On existence of invariant Einstein metrics on a compact homogeneous space

Abstract

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 $C=X_{G,H}^{\Sigma }$ (Böhm polyhedron) introduced by C.Böhm. 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 Böhm existence criteria, and the case of the graph $B$ non-connected is a improved version of the Graph Theorem (C.Böhm, M.Wang, and W.Ziller) actual for any $\mathfrak{z}(\mathfrak{g})$. Moreover, preparation theorems of C. Böhm on retractions are revisited and new constructions of some topologic spaces are suggested.

37 pages, 3 figures