Homotopy representations of SO(7) and Spin(7) at the prime 2

A homotopy (complex) representation of a compact Lie group L at the prime p is a map from BL into the p-completion (in the sense of Bousfield and Kan) of the classifying space of the unitary group image. This paper contains the classification of homotopy representations of SO(7) and image at the prime 2. The motivation for considering this problem is twofold: first, one may hope that it would help to understand maps between classifying spaces. Secondly, the construction of the suitable homotopy representation of image is a crucial step in the construction of a faithful representation of the 2-compact group DI(4) [K. Ziemiański, A faithful unitary representation of the 2-compact group DI (4), Ph.D. Thesis, Uniwersytet Warszawski, 2005].