On the Topology of EIII and EIV

Abstract

Introduction. In [4] we introduced the concept of the root diagram of a compact symmetric triad. Now we consider the symmetric triad (E6; F4, D5 X T1) (where we represent compact Lie groups by the standard symbols for their local structures and take E6 to be simply connected). The corresponding compact symmetric spaces, in E. Cartan's notation, are EIII and EIV. We shall determine the root diagram of the triad and then apply Morse theory to obtain topological information. The natural action of F4 on EIII will be found to have as one of its orbits the Cayley projective plane W= F4/B4, and the action of D5 X T1 on EIV will have an orbit S1 XS9. An analysis of certain spaces of paths will then establish the following theorems and their corollaries.