Abstract : Let H < G be compact Lie groups. C. Böhm introduced a simplicial complex ΔG/H in his paper "Homogeneous Einstein Metrics and Simplicial Complexes" [J. Differential Geom. 67 (2004), no. 1, 79-165; MR2153482]. It is proved that if ΔG/H is non-contractible, then G/H carries a G-invariant Einstein metric. In this talk, it will be shown for which pairs (G,H) with G semisimple and H connected with rank H = rank G the complex ΔG/H is non-contractible, and how a non-trivial homology class of ΔG/H can be found.