The classical topological invariants of homogeneous spaces

Abstract

We study the homogeneous spaces of a simply connected, compact, simple Lie group $G$ through the lens of $\mathrm{K}$-theory. Our general methods apply equally well to the case where $G$ is in one of the four infinite families of classical groups, or one of the five exceptional groups. In this paper we focus on the case of homogeneous spaces $G/K$ where $G$ and $K$ have the same rank. The main examples we study in detail are the three symmetric spaces EIII, EVI, EVIII in Cartan’s list of symmetric spaces. These are, respectively, homogeneous spaces for $E_6$, $E_7$, $E_8$ with dimensions $32$, $64$, $128$ and known as Rosenfeld projective planes.