Geometry & Topology

Saturated fusion systems as idempotents in the double Burnside ring

Kári Ragnarsson and Radu Stancu

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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.

Article information

Source
Geom. Topol., Volume 17, Number 2 (2013), 839-904.

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

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513732568

Digital Object Identifier
doi:10.2140/gt.2013.17.839

Mathematical Reviews number (MathSciNet)
MR3070516

Zentralblatt MATH identifier
1306.20011

Subjects
Primary: 20D20: Sylow subgroups, Sylow properties, $\pi$-groups, $\pi$-structure 55R35: Classifying spaces of groups and $H$-spaces
Secondary: 55P42: Stable homotopy theory, spectra 19A22: Frobenius induction, Burnside and representation rings

Keywords
fusion system Burnside ring finite groups classifying spaces stable splitting

Citation

Ragnarsson, Kári; Stancu, Radu. Saturated fusion systems as idempotents in the double Burnside ring. Geom. Topol. 17 (2013), no. 2, 839--904. doi:10.2140/gt.2013.17.839. https://projecteuclid.org/euclid.gt/1513732568


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