1= 7 3 , λ(7 2 )=7 2 , λ(7 8 ) = 7 4 , λ(7 4 ) = 7i Henceforth we assume that the principal of triality holds and that the above choice of simple roots has been made. Since Spin (8) is simply connected, λ and K induce outer automorphisms of Spin (8); these in turn induce homeomorphisms of B Spin (8), which we continue to denote by λ and /c. In order to determine the cohomology of λ and fc, it will be convenient to use some cohomology classes introduced by Thomas [12]. Let p: B Spin (n) ->

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... be the map defined by the covering homomorphism of Spin (n) over SO(n). Denote by w { the universal Stiefel-Whitney classes, by P, the universal Pontryagin classes, and by X the Euler class of BSO(8). Then H*(BS0(8), Z) = Z[P lf P 2 , P 3 , X] + 2-torsion and H*(BS0(8), Z 2 ) -Z 2 [w 2 , •••, w 8 ]. According to Thomas [12] there exist cohomology classes Q { eiϊ*(J5Spin, Z) (i = 1, 2, 3, 4) and wf eiϊ*(i? Spin, Z 2 ) (i = 4, 6, 7, 8) (where Spin denotes the stable Spin group) such that