Lie Group Actions on a Cayley Projective Plane and a Note on Homogeneous Spaces of Prime Euler Characteristic

In this paper the actions of compact connected Lie groups on rational cohomology Cayley projective planes are studied. In Section 2, the actions of a torus are studied. It is shown that no rank five torus can act effectively on such a space. Then the linear dependence relations among the connected isotropy subgroups of an effective rank four torus action are shown to fall into one of two patterns, the regular and the singular, and for many spaces the singular pattern is shown to be impossible. This information is applied to the actions of the rank four Lie groups (Section 3). For a simple Lie group acting with the regular pattern, the action of F4 on F4/Spin(9) by coset translation is, with two exceptions, a general example, It is also shown that the only rank four groups that cannot act almost effectively on some X are A 2 A2,A C2x A and the groups having G2 as a factor, and a class of actions of A1 x B3 is constructed. Finally the list due to Borel and H. C. Wang of homogeneous spaces G/H with G and H compact connected Lie groups and X(G/H) a prime is extended to include the case of H not necessarily connected (Section 4). We wish to remark that our basic approach is due to Hsiang. In particular, the notions of local weights and F-varieties (Section 1) were first used by him