#### Volume 17, issue 2 (2013)

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Saturated fusion systems as idempotents in the double Burnside ring

Geometry & Topology 17 (2013) 839–904
##### 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.

##### Keywords
fusion system, Burnside ring, finite groups, classifying spaces, stable splitting
##### Mathematical Subject Classification 2010
Primary: 20D20, 55R35
Secondary: 55P42, 19A22
##### Publication
Received: 23 December 2010
Revised: 7 May 2012
Accepted: 12 December 2012
Published: 22 April 2013
Proposed: Jesper Grodal
Seconded: Haynes Miller, Paul Goerss
##### Authors
 Kári Ragnarsson American Institute of Mathematics 360 Portage Ave. Palo Alto, CA 94306 USA Radu Stancu CNRS UMR 7352 - LAMFA Universite de Picardie 33, Rue Saint-Leu 80039 Amiens CX 1 France