Volume 17, issue 2 (2013)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 20
Issue 4, 1807–2438
Issue 3, 1257–1806
Issue 2, 629–1255
Issue 1, 1–627

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Interests
Editorial Procedure
Submission Guidelines
Submission Page
Subscriptions
Author Index
To Appear
Contacts
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Saturated fusion systems as idempotents in the double Burnside ring

Kári Ragnarsson and Radu Stancu

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
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
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