On existence of invariant Einstein metrics on a compact homogeneous space

We prove that the existence of a positively defined, invariant Einstein metric $m$ on a connected homogeneous space $G/H$ of a compact Lie group $G$ is the consequence of non-contractibility of some compact set $C=X_{G,H}^{\Sigma}$ (B\"ohm polyhedron) introduced by C.B\"ohm. There is a natural continuous map of $C$ onto the flag complex $K_B$ of a finite graph $B$. The special case of $C = K_B$, $K_B$ non-contractible, is one of B\"ohm existence criteria, and the case of the graph $B$ non-connected is a improved version of the Graph Theorem (C.B\"ohm, M.Wang, and W.Ziller) actual for any $\mathfrak {z(g)}$. Moreover, preparation theorems of C. B\"ohm on retractions are revisited and new constructions of some topologic spaces are suggested.