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Dwyer, Miller and Wilkerson proved that at the prime 2, the classifyingspaces of SU(2) and SO(3) can be obtained as a homotopy pushout of theclassifying spaces of certain subgroups. In this paper we show explicitly howthese decompositions arise from the fusion systems of SU(2) and SO(3) overmaximal discrete 2-toral subgroups.

A new approach to mod 2 decompositions of BSU(2) and BSO(3)