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Classifying spectra of saturated fusion systems

Kári Ragnarsson

Algebraic & Geometric Topology 6 (2006) 195–252

arXiv: math.AT/0502092

Abstract

The assignment of classifying spectra to saturated fusion systems was suggested by Linckelmann and Webb and has been carried out by Broto, Levi and Oliver. A more rigid (but equivalent) construction of the classifying spectra is given in this paper. It is shown that the assignment is functorial for fusion-preserving homomorphisms in a way which extends the assignment of stable p–completed classifying spaces to finite groups, and admits a transfer theory analogous to that for finite groups. Furthermore the group of homotopy classes of maps between classifying spectra is described, and in particular it is shown that a fusion system can be reconstructed from its classifying spectrum regarded as an object under the stable classifying space of the underlying p–group.

Keywords
fusion systems, p-local finite groups, stable homotopy, transfer
Mathematical Subject Classification 2000
Primary: 55R35
Secondary: 20D20, 55P42
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Publication
Received: 25 March 2005
Revised: 19 January 2006
Accepted: 26 January 2006
Published: 26 February 2006
Authors
Kári Ragnarsson
Department of Mathematical Sciences
University of Aberdeen
Aberdeen AB24 3UE
UK