miller's conjecture on the homotopy characterization of p-local finite groups

Scientific paper

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. This work is partly motivated by a conjecture of Haynes Miller which proposes retractive transfer triples as a purely homotopy-theoretic model for p-local finite groups. We take an important step toward proving this conjecture by showing that a retractive transfer triple gives rise to a p-local finite group when two technical assumptions are made, thus reducing the conjecture to proving those two assumptions.

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