INTRODUCTION

In this paper we describe the cohomological orbit structure of Lie group actions on cohomology projective spaces and show that under quite general conditions an arbitrary action has the same structure as that obtained from an equivariant James' reduced product construction applied to an action of the group on a sphere. Here X is a cohomology projective space if X ∼ Pn(q), i.e. H*(X) = Q[e]/(en+1) as a Q-algebra, where deg e = q is even. Cohomology is always taken with rational coefficients. If q = 2, X is a cohomology complex projective space and if q = 4, X is a cohomology quaternionic projective space. For those cases linear actions on the classical projective spaces demonstrate that tori of large rank can have rich orbit structures (e.g. many non-acyclic components of the fixed point set). The work of Wu Yi Hsiang and Hsiang and Su shows that the cohomogical orbit structure of an arbitrary action is modelled after the linear examples in general. In particular, there is the following theorem of Hsiang and Su: If X ∼ Pn(4) and a torus of rank at least two acts cohomology effectively on X, then at most one component of the fixed point set is a pr(4) with r > O.

Our work is concerned with the higher cohomology projective spaces, i.e. q > 4.