Stable Cohomology of Compact Homogeneous Spaces

The cohomology of certain compact homogeneous spaces is studied. The notion of stable cohomology (invariant under the passage to a finite covering) is introduced; examples of the calculation of this cohomology (Theorem 1) and its application to the study of the structure of compact homogeneous spaces (Theorem 2) are given. Several conjectures about properties of stable cohomology related to various areas of mathematics (such as topology and the cohomology of discrete (in particular, polycyclic) groups) are stated.