Author’s abstract: We study the topology of a complex homogeneous space \(M=G/H\) of complex dimension \(n\), with nonvanishing Euler characteristic and \(G\) of type \(A,D,E\) by means of a topological invariant \(\varphi_2\), which is related to the Poincaré polynomial of \(M\). We introduce the function \(Q=\varphi_2/n\) and we examine how it varies as one passes from a principal orbit of the adjoint representation of a compact Lie group \(G\) to a more singular one. Moreover, it is proved that if \(M\) is a principal orbit \(G/T\) then \(Q\) depends only on the Weyl group of \(G\).