Vol. 34, No. 1, 1970

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Sphere transitive structures and the triality automorphism

Alfred Gray and Paul Stephen Green

Vol. 34 (1970), No. 1, 83–96

Let G be a compact connected Lie group which acts transitively and effectively on a sphere Sn1. A manifold M is said to have a sphere transitive structure if the structure group of the tangent bundle of M can be reduced from 0(n) to G. The study of the existence of such structures is a generalization of the well-known problem of the existence of almost complex structures. We completely solve the question of existence of sphere transitive structures on spheres. For our study of sphere transitive structures we need to know some facts about the triality automorphism λ of Spin (8). We completely determine the cohomology homomorphism induced by λ on the cohomology of the classifying space of Spin (8).

Mathematical Subject Classification
Primary: 57.45
Secondary: 22.00
Received: 27 October 1969
Published: 1 July 1970
Alfred Gray
Paul Stephen Green