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Saturated fusion systems as idempotents in the double Burnside ring

Kári Ragnarsson and Radu Stancu

Geometry & Topology 17 (2013) 839–904

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.

fusion system, Burnside ring, finite groups, classifying spaces, stable splitting
Mathematical Subject Classification 2010
Primary: 20D20, 55R35
Secondary: 55P42, 19A22
Received: 23 December 2010
Revised: 7 May 2012
Accepted: 12 December 2012
Published: 22 April 2013
Proposed: Jesper Grodal
Seconded: Haynes Miller, Paul Goerss
Kári Ragnarsson
American Institute of Mathematics
360 Portage Ave.
Palo Alto, CA 94306
Radu Stancu
Universite de Picardie
33, Rue Saint-Leu
80039 Amiens CX 1