Torus actions on a cohomology product of three odd spheres

Abstract

The main purpose of this paper is to describe how a torus group may act on a space, X X , whose rational cohomology ring is isomorphic to that of a product of three odd-dimensional spheres, in such a way that the fixed point set is nonempty, and X X is not totally nonhomologous to zero in the associated X X -bundle, X T → B T {X_T} \to {B_T} . In the first section of the paper some general results on the cohomology theory of torus actions are established. In the second section the cohomology theory of the above type of action is described; and in the third section the results of the first two sections are used to prove a Golber formula for such actions, which, under certain conditions, bears an interesting interpretation in terms of rational homotopy.