2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/221654The 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.This is the version published by Algebraic & Geometric Topology on 26 February 2006Algebraic TopologyGroup Theory55R35, 20D20, 55P42text