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Saturated fusion systems
as idempotents in the double Burnside ring
Kári Ragnarsson and Radu Stancu |
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Geometry & Topology 17 (2013) 839–904
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Abstract |
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We give a new characterization of saturated fusion systems on a –group in terms of idempotents in the –local double Burnside ring of that satisfy a Frobenius reciprocity relation. Interpreting our results in stable homotopy, we characterize the stable summands of the classifying space of a finite –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 –group. This work is partly motivated by a conjecture of Haynes Miller that proposes –tract groups as a purely homotopy-theoretical model for –local finite groups. We show that a –tract group gives rise to a –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
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Mathematical Subject Classification 2010
Primary: 20D20, 55R35
Secondary: 55P42, 19A22
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References |
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
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Authors |
| Kári Ragnarsson | |
| American Institute of
Mathematics 360 Portage Ave. Palo Alto, CA 94306 USA |
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| Radu Stancu | |
| CNRS UMR 7352 - LAMFA Universite de Picardie 33, Rue Saint-Leu 80039 Amiens CX 1 France |
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