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 BU(n)p∧. This paper contains the classification of homotopy representations of SO(7) and Spin(7) 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 Spin(7) 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].