A remark on the second homotopy groups of compact Riemannian 3-symmetric spaces

Abstract

In order to calculate the second Stiefel-Whitney class of a l-connected compact Riemannian 3-symmetric space $G/K$ by BorelHirzebruch's method, we have to know the second cohomology group $H^{2}(G/K, Z_{2})\cong Hom(\pi_{2}(G/K), Z_{2})$. In this paper, we shall describe precisely the connected Lie subgroup $K$ and calculate explicitly the second homotopy group $\pi_{2}(G/K)$ in terms of the roots of $G$.