## Geometry & Topology

### Saturated fusion systems as idempotents in the double Burnside ring

#### 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
Revised: 7 May 2012
Accepted: 12 December 2012
First available in Project Euclid: 20 December 2017

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

#### 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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