Volume 6, issue 1 (2006)

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Classifying spectra of saturated fusion systems

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