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2013 Saturated fusion systems as idempotents in the double Burnside ring
Kári Ragnarsson, Radu Stancu
Geom. Topol. 17(2): 839-904 (2013). DOI: 10.2140/gt.2013.17.839


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.


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Kári Ragnarsson. Radu Stancu. "Saturated fusion systems as idempotents in the double Burnside ring." Geom. Topol. 17 (2) 839 - 904, 2013.


Received: 23 December 2010; Revised: 7 May 2012; Accepted: 12 December 2012; Published: 2013
First available in Project Euclid: 20 December 2017

zbMATH: 1306.20011
MathSciNet: MR3070516
Digital Object Identifier: 10.2140/gt.2013.17.839

Primary: 20D20, 55R35
Secondary: 19A22, 55P42

Rights: Copyright © 2013 Mathematical Sciences Publishers


Vol.17 • No. 2 • 2013
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